tensor network states, entanglement, and...
TRANSCRIPT
Tensor Network States, Entanglement,
and Anomalies of Topological Phases
of Matters
Yunqin Zheng
A Dissertation
Presented to the Faculty
of Princeton University
in Candidacy for the Degree
of Doctor of Philosophy
Recommended for Acceptance
by the Department of
Physics
Adviser: Bogdan Andrei Bernevig
June 2020
c© Copyright by Yunqin Zheng, 2020.
All rights reserved.
Abstract
This dissertation investigates two aspects of topological phases of matter: 1) the
tensor network state (TNS) representations of the ground states as well as their en-
tanglement entropies of gapped Hamiltonians in diverse dimensions; 2) the anomalies
and dynamics of strongly coupled quantum field theories.
For the first aspect, we first show an efficient method of analytically deriving the
translation invariant TNS and matrix product state (MPS) representation for the
ground state of translation invariant stabilizer code Hamiltonians in both 1d and
higher dimensions. These TNS/MPS states have minimal virtual bond dimension.
Using the TNS, we derive the entanglement entropy for a variety of stabilizer codes,
including the fracton models the Haah code. We further go beyond the stabilizer codes
and study the structure of entanglement entropy for generic 3d gapped Hamiltonians.
In particular, an explicit formula for a universal physical observable – topological
entanglement entropy (TEE) – has been derived, which sharpens previous results.
Our formula shows that the TEE across an arbitrary entanglement surface is linearly
proportional to the TEE across a torus.
For the second aspect, we use the global symmetries and their ’t Hooft anomalies
of the SU(2) Yang-Mills theory with a theta term to constrain its dynamics. In
particular, we point out that there are four different such theories, distinguished
by Lorentz symmetry enrichments of the Wilson loops in the SU(2) fundamental
representation. We further derive a new mixed anomaly between time reversal and
one form symmetry which can only be seen on an unorientable manifold. We further
use the anomalies to explore various possible dynamics, such as nontrivial degrees
of freedom localized on the domain wall due to spontaneously broken time reversal
symmetry, as well as a potentially possible but exotic quantum phase transition —
Gauge Enhanced Quantum Critical Point.
iii
Acknowledgements
First and formost, I would like to thank my advisor Prof. B. Andrei Bernevig
for his guidance and support. His sharp and insightful questions are often the major
driving force which pushes our project to a deeper level. I am significantly influenced
and benefited from his persistence in research: every step should be solid, rigorous
and crystal-clearly presented; never give up easily until we find a satisfying answer.
Nevertheless, I am also extremely grateful that Andrei gives me sufficient freedom to
think independently, and encourages me to seek my own collaborations.
I’d also like to thank Prof. Nicolas Regnault and Dr. Huan He, who guided me to
the world of quantum entanglement. Huan is my major collaborator in the projects
related to quantum entanglements. These projects will never be possible without his
uncountable insightful inputs, his patience in answering my naive questions and his
encouragements from time to time. For Nicolas, I am always impressed by his ability
to detect possible loop-holes in my arguments. Whenever our project got stuck, he
always has enough patience to hearing my confusions and pointing out useful ideas.
I own a lot to Dr. Juven Wang, who I regard as my second advisor. Since the
first time I met him at IAS in my first year, he provided me unprecedented support
and guidance. I benefited significantly from our enormous discussions, and his unique
insight in both condensed matter physics and quantum field theories, his kindness in
sharing numerous unpublished notes, which consequently guides me to the fascinating
world of QFT frontiers. I also thank him for sharing his enthusiasm in arts (piano)
and sports (juggling), his hospitality during my visits in Harvard and kindly providing
me accommodation for free.
I’d like to express my gratitude to my numerous colleagues: Jie Wang and Jingyu
Luo have been my classmate for over 9 years, and our friendship will not terminate
as I graduate. I thank Curt von Keyserlingk, Barry Bradlyn, Titus Neupert and
Jennifer Cano for the kind helps during the early years of my PhD. I thank Ho-Tat
iv
Lam, Zheyan Wan, Jie Wang and Yi-Zhuang You for various simulating discussions at
various stages, it is really enjoyable to discuss and collaborate with you guys and I’ve
learned a lot. My gratitude also goes to my collaborators German Sierra, Kantaro
Ohmori, Pavel Putrov, Meng Guo, Xueda Wen, Apoorv Tiwari and Peng Ye. I thank
the members of the “Bernevig group”: Sanjay Moudgalya, Fang Xie, Zhida Song and
Biao Lian for discussions.
I’d also like to thank all the wonderful guys around Princeton. Ilya Belopolski,
Ksenia Bulycheva, Duyu Chen, Xiaowen Chen, Yiming Chen, Zijia Cheng, Clay Cor-
dova, Dui Da, Trithep Devakul, Yale Fan, Tong Gao, Akash Goel, Pranay Gorantla,
Jung Pyo Hong, Po-Shen Hsin, Yuwen Hu, Luca Iliesiu, Ziming Ji, Jiaqi Jiang, Zhaoqi
Leng, Biao Lian, Sihang Liang, Xinran Li, Yaqiong Li, Jingjing Lin, Yingyu Liu,
Zheng Ma, Kelvin Mei, Alexey Milekhin, Fedor Popov, Hao Qian, Justin Ripley, Shu-
Heng Shao, Yu Shen, Xue Song, Zhida Song, Suerfu, Siwei Wang, Wudi Wang, Yantao
Wu, Yang-Le Wu, Xin Xiang, Jun Xiong, Bin Xu, Zhenbin Yang, Yizhi You, Junyi
Zhang, Sonia Zhang, Bo Zhao, Wenli Zhao, Hao Zheng, Xinan Zhou, Liujun Zou,
thank you all for your friendship and support. Special thank also goes to wonderful
administrative staffs Antonia (Toni) Sarchi, Catherine (Kate) Brosowsky, Barbara
Mooring and Jessica Heslin. I appreciate Prof. Robert Austin for leading me the
experimental project.
Finally, I thank Prof. David Huse to be the second reader, and Prof. Waseem
Bakr and Prof. Silviu Pufu for being the FPO committee members.
v
To my parents.
vi
Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
1 Preliminaries 1
1.1 Representing the Ground State Wavefunctions . . . . . . . . . . . . . 2
1.1.1 Tensor Network States . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Restricted Boltzmann Machine . . . . . . . . . . . . . . . . . 6
1.2 Entanglement Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Anomalies and SPT phases . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Matrix Product State of A Stabilizer Code in 1D 13
2.1 An Example of Stabilizer Codes: ZZXZZ Model . . . . . . . . . . . 14
2.2 MPS for the ZZXZZ Model . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 General Stabilizer Code Convention . . . . . . . . . . . . . . . . . . . 27
2.4 General Algorithm to Construct MPS . . . . . . . . . . . . . . . . . . 29
3 Restricted Boltamann Machine State for Stabilizer Code in 1D 35
3.1 (Restricted) Boltzmann Machine . . . . . . . . . . . . . . . . . . . . 35
3.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1.2 Relation to MPS . . . . . . . . . . . . . . . . . . . . . . . . . 40
vii
3.2 More on ZZXZZ Model . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3 An Inequality for Rank of MPS . . . . . . . . . . . . . . . . . . . . . 43
3.4 Restricted Boltzmann Machine State of a Stabilizer Code . . . . . . . 45
3.4.1 MPS Matrix Rank For Cocycle SPT Models . . . . . . . . . . 46
3.4.2 An Example: ZZXZZ Model Revisited . . . . . . . . . . . . 50
3.4.3 RBM States of Cocycle Hamiltonians . . . . . . . . . . . . . . 54
3.4.4 RBM Construction for Zq−1XZq−1 Model . . . . . . . . . . . 60
4 Tensor Network States, Entanglement Entropy of CSS Stabilizer
Codes and Fracton Models in 3D 65
4.1 Stabilizer Code Tensor Network States . . . . . . . . . . . . . . . . . 65
4.1.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.1.2 CSS Stabilizer Code and TNS Construction . . . . . . . . . . 68
4.2 Entanglement properties of the stabilizer code TNS . . . . . . . . . . 72
4.2.1 TNS as an exact SVD . . . . . . . . . . . . . . . . . . . . . . 72
4.2.2 Summary of the results . . . . . . . . . . . . . . . . . . . . . . 76
4.3 3D Toric Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.3.1 Hamiltonian of 3D Toric Code Model . . . . . . . . . . . . . . 77
4.3.2 TNS for 3D Toric Code . . . . . . . . . . . . . . . . . . . . . . 79
4.3.3 Concatenation Lemma . . . . . . . . . . . . . . . . . . . . . . 84
4.3.4 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.4 Haah Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.4.1 Hamiltonian of Haah code . . . . . . . . . . . . . . . . . . . . 92
4.4.2 TNS for Haah Code . . . . . . . . . . . . . . . . . . . . . . . 93
4.4.3 Entanglement Entropy for SVD Cuts . . . . . . . . . . . . . . 100
4.4.4 Entanglement Entropy for Cubic Cuts . . . . . . . . . . . . . 106
viii
5 Topological Entanglement Entropy of (3+1)D Gapped Phases of
Matter 115
5.1 Reduction formulas for Entanglement Entropy . . . . . . . . . . . . . 115
5.1.1 Strong Sub-Additivity . . . . . . . . . . . . . . . . . . . . . . 116
5.1.2 Topological Entanglement Entropy . . . . . . . . . . . . . . . 120
5.2 Application: Entanglement Entropy of Generalized Walker-Wang The-
ories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.2.1 Wave Function of GWW Models . . . . . . . . . . . . . . . . 127
5.2.2 Entanglement Entropy of GWW Models . . . . . . . . . . . . 135
6 Anomaly and Dynamics of (3 + 1)d SU(2) Yang-Mills Theory 149
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.1.1 Standard Lore of SU(N) Yang-Mills . . . . . . . . . . . . . . . 150
6.1.2 New Aspects: Lorentz Symmetry Enrichments . . . . . . . . . 151
6.2 SU(2) Yang-Mills Theory at θ = π . . . . . . . . . . . . . . . . . . . . 154
6.2.1 Time Reversal Symmetry . . . . . . . . . . . . . . . . . . . . . 154
6.2.2 One-form Symmetry . . . . . . . . . . . . . . . . . . . . . . . 155
6.2.3 Formulating on Unorientable Manifold and Lorentz Symmetry
Fractionalization . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.2.4 Anomaly on an Unorientable Manifold . . . . . . . . . . . . . 158
6.2.5 Low Energy Dynamics: Overview and Questions . . . . . . . . 161
6.3 Domain Wall from Time Reversal Spontaneously Broken . . . . . . . 163
6.3.1 Domain Wall for (K1, K2) = (0, 0): Semion with U2 = 1 . . . . 164
6.3.2 Domain Wall for (K1, K2) = (1, 0): Semion with U2 = −1 . . . 168
6.3.3 Domain Wall for (K1, K2) = (0, 1): Anti-Semion with U2 = 1 . 172
6.3.4 Domain Wall for (K1, K2) = (1, 1): Anti-Semion with U2 = −1 174
6.3.5 Remarks On CP⊥ and T , and Summary . . . . . . . . . . . . 175
6.4 Application I: Domain Wall Theory Nf < NCFT . . . . . . . . . . . . 176
ix
6.4.1 Lorentz Symmetry Fractionalization, K2 = 1 . . . . . . . . . . 177
6.4.2 U Unitary Symmetry Fractionalization . . . . . . . . . . . . . 178
6.5 Deconfined Gapless U(1) Gauge Theory . . . . . . . . . . . . . . . . . 180
6.5.1 U(1) Gauge Theory and Spin Liquids at θ = 0 . . . . . . . . . 181
6.5.2 U(1) Gauge theory and Spin Liquids at θ = 2π . . . . . . . . 184
6.6 Application II: Gauge Enhanced Quantum Critical Point Nf ≥ NCFT 187
6.6.1 SU(2) QCD4 and Higher Order Interactions: U(1) Spin Liquid
Phases From Higgsing . . . . . . . . . . . . . . . . . . . . . . 188
6.6.2 Symmetries Realizations and Symmetry Enriched U(1) Spin
Liquids in the Infrared . . . . . . . . . . . . . . . . . . . . . . 191
6.6.3 Gauge Enhanced Quantum Critical Points . . . . . . . . . . . 196
A Appendices for Chapter 2 199
A.1 Conventions for MPS and Canonical MPS . . . . . . . . . . . . . . . 199
A.1.1 Conventions for MPS and Transfer Matrix . . . . . . . . . . . 199
A.1.2 Review of Canonical MPS . . . . . . . . . . . . . . . . . . . . 200
A.2 Correlation Functions and Transfer Matrix Eigenvalues . . . . . . . . 202
A.3 Stabilizer Operator Acts on MPS Locally . . . . . . . . . . . . . . . . 215
A.4 The Action of L and R Operators on the MPS Matrices . . . . . . . 218
A.5 Commutation Relations of U Operators . . . . . . . . . . . . . . . . . 222
A.6 Linear Equations for Local Tensors . . . . . . . . . . . . . . . . . . . 223
A.7 Virtual U Operators as Tensor Products of Pauli Matrices . . . . . . 227
B Appendices for Chapter 3 230
B.1 Projective Representations and 1D Symmetry Protected Topological
Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
B.1.1 Projective Representations and Cocycles . . . . . . . . . . . . 230
B.1.2 Cocycle States . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
x
B.1.3 Cocycle Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . 232
B.2 Some Useful Identities . . . . . . . . . . . . . . . . . . . . . . . . . . 237
B.3 More Examples of RBM for Cocycle Model . . . . . . . . . . . . . . . 239
C Appendices for Chapter 4 242
C.1 Proof for the Concatenation Lemma for the 3D Toric Code Model . . 242
C.2 Numerics for Haah Code . . . . . . . . . . . . . . . . . . . . . . . . . 245
D Appendices for Chapter 5 248
D.1 Review of Entanglement Entropy and Spectrum . . . . . . . . . . . . 248
D.2 Local Contributions to the Entanglement Entropy . . . . . . . . . . . 249
D.3 Derivation of the Reduction Formula . . . . . . . . . . . . . . . . . . 253
D.3.1 Recurrence for Genus . . . . . . . . . . . . . . . . . . . . . . . 254
D.3.2 Recurrence for b0 . . . . . . . . . . . . . . . . . . . . . . . . . 257
D.4 Vanishing of the Mean Curvature Contribution in KPLW Prescription 260
D.5 Review of Lattice TQFT . . . . . . . . . . . . . . . . . . . . . . . . . 267
D.6 Surfaces in the dual lattice . . . . . . . . . . . . . . . . . . . . . . . . 271
D.7 Mutual and Self-Linking Numbers . . . . . . . . . . . . . . . . . . . . 275
D.7.1 Intersection and Linking . . . . . . . . . . . . . . . . . . . . . 277
D.7.2 Self-linking Number . . . . . . . . . . . . . . . . . . . . . . . . 279
D.8 NA(CE)NAc(CE) is Independent of CE . . . . . . . . . . . . . . . . . . 281
D.9 A Case Study of the Conjecture Between GSD and TEE . . . . . . . 286
Bibliography 291
xi
List of Tables
5.1 Constant part and topological part of the entanglement entropy for
generalized Walker-Wang models. STQFTc is the constant part of the
EE for the TQFT, while Stopo is the TEE for a general theory which
belongs to the same phase of the TQFT. b0 is the zeroth Betti number
of entanglement surface b0 =∑g∗
g=0 ng. χ =∑g∗
g=0(2 − 2g)ng is the
Euler characteristic of the entanglement surface. In particular, we
have Stopo(S2) = Stopo(T 2). . . . . . . . . . . . . . . . . . . . . . . . . 146
6.1 Symmetry fractionalization and anomalies on the domain wall theory
for four siblings of Yang-Mills. . . . . . . . . . . . . . . . . . . . . . 176
C.1 Entanglement entropies for various bipartitions of the |TNS〉 of the
Haah code. The second to fourth column list the coordinates of vertices
in region A. The column of ”Left/Right” labels the spin on the left or
right position on the vertex (x, y, z), where 0 and 1 corresponds to the
left and right position respectively. We used the coordinate frame as
shown in Eq. 4.49 and Fig. 4.6. . . . . . . . . . . . . . . . . . . . . . 247
xii
List of Figures
1.1 Examples of TNS lattice wave functions in 1D and 2D. Each node is a
tensor whose indices are the lines connecting to it. The physical indices
- of the quantum Hilbert space - are the lines with arrows, while the
lines without any arrows are the virtual indices. Connected lines means
the corresponding indices are contracted. Panel (a) is an MPS for 1D
systems. Panel (b) is a PEPS on a 2D square lattice. . . . . . . . . . 3
1.2 A tensor T s~rv1,~rv2,~rv3,~rv4,~ron a 2d square lattice with one physical index
s~r and four virtual indices v1,~rv2,~rv3,~rv4,~r. . . . . . . . . . . . . . . . . 4
1.3 Tensor network on a 2d 2× 2 torus lattice. . . . . . . . . . . . . . . 5
2.1 A graphical representation of the matrix T gr1gr2gr3 . We denote each phys-
ical index by an arrow. The shaded region represents a unit cell, and
the virtual left and right indices are represented by the horizontal line.
The virtual indices are not explicitly shown here. . . . . . . . . . . . 16
2.2 A graph representation of Eq. (2.4). . . . . . . . . . . . . . . . . . . 18
2.3 A graphical representation of Eqs. (2.17), (2.18) and (2.19). . . . . . 20
2.4 Graphical representation of Eq. (2.39). The shaded purple region rep-
resents the operator Orα acting on the physical indices. . . . . . . . . 29
2.5 Graphical representation of Eq. (2.40). The virtual operator U ri,τ and
(U ri,τ )−1 act on the right virtual index between the r + τ − 1 and r-th
unit cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
xiii
2.6 An illustration of the operators Lr−(τ−1)α,τ and Rr−(τ−1)
α,τ with fixed r and
α, and all 1τ ≤ Pα−1. The blue blocks represent unit cells. The purple
blocks represent the operators Lr−(τ−1)α,τ , and the operators Rr−(τ−1)
α,τ . 31
3.1 An example of RBM state corresponding to q = 3,M = 2, M = 2. The
red circles represent visible spins. The black rectangles are the hidden
spins connecting visible spin belonging to different unit cells, which are
linked to the purple and orange lines representing the weights Aia and
Bia respectively. The black triangles are the hidden spins connecting
visible spins within the same unit cell, which are linked to the green
lines representing the weights Cib. The blue region represents a unit
cell. Notice that the nonzero weights are only between the hidden spins
and the visible spins. . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Graphical representation of the RBM state of the ZZXZZ model. . 54
3.3 Graphical representation of the RBM state of the ZXZ model. The
red circles represent visible spins, the black rectangles represent the
hidden spins connecting visible spin belonging to different unit cells,
and the black triangles represent the hidden spins connecting visible
spins within the same unit cell. . . . . . . . . . . . . . . . . . . . . . 63
4.1 TNS gauge in MPS. (a) A part of an MPS. A1 and A2 are two local
tensors contracted together. (b) We insert the identity operator I =
UU−1 at the virtual level - it acts on the virtual bonds. The tensor
contraction of A1 and A2 does not change. (c) We further multiply U
with A1 and U−1 with A2, resulting in A1 and A2 respectively in Panel
(d). The tensor contraction of A1 and A2 is the same as the tensor
contraction of A1 and A2. The TNS wave function does not change as
well. Similar TNS gauges also appear in other TNS such as PEPS. . . 67
xiv
4.2 (a) A plane of TNS on a cubic lattice. (b) TNS on a cube. The
lines with arrows are the physical indices. The connected lines are
the contracted virtual indices, while the open lines are not contracted.
On each vertex, there lives a T tensor, and on each bond, we have a
projector g tensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.3 The Hamiltonian terms of the 3D toric code model. Panel (a) is Av
which is a product of 6 Z operators, and Panel (b) is Bp which is a
product of 4 X operators. The circled X and Z represent the Pauli
matrices acting on the spin-1/2’s. The toric code Hamiltonian includes
Av terms on all vertices v and Bp terms on all plaquettes p. . . . . . . 78
4.4 Contraction of two local T tensors in the z-direction. . . . . . . . . . 85
4.5 The splitting of tensors near the entanglement cut. . . . . . . . . . . 87
4.6 Tensor contraction for the Haah Code TNS. (a) The lattice size is
2× 3× 3. (b) The lattice size is 3× 3× 3 . . . . . . . . . . . . . . . 99
4.7 Region A contains all the spins connecting with l− 1 T tensors which
are contracted along z direction. The figure shows an example with
l = 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.8 Region A contains all the spins connecting with T tensors which are
contracted in a “tripod-like” shape, where three legs extend along
x, y, z directions. There are three legs extending along x, y, z direc-
tions respectively. In general, three legs can have different length, each
with lx−1, ly−1, lz−1 cubes along three directions. This figure shows
an example where lx = ly = lz = 3. . . . . . . . . . . . . . . . . . . . 101
4.9 Transferring the Pauli X operators of the Bc operator from the region
A (a) to the region A (b). . . . . . . . . . . . . . . . . . . . . . . . . 108
xv
5.1 KPLW prescription of entanglement surface T 2. The space inside the
two torus is divided into three regions, A, B and C, each being a solid
torus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.2 A schematic figure of the topology of spacetime M4 and space S3.
Inside S3, we schematically draw a loop l representing the loop con-
figurations C of the B field in the dual lattice. The dashed surface S
bounding the loop l extends into the spacetime bulkM4, representing
the B field configuration in the dual lattice of spacetime. S ′ repre-
sents the B field configurations that form closed surfaces away from
the boundary of the spacetime ∂M4. The boundary condition in the
path integral Eq. (5.23) is specified by a fixed B configuration C on
S3. The path integral should integrate over all the configurations in
the spacetime bulk M4 with the boundary configuration C on S3 fixed. 128
5.3 A tetrahedron is drawn with solid lines, and its dual is drawn in dash
and gray lines. The 2-simplex (ijk) in the original lattice is dual to
the 1-simplex (ab) in the dual lattice. Similarily, (ikl) is dual to (ad),
(ijl) is dual to (ca) and (jkl) is dual to (ea). The colored dash ar-
rows indicate the orientations of the four 2-simplices, where (ijk) and
(ikl) share the same orientation, and (ijl) and (jkl) share the opposite
orientation. The orientations of the dual-lattice 1-simplices are also
indicated by the arrows on the grey/dashed lines. . . . . . . . . . . . 130
xvi
5.4 An example of the lattice structure of an entanglement cut in (2+1)D.
The green simplices form the entanglement cut Σ, which partitions the
lattice into region A and region Ac. We include Σ as part of region A.
B = π on the red simplices, while B = 0 elsewhere. The dotted loop is
the dual lattice configuration of the red simplices. In this example, the
configuration CE contains two B = π 1-simplices at the entanglement
cut Σ, which are the fourth and eighth 1-simplices of Σ (counting from
the left side) as shown in the figure. . . . . . . . . . . . . . . . . . . . 136
5.5 A particular spatial configuration with one loop γ1 (dashed line)
threading through the hole (the hole itself belongs to region Ac) inside
the region A and one loop γ2 (grey line) threading through the hole
inside the region Ac. γ3 and γ4 are two linked contractible loops, where
γ3 locates inside region A, and γ4 locates both in region A and Ac.
The two blue points are the intersection of l4 with Σ. The simplices
(gray triangles) are living in the real lattice where B = π. The lines
perpendicular to the simplices are living in the dual lattice where
B = π and they form loops in the dual lattice. This configuration
corresponds to α = o, β = o. . . . . . . . . . . . . . . . . . . . . . . . 138
6.1 Lorentz symmetry fractionalization on the Wilson line. The left panel
is the Wilson line with K1 = K2 = 0. When the background field B
for the one-form symmetry is activated, the Wilson line is attached
to a surface Σ bounded by γ. This means that the Wilson line carries
charge 1 under Z2,[1]. K1 = K2 = 0 implies that W1/2 is the worldline of
a boson and a Kramers singlet. The right panel is the Wilson line with
nontrivial (K1, K2). The quantum number of the Lorentz symmetry is
shown in (6.17). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
xvii
6.2 When time reversal is spontaneously broken, there are two vacua. We
consider a configuration where each vacuum occupies half of the space,
and there is a domain wall in between. Time reversal exchanges the
two vacua. The anomaly (6.23) in the bulk induces an anomaly (6.24)
on the domain wall, which consequently constrains that there is an
Abelian semion TQFT on the wall. . . . . . . . . . . . . . . . . . . . 165
6.3 Schematic RG flow diagram around the QCD4 fixed point for odd Nf
and Nf > 11. Possible IR fates are listed for completeness, although
some (such as the U(1) SL on the θ = 0 side) may be extremely unlikely.189
A.1 Graphical representation of Eq. (A.7). . . . . . . . . . . . . . . . . . 202
A.2 Graphical representation of (a) Eq. (A.64) and (b) Eq. (A.67). . . . 217
A.3 Graphical representation of (a) Eq. (A.70) and (b) the virtual operator
U r1,1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
A.4 Graphical representation of (a) Eq. (A.86) and (b) Eq. (A.88), and (c)
Eq. (A.91). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
B.1 RBM network for cocycle model with q = 3, P12 = P13 = 1, P23 = 0. . 241
B.2 RBM network for cocycle model with q = 4, P12 = P13 = P14 = 1, P23 =
P24 = P34 = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
D.1 Entanglement surfaces used in the application of strong sub-additivity
to derive the recurrence relation Eq. (D.17). In (a), A is a general
3-manifold (as an example, we draw A with 1 genus 3 surface and 2
genus 0 surfaces), B is 3-ball and C is a solid torus. In (b), A′ is a
general 3-manifold (as an example, we draw A′ with 1 genus 3 surface
and 2 genus 0 surfaces), B′ is a solid torus, and C′ is a 3-ball, which is
located exactly at the hole of B′. . . . . . . . . . . . . . . . . . . . . 254
xviii
D.2 Entanglement surfaces used in the application of strong sub-additivity
to derive Eq. (D.22). In (a), A is a 3-manifold with multiple genus
zero surfaces, B is a 3-ball, C is a 3-ball with small 3-ball removed. In
(b), A′ is an open 3-manifold with multiple genus zero surfaces, B′ is
a 3-ball with a small 3-ball removed and C′ is a 3-ball located exactly
in the empty 3-ball inside B′. . . . . . . . . . . . . . . . . . . . . . . 257
D.3 KPLW prescription of regularized entanglement surface T 2. . . . . . . 260
D.4 Left: Regularization of a rectangular hinge with small arcs. Right:
One choice of regularization of each hinge in Fig. D.3. The numbers
label various hinges. . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
D.5 Dual lattice of a tetrahedron (ijkl). (ijkp), (ijlq), (iklr), (jkls) are
four adjacent tetrahedra to (ijkl), which are dual to (b), (c), (d), (e), (a)
respectively. The red dots are the intersection between 2-simplices in
the real lattice and the 1-simplices in the dual lattice. For example,
the red dot on (ab) is the intersection point of (ab) and (ijk). . . . . . 271
xix
D.6 We illustrate the geometric meaning of the Hodge dual in a two-
dimensional space example. Suppose A is a 1-cochain, which equals
π on 1-simplices in the dual lattice and 0 elsewhere. A = π ∗2 Σ(l1) +
π ∗2 Σ(l2), where l1 and l2 are loops in the dual lattice drawn in dashed
lines. Σ(l1) and Σ(l2) are 1-cochains living on the 1-simplices in the
dual lattice. ∗2 is a lattice version of Hodge star, which transforms
the 1-cochain living on the dual lattice (dashed lines) to a 1-cochain
living on the lattice (green and purple bold lines). Correspondingly,
A = π ∗2 Σ(l1) + π ∗2 Σ(l2) is a 1-cochain living on the green and pur-
ple bold lines. We use the dual lattice configuration Si, li to label the
B,A-cochains because the dual lattice configurations are easier to vi-
sualize. The interpretation of the 2-cochain B can be straightforwardly
generalized to three spatial dimensions. . . . . . . . . . . . . . . . . . 276
D.7 Regularization of a spatial lattice. The blue arrow represents the con-
stant vector (ax, ay, az). The dashed lattice is obtained from the solid
lattice by the translation (x, y, z)→ (x+ ax, y + ay, z + az). . . . . . 280
D.8 An example of lattice regularization of a trefoil knot. l is a knot (drawn
in the dual lattice), while la is the knot obtained by lattice regulariza-
tion. The underlying lattice is omitted for clarity. . . . . . . . . . . . 280
D.9 A configuration associated with nontrivial CE (on panel (a)) can be
reduced to a configuration associated with trivial CE (on panel (b)). 283
D.10 A configuration associated with trivial CE (on panel (a)) can be reduced
to a configuration associated with a nontrivial CE (on panel (b)). . . 283
xx
D.11 Configurations on a 2 × 2 lattice with periodic boundary conditions.
There are two entanglement cuts, denoted by two green lines. The oc-
cupied bonds in the real lattice are shown in red, and occupied bonds in
the dual lattice are shown as dotted lines. (a), (b), (c), (d) are configu-
rations with no bonds occupied on the entanglement cut. (e), (f), (g),
(h) are configurations with two bonds occupied on the entanglement
cut. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
xxi
Chapter 1
Preliminaries
Topological phases of matter has recently emerged as one of the central themes of con-
densed matter physics[1, 2, 3, 4, 5]. Prior to the discovery of topological phases, the
consensus in the physics community (dubbed the Landau Paradigm) was that gapped
phases could be classified by symmetry breaking order parameters[6, 7] where the
ground states spontaneously break the global symmetry. The discovery of topological
order[8, 9, 10] revealed that two gapped systems can reside in distinct phases ab-
sent any global symmetries. The discovery of symmetry protected topological (SPT)
order[11, 12, 13, 2, 14, 15] further enriched the family of topological phases of matter:
two systems with the same global symmetry can be in different phases even with
trivial topological order. Both the topological order and the SPT phases (which de-
scribe the phases of gapped Hamiltonians) belong to the family of topological phases
of matter. There are other candidates of topological phases of matters that describe
the phases of gapless Hamiltonians[16, 17, 18, 19], which will not be discussed in this
thesis.
1
1.1 Representing the Ground State Wavefunctions
The ground state encodes the key information of the topological phases of the gapped
Hamiltonian. For instance, in two spatial dimensions, the subleading term of the
entanglement entropy of the ground state (i.e., the topological entanglement entropy
(TEE)) reveals the total quantum dimension D of the topological order[20, 21]. For
example, when D = 2, there are only two possible topological orders: Z2 topological
order (described by the toric code model) [22], and twisted Z2 topological order
(described by the double semion model)[23]. The overlap between different ground
states encodes the information of quasi-particle statistics. [24]
In this thesis, we will study the two representations of the ground state wavefunc-
tions, i.e., the Tensor Network State (TNS) and the Restricted Boltzmann Machine
(RBM). In particular, when a wavefunction describes the ground state of a one di-
mensional Hamiltonian, the TNS is usually dubbed the Matrix Product State (MPS).
1.1.1 Tensor Network States
TNS has been heavily used in condensed matter physics in the past decade, especially
in the study of 1D and 2D topological phases[25].1 Amongst many examples,
1. Numerical simulations of the 1D Haldane chain led to the discovery of symmetry
protected topological phases (SPT)[26].
2. Fractional quantum Hall states can be exactly written as MPS[27, 28, 29, 30,
31, 32, 33, 34, 35, 36, 37, 38] which allows performing numerical calculations
not accessible by exact diagonalization techniques.
3. A large class of spin liquids wave functions can be constructed using TNS with
global spin rotation symmetries and lattice symmetries[39, 40, 41, 42, 43, 44, 45].
1We use nD to denote n spatial dimension, and nd to label n spacetime dimension. We will alsosometimes use (n+ 1)d to label n spatial dimension and 1 time dimension.
2
(a)
(b)
Figure 1.1: Examples of TNS lattice wave functions in 1D and 2D. Each node is a
tensor whose indices are the lines connecting to it. The physical indices - of the quan-
tum Hilbert space - are the lines with arrows, while the lines without any arrows are
the virtual indices. Connected lines means the corresponding indices are contracted.
Panel (a) is an MPS for 1D systems. Panel (b) is a PEPS on a 2D square lattice.
The definition of the TNS is as follows. For convenience, we will focus on the TNS
defined on a 2d square lattice M, see (b) in Figure 1.1. M can be the infinite 2d
plane M = R2, or 2d torus M = T 2. (The definitions of TNS in other dimensions
and in other lattices are similar.) Each vertex supports a physical degree of freedom
(dof) labeled as s~r where ~r denotes the location of the site.2 For instance, if the dof at
each site is a spin-S, then s~r can take 2S+1 values, which can be conveniently labeled
as s~r ∈ {0, 1, ..., 2S}. We denote 2S+ 1 as the physical bond dimension. Graphically,
we use an arrow to represent this physical dof at each site as in Figure 1.1.
2The physical dof can also be chosen to define on links, which will be the case in Chapter 4.
3
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v4,~r<latexit sha1_base64="aYxE4WumYiNWoWicI33SgRE0GEA=">AAAB83icbZBLSwMxFIXv1Fetr6pLN8EiuJAyIwVdFty4rGBbsTOUTHqnDc1khiRTKEPBX+HGhSJu/TPu/Demj4W2Hgh8nHNDbk6YCq6N6347hbX1jc2t4nZpZ3dv/6B8eNTSSaYYNlkiEvUQUo2CS2wabgQ+pAppHApsh8Obad4eodI8kfdmnGIQ077kEWfUWMsfdfPahT9CRtSkW664VXcmsgreAiqwUKNb/vJ7CctilIYJqnXHc1MT5FQZzgROSn6mMaVsSPvYsShpjDrIZztPyJl1eiRKlD3SkJn7+0ZOY63HcWgnY2oGejmbmv9lncxE10HOZZoZlGz+UJQJYhIyLYD0uEJmxNgCZYrbXQkbUEWZsTWVbAne8pdXoXVZ9Szf1Sr1x6d5HUU4gVM4Bw+uoA630IAmMEjhGV7hzcmcF+fd+ZiPFpxFhcfwR87nD6L9keM=</latexit><latexit sha1_base64="aYxE4WumYiNWoWicI33SgRE0GEA=">AAAB83icbZBLSwMxFIXv1Fetr6pLN8EiuJAyIwVdFty4rGBbsTOUTHqnDc1khiRTKEPBX+HGhSJu/TPu/Demj4W2Hgh8nHNDbk6YCq6N6347hbX1jc2t4nZpZ3dv/6B8eNTSSaYYNlkiEvUQUo2CS2wabgQ+pAppHApsh8Obad4eodI8kfdmnGIQ077kEWfUWMsfdfPahT9CRtSkW664VXcmsgreAiqwUKNb/vJ7CctilIYJqnXHc1MT5FQZzgROSn6mMaVsSPvYsShpjDrIZztPyJl1eiRKlD3SkJn7+0ZOY63HcWgnY2oGejmbmv9lncxE10HOZZoZlGz+UJQJYhIyLYD0uEJmxNgCZYrbXQkbUEWZsTWVbAne8pdXoXVZ9Szf1Sr1x6d5HUU4gVM4Bw+uoA630IAmMEjhGV7hzcmcF+fd+ZiPFpxFhcfwR87nD6L9keM=</latexit><latexit sha1_base64="aYxE4WumYiNWoWicI33SgRE0GEA=">AAAB83icbZBLSwMxFIXv1Fetr6pLN8EiuJAyIwVdFty4rGBbsTOUTHqnDc1khiRTKEPBX+HGhSJu/TPu/Demj4W2Hgh8nHNDbk6YCq6N6347hbX1jc2t4nZpZ3dv/6B8eNTSSaYYNlkiEvUQUo2CS2wabgQ+pAppHApsh8Obad4eodI8kfdmnGIQ077kEWfUWMsfdfPahT9CRtSkW664VXcmsgreAiqwUKNb/vJ7CctilIYJqnXHc1MT5FQZzgROSn6mMaVsSPvYsShpjDrIZztPyJl1eiRKlD3SkJn7+0ZOY63HcWgnY2oGejmbmv9lncxE10HOZZoZlGz+UJQJYhIyLYD0uEJmxNgCZYrbXQkbUEWZsTWVbAne8pdXoXVZ9Szf1Sr1x6d5HUU4gVM4Bw+uoA630IAmMEjhGV7hzcmcF+fd+ZiPFpxFhcfwR87nD6L9keM=</latexit><latexit sha1_base64="aYxE4WumYiNWoWicI33SgRE0GEA=">AAAB83icbZBLSwMxFIXv1Fetr6pLN8EiuJAyIwVdFty4rGBbsTOUTHqnDc1khiRTKEPBX+HGhSJu/TPu/Demj4W2Hgh8nHNDbk6YCq6N6347hbX1jc2t4nZpZ3dv/6B8eNTSSaYYNlkiEvUQUo2CS2wabgQ+pAppHApsh8Obad4eodI8kfdmnGIQ077kEWfUWMsfdfPahT9CRtSkW664VXcmsgreAiqwUKNb/vJ7CctilIYJqnXHc1MT5FQZzgROSn6mMaVsSPvYsShpjDrIZztPyJl1eiRKlD3SkJn7+0ZOY63HcWgnY2oGejmbmv9lncxE10HOZZoZlGz+UJQJYhIyLYD0uEJmxNgCZYrbXQkbUEWZsTWVbAne8pdXoXVZ9Szf1Sr1x6d5HUU4gVM4Bw+uoA630IAmMEjhGV7hzcmcF+fd+ZiPFpxFhcfwR87nD6L9keM=</latexit>
~r<latexit sha1_base64="rpC3ISyP1CWO+PaHVq+XM6lAlP0=">AAAB7XicbZDLSgMxFIbP1Futt6pLN8EiuCozIuiy4MZlBXvBdiiZ9LSNzSRDkimUoeAjuHGhiFvfx51vY3pZaOsPgY//nHDO+aNEcGN9/9vLra1vbG7ltws7u3v7B8XDo7pRqWZYY0oo3YyoQcEl1iy3ApuJRhpHAhvR8GZab4xQG67kvR0nGMa0L3mPM2qdVW+PkBHdKZb8sj8TWYVgASVYqNopfrW7iqUxSssENaYV+IkNM6otZwInhXZqMKFsSPvYcihpjCbMZttOyJlzuqSntHvSkpn7+0dGY2PGceQ6Y2oHZrk2Nf+rtVLbuw4zLpPUomTzQb1UEKvI9HTS5RqZFWMHlGnudiVsQDVl1gVUcCEEyyevQv2iHDi+uyxVHp7mceThBE7hHAK4ggrcQhVqwOARnuEV3jzlvXjv3se8NectIjyGP/I+fwBmwI96</latexit><latexit sha1_base64="rpC3ISyP1CWO+PaHVq+XM6lAlP0=">AAAB7XicbZDLSgMxFIbP1Futt6pLN8EiuCozIuiy4MZlBXvBdiiZ9LSNzSRDkimUoeAjuHGhiFvfx51vY3pZaOsPgY//nHDO+aNEcGN9/9vLra1vbG7ltws7u3v7B8XDo7pRqWZYY0oo3YyoQcEl1iy3ApuJRhpHAhvR8GZab4xQG67kvR0nGMa0L3mPM2qdVW+PkBHdKZb8sj8TWYVgASVYqNopfrW7iqUxSssENaYV+IkNM6otZwInhXZqMKFsSPvYcihpjCbMZttOyJlzuqSntHvSkpn7+0dGY2PGceQ6Y2oHZrk2Nf+rtVLbuw4zLpPUomTzQb1UEKvI9HTS5RqZFWMHlGnudiVsQDVl1gVUcCEEyyevQv2iHDi+uyxVHp7mceThBE7hHAK4ggrcQhVqwOARnuEV3jzlvXjv3se8NectIjyGP/I+fwBmwI96</latexit><latexit sha1_base64="rpC3ISyP1CWO+PaHVq+XM6lAlP0=">AAAB7XicbZDLSgMxFIbP1Futt6pLN8EiuCozIuiy4MZlBXvBdiiZ9LSNzSRDkimUoeAjuHGhiFvfx51vY3pZaOsPgY//nHDO+aNEcGN9/9vLra1vbG7ltws7u3v7B8XDo7pRqWZYY0oo3YyoQcEl1iy3ApuJRhpHAhvR8GZab4xQG67kvR0nGMa0L3mPM2qdVW+PkBHdKZb8sj8TWYVgASVYqNopfrW7iqUxSssENaYV+IkNM6otZwInhXZqMKFsSPvYcihpjCbMZttOyJlzuqSntHvSkpn7+0dGY2PGceQ6Y2oHZrk2Nf+rtVLbuw4zLpPUomTzQb1UEKvI9HTS5RqZFWMHlGnudiVsQDVl1gVUcCEEyyevQv2iHDi+uyxVHp7mceThBE7hHAK4ggrcQhVqwOARnuEV3jzlvXjv3se8NectIjyGP/I+fwBmwI96</latexit><latexit sha1_base64="rpC3ISyP1CWO+PaHVq+XM6lAlP0=">AAAB7XicbZDLSgMxFIbP1Futt6pLN8EiuCozIuiy4MZlBXvBdiiZ9LSNzSRDkimUoeAjuHGhiFvfx51vY3pZaOsPgY//nHDO+aNEcGN9/9vLra1vbG7ltws7u3v7B8XDo7pRqWZYY0oo3YyoQcEl1iy3ApuJRhpHAhvR8GZab4xQG67kvR0nGMa0L3mPM2qdVW+PkBHdKZb8sj8TWYVgASVYqNopfrW7iqUxSssENaYV+IkNM6otZwInhXZqMKFsSPvYcihpjCbMZttOyJlzuqSntHvSkpn7+0dGY2PGceQ6Y2oHZrk2Nf+rtVLbuw4zLpPUomTzQb1UEKvI9HTS5RqZFWMHlGnudiVsQDVl1gVUcCEEyyevQv2iHDi+uyxVHp7mceThBE7hHAK4ggrcQhVqwOARnuEV3jzlvXjv3se8NectIjyGP/I+fwBmwI96</latexit>
Figure 1.2: A tensor T s~rv1,~rv2,~rv3,~rv4,~ron a 2d square lattice with one physical index s~r
and four virtual indices v1,~rv2,~rv3,~rv4,~r.
To write down the wavefunction, one associates a tensor T s~rv1,~rv2,~rv3,~rv4,~rwith one
physical index s~r and four virtual indices v1,~rv2,~rv3,~rv4,~r on each site, see Figure 1.2.
Suppose each virtual index can be valued in P distinct values, e.g. va,~r ∈ {1, ..., P},
we denote P as the virtual bond dimension. The tensor network state is
|TNS〉 =∑
{s~r}
CM(∏
~r∈M
T s~r
)|{s~r}〉 (1.1)
Let us explain the notations in Eq. (1.1).
1. As explained above, M can be for instance the infinite 2d plane R2 or a 2d
torus T 2 depending on the choice of boundary conditions.
2. |{s~r}〉 is the short hand notation of the direct product state ⊗~r∈M|s~r〉, which
serves as a set of complete, orthogonal and normalized basis of the TNS.
3.∑{s~r} represents the sum over all possible configurations of the spin configura-
tions on the lattice M.
4. CM(∏
~r∈M T s~r)
is the coefficient of the basis |{s~r}〉, which represents the con-
traction over all the virtual indices shared by the neighorhood tensors. As a
simple example, let us consider a 2d torus with two sites along each direction,
4
see figure 1.3. Then
CT 2
(∏
~r∈M
T s~r
)=
P∑
v1,v2,...,v8=1
Ts~0v1v3v2v4T
s~xv2v4v1v8
Ts~yv5v7v6v3T
s~x+~yv6v8v5v4 (1.2)
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Figure 1.3: Tensor network on a 2d 2× 2 torus lattice.
In this thesis, we will study the TNS and MPS representation of the ground state
of a particular class of Hamiltonians: the translation invariant stabilizer codes. The
stabilizer codes are a class of spin Hamiltonians where every term mutually commute.
These are the simplest class of Hamiltonians with a finite energy gap. We will address
the following question: Given a stabilizer code Hamiltonian, how do we find the
MPS/TNS of its ground state? We will address this question in Chapter 2 for an
arbitrary 1D translation invariant stabilizer code based on various nice properties of
1D MPS, following [46]. In Chapter 4, following [47], we work out the TNS for a large
class of stabilizer codes in higher dimensions — the CSS codes — whose Hamiltonian
terms are either products of purely Pauli X or purely Pauli Z operators (and there
are no mixing terms). We will find simple algorithms to find the tensors/matrices
directly from the Hamiltonian whose virtual bond dimension is minimal.
5
1.1.2 Restricted Boltzmann Machine
Restricted Boltzmann machines (RBM)[48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59,
60, 61, 62, 63, 64] and more generally neural networks, have recently gained lots
of attention as numerical tools for studying quantum many-body physics, boosted
by the fast paced progress in machine learning. An RBM is a restriction from a
Boltzmann machine (BM). The BM is defined on a bipartite graph, whose vertices
are grouped into two classes: the visible vertices and the hidden vertices. Suppose
there are n visible vertices andm hidden vertices, and we associate the visible variables
g ∈ {0, 1}n and the hidden variables h ∈ {0, 1}m on the visible and hidden vertices
respectively. The variables {g, h} obey the Boltzmann distribution,
P (g, h) =1
Zexp (−E(g, h)) , (1.3)
where E(g, h) is a real function mimicking the “energy” in the Boltzmann distribution,
and Z =∑
g,h exp (−E(g, h)) is the partition function. As the name suggests, only the
visible variables will show up in the physical probability distribution, while the hidden
variables are summed over and thus hidden. Given Eq. (1.3), the BM is defined to be
the marginal distribution P (g) over the visible variables g by summing over all the
hidden variables {h}
P (g) =∑
h
P (g, h) =1
Z
∑
h
exp (−E(g, h)) . (1.4)
The RBM further requires that the “energy” function E(g, h) depends linearly on g
and h. The most important property of RBM is its representing power. It has been
proven[65] in the machine learning context that any probability distribution P0(g) of
an n number of Z2 variables, i.e., g ∈ {0, 1}n, can be approximated arbitrarily well
by an RBM P (g) given enough number of hidden spins. See Ref. [65] for details.
6
For the purposes of the quantum physics, it is natural to change the “energy”
function E(g, h) from a real function to a complex one. Then we can interpret the
“complex probability distribution” P (g) as the coefficients of a quantum many-body
wave function:3 4
|Ψ〉 =∑
g
P (g)|g〉, (1.5)
where |g〉 is the basis to expand the quantum states |Ψ〉. The RBM state refers to
the ansatz in Eq. (1.5).
In chapter 3, following [46], we study the RBM representation of the ground state
of the 1D translation invariant stabilizer code, the simplest class of models one can
possibly study. We make progresses toward answering the following questions, when
the stabilizer codes have one ground state with periodic boundary condition (PBC):
1. How to map a translational invariant and finitely connected RBM to an MPS?
2. Given a stabilizer code, can we cast the ground state as an RBM state mini-
mizing the number of hidden spins?
For the first question, we give a necessary condition for a MPS that can be written as
an RBM state. For the second one, we find a sufficient condition where the ground
state of a stabilizer code can be exactly written as an RBM state with minimal hidden
spins.
1.2 Entanglement Entropy
Entanglement entropy is a key ingredient in characterizing the “nontrivialness” of
the ground state wavefunction. If a ground state of a gapped Hamiltonian does not
have entanglement entropy, it means that the state is a product state, hence by
3To obtain a normalized state, we need to rescale P (g) by a common factor irrelevant of g.4The construction of the quantum wave-function from a classical Hamiltonian has been discussed
in Ref. [66] following the work of Rokhsar and Kivelson[67].
7
definition the Hamiltonian belongs to a trivial phase. Moreover, one can also use
the entanglement entropy to probe the topological phase, which will be discussed in
detail in this thesis.
Let us define the entanglement entropy for a generic state |ψ〉. To define the
entanglement unambiguously, one needs to discretize the space M into a lattice. As
an example, we consider a 2D lattice in (b) of Figure 1.1 where each site supports
some dof which expand a local Hilbert space H~r. (When the dof in each site is a
spin 12, then the local Hilbert space is two dimensional, expanded by | ↑〉, | ↓〉. ) The
entire Hilbert space HM is the tensor product of the local Hilbert spaces at each site:
HM = ⊗~rH~r (1.6)
We further divide the lattice into two dis-adjoint regions, denoted as A and B respec-
tively. M = A∪B, where ∪ represents ”union”. Hence a given site either belongs to
region A or region B. Then the total Hilbert space HM splits into the direct product
of the Hilbert space in each region:
HM = ⊗~r∈A ⊗~r∈B H~r = HA ⊗HB (1.7)
where HA = ⊗~r∈AH~r and HB = ⊗~r∈BH~r. The density matrix ρ of the state |ψ〉 is
simply ρ = |ψ〉〈ψ|. The reduced density matrix of region A is given by the trace of ρ
on HB.
ρA = TrHB |ψ〉〈ψ| (1.8)
8
The entanglement entropy of region A is given by the Von-Neumann entropy of the
reduced density matrix ρA
SA = −TrρA log ρA (1.9)
In particular, when |ψ〉 is the ground state of a gapped Hamiltonian, the entan-
glement entropy SA typically obeys the area law [68]: the entanglement entropy of
the ground states with respect to a subregion A grows linearly with the area of the
boundary Area(∂A) of subregion A:
SA ∼ Area(∂A).
Specifically, in 1D, the entanglement entropy of the subsystem A is a constant (which
is proven in [68]):
Sl ∼ Const,
since the boundary of A contains only two points. In 2D, the entanglement entropy
of the subsystem A obeys:
SA ∼ l,
where l is the perimeter length of the boundary of A. For topological ordered phases
in 2D, the area law gets supplemented by a sub-leading constant contribution dubbed
topological entanglement entropy (TEE) [69, 70], which contains the total quantum
dimension of the ground state. An important feature of the TEE is that it is invariant
under the renormalization group flow of the Hamiltonian as well as the deformation
of the entanglement cut. Therefore. the TEE is one of the universal characterizations
of phases of matters.
In higher spatial dimensions than 2D, SA exhibits more exotic properties:
9
1. For a Hamiltonian with conventional topological order (where the ground state
degeneracy on a 3-torus is finite), the subleading terms to the area law term
can host non-topological information of the system. This is in contrast to the
entanglement entropy of the 2D ground state, where the subleading term is
topological. Hence it is more involved to understand the 3D topological entan-
glement entropy. This aspect will be addressed in chapter 5.
2. There are more exotic topological orders in 3D, where the ground state degen-
eracy on a 3-torus depends on the system size. Recently, 3D so-called fracton
models[71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90,
91, 92] represented by Haah code[71] and X-cube model have been proposed,
attracting the attention of both quantum information[93] and condensed matter
community[94, 95, 96, 97, 98]. They can be realized by stabilizer code Hamilto-
nians, whose fundamental property is that they consist solely of sums of terms
that commute with each other. They are hence exactly solvable. The defining
features of fracton models include (but are not restricted to):
(a) Fracton models have an energy gap, since they can be realized by com-
muting Hamiltonian terms.
(b) The ground state degeneracy on the torus changes as the system size
changes. Hence, fracton models seem not to have thermodynamic limits.
(c) The low energy excitations can have fractal shapes, other than only points
and loops available in conventional topological phases.
(d) The excitations of fracton models are not fully mobile: they can only
move either along submanifold of the 3D lattice (Type I fracton model), or
completely immobile without energy dissipation (Type II fracton model).
We will discuss the entanglement entropy of 3D stabilizer codes extensively in chapter
4 using the TNS, following [47].
10
1.3 Anomalies and SPT phases
Another interesting aspect of topological phase of matter is that when the space
has a boundary, there can be interesting phenomena (such as gapless mode, sponta-
neous symmetry breaking, nontrivial topological order, etc) localized on the boundary.
[2, 14, 99, 100] This relation between the topological phases and boundary phenomena
has been established for a particular subclass of topological phases: the SPT phases.
Concretely, the boundary of the (d+ 1) dimensional system in a SPT phase can not
be a trivially gapped insulator (whose ground state can be adiabatically connected
to a product state). The nontrivial degree of freedom on the boundary should be
anomalous in order to componsate the nontrivial anomaly inflow from the SPT in
the bulk [101, 102, 5, 103, 104]. Recently there are extensive progress toward under-
standing the boundary of topological order, which the boundary can support degrees
of freedom with “non-invertible anomaly”[105, 106, 107].
A canonical example in condensed matter physics is the topological insulator in
(3+1)d. [2, 14, 99, 100] A topological insulator in (3+1)d is within a nontrivial SPT
phase protected by (U(1)oZT4 )/Z2, where U(1) is the charge conservation symmetry,
ZT4 is the time reversal symmetry which satisfies T 2 = (−1)F , and the Z2 in the
denominator means that the subgroup Z2 ⊂ U(1) and the subgroup ZT2 ⊂ ZT4 are
identical, which is precisely the fermi parity generated by (−1)F . On the boundary,
it is well known that the bands have to cross and the vicinity of the crossing point
can be described by a Dirac fermion. The symmetry (U(1)oZT4 )/Z2 in such a Dirac
fermion is know to be anomalous: gauging the U(1) symmetry of a Dirac fermion
necessarily breaks time reversal symmetry. If we want to gauge U(1) while preserving
the time reversal, we need to consider the combined system of (2 + 1)d Dirac fermion
and (3+1)d topological insulator in the bulk. For this combined system, gauging U(1)
symmetry does not break the time reversal, hence the combined system is anomaly
free.
11
In this thesis, following [108, 109], we study another example of the correspondence
“anomalous dof on the boundary ←→ SPT in the bulk”. The “anomalous dof” will
be (3 + 1)d Yang-Mills (YM) theory with a theta term at (θ = π), which has a Z2,[1]
one-form symmetry and time reversal symmetry ZT2 . We will study the anomaly of
the global symmetry Z2,[1] × ZT2 , and identify the SPT in the (4 + 1)d bulk. We will
also discuss the dynamics of this (3 + 1)d YM theory, which is constrained by the
anomaly.
12
Chapter 2
Matrix Product State of A
Stabilizer Code in 1D
In this chapter, we present our algorithm to find an MPS for the ground state of a
translational invariant stabilizer code. We illustrate our algorithm using an example,
the ZZXZZ model, and then discuss of the general case. Each step of the algorithm
is proven in App. A.2, A.3, A.4, A.5 and A.6. We derive and prove our results based
on the following assumptions throughout this chapter:
Assumption 2.0.1. We only consider the translational invariant stabilizer codes that
have a unique ground state with periodic boundary condition (PBC).
Assumption 2.0.2. The MPS matrices of the translational invariant stabilizer codes
become independent of the system size for sufficiently large system sizes.
Assumption 2.0.3. The MPS matrices are independent of the boundary condition.
In other words, in the bulk, the MPS matrices for PBC are the same as those for the
open boundary condition (OBC).
We begin by stating the notations of spin chains. We mainly consider spin models
defined on a finite chain with L unit cells and PBC. Each unit cell contains q spin-12’s.
13
For the i-th spin (i = 1, . . . , q) in the r-th unit cell (r = 0, . . . , L− 1), we associate a
two dimensional Hilbert space spanned by |gri 〉, where gri = 0, 1 corresponds to spin
up and spin down respectively. |gri 〉 satisfies
Zri |gri 〉 = (−1)g
ri |gri 〉, Xr
i |gri 〉 = |1− gri 〉, (2.1)
where Zri and Xr
i are Pauli Z and X matrices acting on |gri 〉.
2.1 An Example of Stabilizer Codes: ZZXZZ
Model
To define the ZZXZZ model, we place 3 physical spin-12’s in each unit cell, i.e.,
q = 3. (We choose q = 3 since it fits naturally into the discussion of general cocycle
models. See App. B.1 for details.) We introduce three sets of commuting operators
Orα (α = 1, 2, 3) defined as
Or1 = Zr2Z
r3X
r+11 Zr+1
2 Zr+13
Or2 = Zr3Z
r+11 Xr+1
2 Zr+13 Zr+2
1
Or3 = Zr1Z
r2X
r3Z
r+11 Zr+1
2 ,
(2.2)
where r is defined modulo L due to PBC. Using these operators, the Hamiltonian
reads
HZZXZZ = −L−1∑
r=0
(Or1 +Or2 +Or3). (2.3)
All the terms in the Hamiltonian Eq. (2.3) mutually commute, and have eigenvalues
±1. Thus its ground state is the common positive eigenstate of Orα for any r and α,
14
i.e.,
Orα|GS〉 = |GS〉, α = 1, 2, 3, r = 0, 1, . . . , L− 1. (2.4)
For example, one can construct |GS〉 as
|GS〉 =L−1∏
r=0
3∏
α=1
(1 +Orα
2
)|0〉, (2.5)
where
|0〉 =L−1⊗
r=0
3⊗
i=1
|0ri 〉. (2.6)
It is straightforward to verify that the |GS〉 in Eq. (2.5) satisfies Eq. (2.4).
Our goal in this section is to express the ground state |GS〉 as an MPS
|GS〉 =∑
{gri }
Tr
( L−1∏
r=0
T gr1gr2gr3
)|{gri }〉, (2.7)
where
|{gri }〉 ≡L−1⊗
r=0
3⊗
i=1
|gri 〉. (2.8)
The matrix T gr1gr2gr3 is labeled with three physical indices gr1, g
r2 and gr3 in the r-th
unit cell. The left and right virtual indices of T gr1gr2gr3 and matrix elements will be
solved in Sec. 2.2. The product of two T matrices amounts to contracting the pair of
virtual indices between them. The coefficient of |{gri }〉 is determined by contracting
all virtual indices with the same configuration of physical spins {gri }.
15
Figure 2.1: A graphical representation of the matrix T gr1gr2gr3 . We denote each physical
index by an arrow. The shaded region represents a unit cell, and the virtual left
and right indices are represented by the horizontal line. The virtual indices are not
explicitly shown here.
The matrix T gr1gr2gr3 is graphically represented in Fig. 2.1. Some notations and general
properties of MPS for stabilizer codes are given in App. A.1.
2.2 MPS for the ZZXZZ Model
To derive the MPS for the ground state |GS〉 of the ZZXZZ model Eq. (2.3), we
start with Eq. (2.4). Ori acts on the basis |{gri }〉 in each summand of |GS〉 in Eq. (2.7).
By re-arranging the summation, we derive the action of Ori on the T -matrices. For
16
example, let us consider the action of Or1,
Or1|GS〉 =∑
{gr′i }
Tr
( L−1∏
r′=0
T gr′1 g
r′2 g
r′3
)Or1|{gr
′
i }〉
=∑
{gr′i }
Tr
( L−1∏
r′=0
T gr′1 g
r′2 g
r′3
)(−1)g
r2+gr3+gr+1
2 +gr+13
×∣∣∣∣{gr
′
i }|r′≤r, {(1− gr+11 )gr+1
2 gr+13 }, {gr′′i }|r′′>r+1
⟩
=∑
{gr′i }
Tr
((∏
r′<r
T gr′1 g
r′2 g
r′3
)· T gr1 gr2 gr3 · T (1−gr+1
1 )gr+12 gr+1
3
·( ∏
r′>r+1
T gr′1 g
r′2 g
r′3
))(−1)g
r2+gr3+gr+1
2 +gr+13 |{gr′i }〉
≡∑
{gr′i }
Tr
((∏
r′<r
T gr′1 g
r′2 g
r′3
)· Or1 ◦
(T g
r1 gr2 gr3 · T gr+1
1 gr+12 gr+1
3
)
·( ∏
r′>r+1
T gr′1 g
r′2 g
r′3
))|{gr′i }〉.
(2.9)
In the second equality, we use the definition of Or1 in Eq. (2.2), and in the last equality
we defined the action of Or1 on T gr1gr2gr3 · T gr+1
1 gr+12 gr+1
3 , via
Or1 ◦(T g
r1gr2gr3 ·T gr+1
1 gr+12 gr+1
3
)≡ (−1)g
r2+gr3+gr+1
2 +gr+13
(T g
r1gr2gr3 ·T (1−gr+1
1 )gr+12 gr+1
3
). (2.10)
For Or2 and Or3, we similarly define
Or2 ◦(T g
r1gr2gr3 · T gr+1
1 gr+12 gr+1
3 · T gr+21 gr+2
2 gr+23
)
≡ (−1)gr3+gr+1
1 +gr+13 +gr+2
1
(T g
r1gr2gr3 · T gr+1
1 (1−gr+12 )gr+1
3 · T gr+21 gr+2
2 gr+23
)
Or3 ◦(T g
r1gr2gr3 · T gr+1
1 gr+12 gr+1
3
)≡ (−1)g
r1+gr2+gr+1
1 +gr+12
(T g
r1gr2(1−gr3) · T gr+1
1 gr+12 gr+1
3
).
(2.11)
17
Figure 2.2: A graph representation of Eq. (2.4).
Using the definitions Eqs. (2.10) and (2.11), we find that a sufficient condition for the
stabilizer condition Eq. (2.4) is
Or1 ◦(T g
r1gr2gr3 · T gr+1
1 gr+12 gr+1
3
)=(T g
r1gr2gr3 · T gr+1
1 gr+12 gr+1
3
)
Or2 ◦(T g
r1gr2gr3 · T gr+1
1 gr+12 gr+1
3 · T gr+21 gr+2
2 gr+23
)=(T g
r1gr2gr3 · T gr+1
1 gr+12 gr+1
3 · T gr+21 gr+2
2 gr+23
)
Or3 ◦(T g
r1gr2gr3 · T gr+1
1 gr+12 gr+1
3
)=(T g
r1gr2gr3 · T gr+1
1 gr+12 gr+1
3
).
(2.12)
We prove in App. A.2 and A.3 that Eq. (2.12) is also a necessary condition for
Eq. (2.4). A graphical representation of these equations is given in Fig. 2.2.
We now construct a solution of T matrices from Eq. (2.12). It is difficult to solve
Eq. (2.12) directly, as it is a set of nonlinear equations of the T matrices. In the
following, we will derive a new set of equations equivalent to Eq. (2.12), which are
linear in the T matrices and only contain the matrices in the r-th unit cell.
18
The idea to derive the equations linear in the T matrices is to decompose the
Hamiltonian terms Orα into separate parts, where each part acts only on one unit cell,
and then derive their action on a single T matrix. (Notice that we are only allowed
to cut in between the unit cells, not inside one unit cell.) As a first step, we start by
cutting the operator Orα into two parts: Or1 and Or3 can be cut into Or1 = Lr1,1Rr1,1
and Or3 = Lr3,1Rr3,1, such that the operators
Lr1,1 = Ir1 ⊗ Zr2 ⊗ Zr
3
Rr1,1 = Xr+1
1 ⊗ Zr+12 ⊗ Zr+1
3
(2.13)
and
Lr3,1 = Zr1 ⊗ Zr
2 ⊗Xr3
Rr3,1 = Zr+1
1 ⊗ Zr+12 ⊗ Ir+1
3
(2.14)
only act on a given unit cell. The second subscript τ of Lrα,τ labels the position of
bipartition of Orα. Since Or1 and Or3 are supported on two unit cells, there is only
one way to cut them into two parts and hence τ only takes one value, i.e., τ = 1.
Such a unique bipartition is not possible for Or2 since Or2 is supported on 3 unit cells.
Nevertheless, we can define two bipartitions as follows:
Or2 = Lr2,1Rr2,1, Or2 = Lr2,2Rr
2,2, (2.15)
where Lr2,τ ,Rr2,τ (τ = 1, 2) are
Lr2,1 = Ir1 ⊗ Ir2 ⊗ Zr3
Rr2,1 = Zr+1
1 ⊗Xr+12 ⊗ Zr+1
3 ⊗ Zr+21 ⊗ Ir+2
2 ⊗ Ir+23
Lr2,2 = Ir1 ⊗ Ir2 ⊗ Zr3 ⊗ Zr+1
1 ⊗Xr+12 ⊗ Zr+1
3
Rr2,2 = Zr+2
1 ⊗ Ir+22 ⊗ Ir+2
3 .
(2.16)
19
Figure 2.3: A graphical representation of Eqs. (2.17), (2.18) and (2.19).
Notice that Rr2,1 and Lr2,2 have a support over two unit cells while Rr
2,2 and Lr2,1 have
a support over a single unit cell.
Let us consider the action of Lrα,τ ’s and Rrα,τ ’s on the T matrices. First we focus
on Or1. The product of two matrices T gr1gr2gr3 · T gr+1
1 gr+12 gr+1
3 should be invariant under
the combined action of Lr1,1Rr1,1, where Lr1,1 acts only on T g
r1gr2gr3 while Rr
1,1 only on
T gr+11 gr+1
2 gr+13 . In App. A.4, we prove in a general setting of stabilizer codes that the
following condition is both necessary and sufficient of Eq. (2.12): the action of Lr1,1 on
T gr1gr2gr3 can be encoded by a transformation U r
1,1 on the right virtual index of T gr1gr2gr3 ,
while the action of Rr1,1 on T g
r+11 gr+1
2 gr+13 can be encoded by the inverse of the same
transformation (U r1,1)−1 on the left virtual index of T g
r+11 gr+1
2 gr+13 . Concretely, we have
20
Lr1,1 ◦ T gr1gr2gr3 = T g
r1gr2gr3 · U r
1,1
Rr1,1 ◦ T g
r+11 gr+1
2 gr+13 = (U r
1,1)−1 · T gr+11 gr+1
2 gr+13 ,
(2.17)
where ◦ represents the action on the physical indices (see Eqs. (2.10) and (2.11)), and
· represents the matrix multiplication (i.e., the contraction over the virtual indices).
See Fig. 2.3. Similarly, for Lr2,1,Rr2,1 and Lr2,2,Rr
2,2,
Lr2,1 ◦ T gr1gr2gr3 = T g
r1gr2gr3 · U r
2,1 (2.18a)
Rr2,1 ◦ (T g
r+11 gr+1
2 gr+13 · T gr+2
1 gr+22 gr+2
3 )
= (U r2,1)−1 · (T gr+1
1 gr+12 gr+1
3 · T gr+21 gr+2
2 gr+23 ) (2.18b)
Lr2,2 ◦ (T gr1gr2gr3 · T gr+1
1 gr+12 gr+1
3 )
= (T gr1gr2gr3 · T gr+1
1 gr+12 gr+1
3 ) · U r2,2 (2.18c)
Rr2,2 ◦ T g
r+21 gr+2
2 gr+23
= (U r2,2)−1 · T gr+2
1 gr+22 gr+2
3 . (2.18d)
For Lr3,1,Rr3,1, we get
Lr3,1 ◦ T gr1gr2gr3 = T g
r1gr2gr3 · U r
3,1
Rr3,1 ◦ T g
r+11 gr+1
2 gr+13 = (U r
3,1)−1 · T gr+11 gr+1
2 gr+13 .
(2.19)
Eqs. (2.17), (2.18) and (2.19) are graphically represented in Fig. 2.3.
Let us use the translational invariance and focus on the operators that act on the
virtual indices between the r-th and (r+1)-th unit cells, i.e., U r1,1, U
r2,1, U
r−12,2 and U r
3,1.
For Eqs. (2.17), (2.18) and (2.19) being consistent, the virtual U r′
α′,τ ′ and (U r′
α′,τ ′)−1
operators on the right hand side (RHS) should satisfy the same commutation relations
as the physical Lr′α′,τ ′ and Rr′
α′,τ ′ operators on the left hand side (LHS) respectively.
21
As a result, Lr′α′,τ ′ and Rr′
α′,τ ′ share the same commutation relation.1 Here,
Lr′α′,τ ′ ∈ {Lr1,1,Lr2,1,Lr−12,2 ,Lr3,1},
Rr′
α′,τ ′ ∈ {Rr1,1,Rr
2,1,Rr−12,2 ,Rr
3,1}
U r′
α′,τ ′ ∈ {U r1,1, U
r2,1, U
r−12,2 , U
r3,1}
(2.20)
This statement is proved in App. A.5 in a general setting of stabilizer codes. The
commutation relations can be encoded using the compact notations:
Lr′α′,τ ′Lr′′
α′′,τ ′′ = (−1)tr′,r′′
(α′τ ′),(α′′τ ′′)Lr′′α′′,τ ′′Lr′
α′,τ ′ ,
Rr′
α′,τ ′Rr′′
α′′,τ ′′ = (−1)tr′,r′′
(α′τ ′),(α′′τ ′′)Rr′′
α′′,τ ′′Rr′
α′,τ ′ ,
U r′
α′,τ ′Ur′′
α′′,τ ′′ = (−1)tr′,r′′
(α′τ ′),(α′′τ ′′)U r′′
α′′,τ ′′Ur′
α′,τ ′ ,
(2.21)
Since R operators obey the same commutation relations as the L’s, we just focus on
the L operators. The coefficients tr′,r′′
(α′τ ′),(α′′τ ′′) form an anti-symmetric t matrix, which
under the basis (Lr1,1,Lr2,1,Lr−12,2 ,Lr3,1)T is given by2
t =
0 0 1 1
0 0 0 1
−1 0 0 0
−1 −1 0 0
. (2.22)
We first determine the dimension of irreducible representation of the algebra
Eq. (2.21) that {Lr1,1,Lr2,1,Lr−12,2 ,Lr3,1} and {U r
1,1, Ur2,1, U
r−12,2 , U
r3,1} obey. It is conve-
1This is because the commutation relations of L’s and those of the R’s are both related to thecommutation relations of U ’s.
2We only focus on the commutation relations between the virtual U operators which act onthe same virtual bond, e.g. {Ur1,1, Ur2,1, Ur−1
2,2 , Ur3,1}. According to Eq. (2.21), they have the same
commutation relations as {Lr1,1,Lr2,1,Lr−12,2 ,Lr3,1}. These commutation relations constrain the rep-
resentation of the virtual U operators. The commutation relations between {Lr1,1,Lr2,1,Lr−12,2 ,Lr3,1}
and other L operators, e.g. Lr+11,1 do not constrain the representation of the virtual U operators,
thus we do not consider them.
22
nient to introduce:
L1 = Lr3,1, U1 = U r3,1,
L2 = Lr2,1, U2 = U r2,1
L3 = Lr−12,2 , U3 = U r−1
2,2
L4 = Lr1,1Lr2,1, U4 = U r1,1U
r2,1.
(2.23)
The new operators {Lα, Uα, α = 1, 2, 3, 4} satisfy a simpler algebra,
LαLβ = (−1)tαβ LβLα
UαUβ = (−1)tαβ UβUα,
(2.24)
with
t =
0 1 0 0
−1 0 0 0
0 0 0 1
0 0 −1 0
. (2.25)
The algebra of {Lα} (or {Uα}) is decoupled into two subalgebras, generated by
{L1, L2} (or {U1, U2}) and {L3, L4} (or {U3, U4}) respectively. Each subalgebra has
a two dimensional irreducible representation. Hence the total dimension of the ir-
reducible representation of {Lα} (or {Uα}) is 2 × 2 = 4. Finally, noticing that
the transformation between L’s and L’s is invertible. The inverse transformation of
Eq. (2.23) is
Lr3,1 = L1, Ur3,1 = U1,
Lr2,1 = L2, Ur2,1 = U2
Lr−12,2 = L3, U
r−12,2 = U3
Lr1,1 = L4(L2)−1, U r1,1 = U4(U2)−1
(2.26)
23
Since L’s form a irreducible representation, the physical operators Lr+11,1 ,Lr+1
2,1 ,Lr2,2and Lr3,1 (and thus by Eq. (2.21), U r+1
1,1 , Ur+12,1 , U
r2,2 and U r
3,1 as well) also provide a 4
dimensional irreducible representation.
For the rest of the task, we need to first find a matrix representation of
U r1,1, U
r2,1, U
r−12,2 and U r
3,1 satisfying the algebra Eq. (2.21), and then solve for T gr1gr2gr3
in Eqs. (2.17), (2.18) and (2.19). Since the irreducible representation of the algebra
is 4-dimensional, the virtual U operators should be 4× 4 matrices.
The matrix representations for the U operators are not unique. If U r′
α′,τ ′ satisfies
the algebra Eq. (2.21) and T gr′1 g
r′2 g
r′3 is the solution of Eqs. (2.17), (2.18) and (2.19),
then S · U r′
α′,τ ′ · S−1, with S independent of r′, α′, τ ′, also satisfies Eq. (2.21) and the
corresponding solution for the T matrices is given by S ·T gr′
1 gr′2 g
r′3 ·S−1. Hence without
loss of generality, let us choose the virtual U operators {U r′
α′,τ ′} to be:3
U r1,1 = (X ⊗X)r,
U r2,1 = (I ⊗X)r,
U r−12,2 = ((−Y )⊗ I)r.
U r3,1 = (I ⊗ (−Y ))r.
(2.27)
where the superscript r on the RHS indicates that the operators acts on the virtual
bonds connecting the the r-th and r + 1-th unit cell. We denote the corresponding
MPS matrix elements as Tgr1g
r2gr3
h1h2,h3h4where h1, h2 ∈ {0, 1} represent the left virtual
indices and h3, h4 ∈ {0, 1} represent the right virtual indices. In Eq. (2.27), the first
Pauli matrices act on the virtual indices h1 and h3, while the second Pauli matrices
act on the virtual indices h2 and h4.
So far, we have only considered bipartition of the Hamiltonian terms Orα. For
the operators that have support over three or more unit cells, we can take the com-
3We choose these U operators in order to compare with the MPS matrices derived from the RBMstates in Sec. 3.4.
24
binations of L and R operators so that Orα can be decomposed into a product of
operators which only act within a single unit cell. We enumerate the decompositions
for all three types of operators:
Or1,1 = Lr1,1 · Rr1,1
Or2 = Lr2,1 ·(Rr
2,1(Rr2,2)−1
)· Rr
2,2
= Lr2,1 ·((Lr2,1)−1Lr2,2
)· Rr
2,2
Or3,1 = Lr3,1 · Rr3,1
(2.28)
where all the terms on the RHS of Eq. (2.28), in particular(Rr
2,1(Rr2,2)−1
)and
((Lr2,1)−1Lr2,2
), are supported within one unit cell. In App. A.6, we show in a general
setting of stabilizer codes, that Eq. (2.18) is equivalent to the following equations
linear to the T matrix in the r-th unit cell:
Lr1,1 ◦ T gr1gr2gr3 = T g
r1gr2gr3 · U r
1,1
Rr−11,1 ◦ T g
r1gr2gr3 = (U r−1
1,1 )−1 · T gr1gr2gr3
Lr2,1 ◦ T gr1gr2gr3 = T g
r1gr2gr3 · U r
2,1
((Lr−12,1 )−1Lr−1
2,2 ) ◦ T gr1gr2gr3 = (U r−12,1 )−1 · T gr1gr2gr3 · U r−1
2,2
(Rr−12,1 (Rr−1
2,2 )−1) ◦ T gr1gr2gr3 = (U r−12,1 )−1 · T gr1gr2gr3 · U r−1
2,2
Rr−22,2 ◦ T g
r1gr2gr3 = (U r−2
2,2 )−1 · T gr1gr2gr3
Lr3,1 ◦ T gr1gr2gr3 = T g
r1gr2gr3 · U r
3,1
Rr−13,1 ◦ T g
r1gr2gr3 = (U r−1
3,1 )−1 · T gr1gr2gr3 ,
(2.29)
25
where by translational invariance
U r1,1 = (X ⊗X)r, U r−1
1,1 = (X ⊗X)r−1,
U r2,1 = (I ⊗X)r, U r−1
2,1 = (I ⊗X)r−1,
U r−12,2 = ((−Y )⊗ I)r, U r−2
2,2 = ((−Y )⊗ I)r−1,
U r3,1 = (I ⊗ (−Y ))r, U r−1
3,1 = (I ⊗ (−Y ))r−1.
(2.30)
More concretely, the equations in (2.29) are
Tgr1g
r2gr3
h1h2,h3h4(−1)g
r2+gr3 = T
gr1gr2gr3
h1h2,(1−h3)(1−h4)
T(1−gr1)gr2g
r3
h1h2,h3h4(−1)g
r2+gr3 = T
gr1gr2gr3
(1−h1)(1−h2),h3h4
Tgr1g
r2gr3
h1h2,h3h4(−1)g
r3 = T
gr1gr2gr3
h1h2,h3(1−h4)
Tgr1(1−gr2)gr3h1h2,h3h4
(−1)gr1+gr3 = −iT g
r1gr2gr3
h1(1−h2),(1−h3)h4(−1)1−h3
Tgr1(1−gr2)gr3h1h2,h3h4
(−1)gr1+gr3 = −iT g
r1gr2gr3
h1(1−h2),(1−h3)h4(−1)1−h3
Tgr1g
r2gr3
h1h2,h3h4(−1)g
r1 = −iT g
r1gr2gr3
(1−h1)h2,h3h4(−1)h1
Tgr1g
r2(1−gr3)
h1h2,h3h4(−1)g
r1+gr2 = −iT g
r1gr2gr3
h1h2,h3(1−h4)(−1)1−h4
Tgr1g
r2gr3
h1h2,h3h4(−1)g
r1+gr2 = −iT g
r1gr2gr3
h1(1−h2),h3h4(−1)h2 .
(2.31)
Thus we have derived a set of linear equations Eq. (2.31) of the T -matrices from the
non-linear ones Eq. (2.12). Solving Eq. (2.31), we obtain a solution up to a total scale
26
factor:
T 000 =
1 1 1 1
i i i i
i i i i
−1 −1 −1 −1
, T 001 =
i −i i −i
−1 1 −1 1
−1 1 −1 1
−i i −i i
,
T 010 =
−1 −1 1 1
i i −i −i
−i −i i i
−1 −1 1 1
, T 011 =
i −i −i i
1 −1 −1 1
−1 1 1 −1
i −i −i i
,
T 100 =
−1 −1 −1 −1
i i i i
i i i i
1 1 1 1
, T 101 =
i −i i −i
1 −1 1 −1
1 −1 1 −1
−i i −i i
,
T 110 =
1 1 −1 −1
i i −i −i
−i −i i i
1 1 −1 −1
, T 111 =
i −i −i i
−1 1 1 −1
1 −1 −1 1
i −i −i i
.
(2.32)
2.3 General Stabilizer Code Convention
We will generalize the method in Sec. 2.2 to a generic 1d translation invariant stabilizer
code. A generic translational invariant stabilizer code is described by the Hamiltonian
H = −L−1∑
r=0
t∑
α=1
Orα (2.33)
where t is the total number of types of interactions and α ∈ 1, . . . , t labels the type.
Each unit cell contains q spin-12’s. Orα is a product of Pauli X and Z operators such
27
that (1) Orα is Hermitian and (Orα)2 = 1 for any r, α; and (2) Orα and Or′α′ commute
for any r, r′, α, α′, i.e.,
[Orα,Or′
α′ ] = 0, ∀ α, α′, r, r′. (2.34)
Each interaction term Orα is supported over the unit cells r, r + 1, ..., r + Pα − 1, and
can be written as an ordered product of Pα number of local operators orα,τ
Orα =Pα∏
τ=1
orα,τ , (2.35)
where orα,τ is a product of Pauli matrices only supported on the (r + τ − 1)-th unit
cell. For convenience, we define Lrα,τ and Rrα,τ as follows
Lrα,τ =τ∏
µ=1
orα,µ,
Rrα,τ =
Pα∏
µ=τ+1
orα,µ, τ = 1, . . . , Pα − 1.
(2.36)
By Assumption 2.0.1, we consider only cases where Eq. (2.33) has a unique ground
state when using PBC. The ground state |GS〉 is the common eigenstate of all Orα’s
with eigenvalue 1 for all r and α,
Orα|GS〉 = |GS〉, ∀ r, α. (2.37)
Our goal is to express the ground state |GS〉 as an MPS
|GS〉 =∑
{gri }
Tr
( L−1∏
r=0
T gr1 ...g
rq
)|{gri }〉. (2.38)
where grα, (α = 1, ..., q), labels the value of the α-th physical spin in the r-th unit cell.
28
2.4 General Algorithm to Construct MPS
The calculation algorithm to construct an MPS representation is divided into in 4
steps. These 4 steps will follow those of Sec. 2.2 for the ZZXZZ model.
1. We start with the stabilizer condition Eq. (2.37). A sufficient condition for the
MPS of Eq. (2.38) to satisfy Eq. (2.37) is
Orα ◦( r+Pα−1∏
r′=r
T gr′1 ...g
r′q
)=( r+Pα−1∏
r′=r
T gr′1 ...g
r′q
). (2.39)
Eq. (2.39) is graphically represented as Fig. 2.4. In fact, Eq. (2.39) is not only
sufficient, but also necessary for Eq. (2.37), as derived in App. A.3.
2. To find a solution of Eq. (2.39), we consider a bipartition of the Hamiltonian
term Orα into the product of the left and right part, i.e., Lrα,τ and Rrα,τ . The two
Figure 2.4: Graphical representation of Eq. (2.39). The shaded purple region repre-sents the operator Orα acting on the physical indices.
29
parts act solely on two disjoint and contiguous sets of unit cells. For Pα > 1, since
Orα is supported on the unit cells between r and r + Pα − 1, Lrα,τ is chosen to be
supported from r to r + τ − 1-th unit cell, and Rrα,τ is supported from (r + τ) to
(r + Pα − 1)-th unit cell. The definitions of Lrα,τ and Rrα,τ are in given Eq. (2.36).
Either Lrα,τ or Rrα,τ can act nontrivially on the MPS, although their product leaves
the MPS invariant. This nontrivial action can be captured by a transformation on
the virtual index exactly across the cut (between the (r+ τ −1)-th and the (r+ τ)-th
unit cell). From Eq. (2.39), we find
Lrα,τ ◦( r+τ−1∏
r′=r
T gr′1 ...g
r′q
)=( r+τ−1∏
r′=r
T gr′1 ...g
r′q
)· U r
α,τ (2.40)
Rrα,τ ◦
( r+Pα−1∏
r′=r+τ
T gr′1 ...g
r′q
)= (U r
α,τ )−1 ·
( r+Pα−1∏
r′=r+τ
T gr′1 ...g
r′q
).
Eq. (2.40) is graphically represented as Fig. 2.5. We prove in App. A.4 that Eq. (2.40)
is both necessary and sufficient for Eq. (2.39).
Figure 2.5: Graphical representation of Eq. (2.40). The virtual operator U ri,τ and
(U ri,τ )−1 act on the right virtual index between the r + τ − 1 and r-th unit cell.
30
For convenience, for each choice of (α, τ) we can shift r to r− τ + 1 in Eq. (2.40)
by translational invariance such that the U r−τ+1α,τ acts on the virtual bond between
the r-th and (r+1)-th unit cell. See Fig. 2.6 for the operators that are obtained from
shifting Orα. Under the shift r → r − τ + 1, Eq. (2.40) becomes
Lr−τ+1α,τ ◦
( r∏
r′=r−τ+1
T gr′1 ...g
r′q
)=( r∏
r′=r−τ+1
T gr′1 ...g
r′q
)· U r−τ+1
α,τ
Rr−τ+1α,τ ◦
( r+Pα−τ∏
r′=r+1
T gr′1 ...g
r′q
)= (U r−τ+1
α,τ )−1 ·( r+Pα−τ∏
r′=r+1
T gr′1 ...g
r′q
),
1 ≤ α ≤ t, 1 ≤ τ ≤ Pα − 1
(2.41)
Figure 2.6: An illustration of the operators Lr−(τ−1)α,τ and Rr−(τ−1)
α,τ with fixed r and α,
and all 1τ ≤ Pα−1. The blue blocks represent unit cells. The purple blocks represent
the operators Lr−(τ−1)α,τ , and the operators Rr−(τ−1)
α,τ .
When the operator Orα is supported only over 1 unit cell, i.e., Pα = 1, Eq. (2.39)
is already linear. Without loss of generality, we take Lrα,1 = Orα, Rrα,1 = Ir and
U rα,1 = Ir where Ir is an identity operator acting on the r-th unit cell.
31
3. We further determine the minimal bond dimension of T gr1 ...g
rq . We prove in
App. A.5 that the commutation and anti-commutation relations of the virtual U op-
erators on RHS of Eq. (2.41) should match those of the physical L and R operators
on the LHS,
Lr−(τ ′−1)α′,τ ′ Lr−(τ ′′−1)
α′′,τ ′′ = (−1)tr−(τ ′−1),r−(τ ′′−1)
(α′τ ′),(α′′τ ′′) Lr−(τ ′′−1)α′′,τ ′′ Lr−(τ ′−1)
α′,τ ′ ,
Rr−(τ ′−1)α′,τ ′ Rr−(τ ′′−1)
α′′,τ ′′ = (−1)tr−(τ ′−1),r−(τ ′′−1)
(α′τ ′),(α′′τ ′′) Rr−(τ ′′−1)α′′,τ ′′ Rr−(τ ′−1)
α′,τ ′ ,
Ur−(τ ′−1)α′,τ ′ U
r−(τ ′′−1)α′′,τ ′′ = (−1)
tr−(τ ′−1),r−(τ ′′−1)
(α′τ ′),(α′′τ ′′) Ur−(τ ′′−1)α′′,τ ′′ U
r−(τ ′−1)α′,τ ′ ,
1 ≤ α′, α′′ ≤ t, 1 ≤ τ ′ ≤ Pα′ − 1, 1 ≤ τ ′′ ≤ Pα′′ − 1
(2.42)
The parameter
tr−(τ ′−1),r−(τ ′′−1)(α′τ ′),(α′′τ ′′) = 0, 1 mod 2 (2.43)
encodes whether Ur−(τ ′−1)α′,τ ′ and U
r−(τ ′′−1)α′′,τ ′′ commute or anti-commute. We ensemble
the parameters tr−(τ ′−1),r−(τ ′′−1)(α′τ ′),(α′′τ ′′) into an anti-symmetric matrix t. 4
The algebra Eq. (2.42) is a generalization of the Clifford algebra, where the
standard Clifford algebra is generated by mutually anti-commuting operators. In
Ref. [110], it was shown that any integer-valued antisymmetric matrix t can be block
diagonalized by a unimodular integer matrix V , such that each nontrivial block is a
2 × 2 anti-symmetric matrix with integer off-diagonal elements. Due to Eq. (2.43),
only the modulo 2 values of the off-diagonal elements of the nontrivial 2 × 2 blocks
4In general, if the set of operators {Uλ} satisfy Uλ′Uλ′′ = eitλ′,λ′′Uλ′Uλ′′ , where tλ′,λ′′ is a realnumber characterizing the commutation relations between {Uλ}, tλ′,λ′′ is antisymmetric: tλ′,λ′′ =−tλ′′,λ′ . This is because from the above commutation relation, one move the phase to the left handside as e−itλ′,λ′′Uλ′Uλ′′ = Uλ′Uλ′′ . Combining with the definition Uλ′′Uλ′ = eitλ′′,λ′Uλ′′Uλ′ , wederive that e−itλ′,λ′′ = eitλ′′,λ′ . This yields that tλ′′,λ′ = −tλ′,λ′′ mod 2π. Applying to our case,
tr−(τ ′−1),r−(τ ′′−1)(α′τ ′),(α′′τ ′′) = −tr−(τ ′′−1),r−(τ ′−1)
(α′′τ ′′),(α′τ ′) mod 2.
32
matter. The nontrivial blocks can therefore be written as follows:5
V tV T =
0 1
−1 0
⊕
0 1
−1 0
· · · ⊕
0 1
−1 0
⊕ 0 · · · . (2.44)
Here we explicitly keep the minus signs to make the antisymmetry manifest. In the
new basis, the operators Lrα,τ become decoupled pairs of anti-commuting operators
(such as Eq. (2.23)); there are rank(t)2
such pairs. Since each pair provides a two
dimensional irreducible representation, the dimension of the irreducible representation
of the generalized Clifford algebra Eq. (2.42) is given by
D = 2rank(t)
2 . (2.45)
Since the dimension of an irreducible representation of the algebra Eq. (2.42) is D, the
matrices of the U rα,τ operators, as well as the MPS matrix T g
r1 ...g
rq under the irreducible
representation should be D×D matrices.6 Since the representation is irreducible, D
is also the minimal bond dimension. For the ZZXZZ model with 3 spins per unit
cell discussed in Sec. 2.2, the t matrix is given by Eq. (2.22), which is of rank 4. By
Eq. (2.45), the minimal bond dimension of the MPS is D = 24/2 = 4, which matches
the MPS explicitly derived in Eq. (2.32).
4. We solve Eq. (2.41) for the MPS matrices T with the minimal bond dimension D.
Let us first determine the form of U . The matrix elements of U can be obtained by
finding the representation of the algebra Eq. (2.42). Here we focus only on irreducible
5When there is an operator Lr−τ+1α,τ commuting with all other Lr−τ ′+1
α′,τ ′ for any (τ ′, α′), the t-matrix is not full rank. For instance, when there is an operator Orα which is only supported overone unit cell, i.e., Pα = 1, then by our convention in the previous paragraph, Lrα,1 = Orα,Rrα,1 = 1.
Then Lrα,1 commutes with Lr−τ ′+1α′,τ ′ for any (τ ′, α′). Hence there are 0 blocks in the decomposition
Eq. (2.44), i.e., t matrix is not full rank.6One may consider Urα,τ = 0 (for all α, τ and r) to be a solution of Eq. (2.42). However, due to
Eq. (2.41), the T gr1 ...g
rq would be zero, hence the MPS is a null state. So we do not consider this
solution.
33
representations such that the bond dimension is minimal. Notice that there exist
multiple choices of U operators satisfying the same algebra Eq. (2.42). However,
since we only consider models with a single ground state, different solutions of T
from different choices of U should correspond to the same ground state. Hence it
is sufficient to work with one choice of U . As shown in App. A.7, U can always
be constructed as a tensor product of rank(t)2
Pauli matrices. After specifying the
virtual U operators, we manipulate the equations in Eq. (2.41) such that all the
physical operators on the LHS only act on the r-th unit cell, and all the equations are
linear in T gr1 ···grq . For instance, using the definition Eq. (2.36), Lr−τ+1
α,τ =∏τ
µ=1 or−τ+1α,µ
and Lr−τ+1α,τ−1 =
∏τ−1µ=1 o
r−τ+1α,µ , the combination
((Lr−τ+1
α,τ−1 )−1Lr−τ+1α,τ
)= or−τ+1
α,τ is only
supported on the r-th unit cell. In App. A.6, we show that Eq. (2.41) are equivalent
to
Lrα,1 ◦ T gr1 ...g
rq = T g
r1 ...g
rq · U r
α,1
((Lr−τ+1
α,τ−1 )−1Lr−τ+1α,τ
)◦ T gr1 ···grq = (U r−τ+1
α,τ−1 )−1 · T gr1 ···grq · U r−τ+1α,τ
(Rr−τ+1α,τ−1 (Rr−τ+1
α,τ )−1)◦ T gr1 ...grq = (U r−τ+1
α,τ−1 )−1 · T gr1 ...grq · U r−τ+1α,τ
Rr−(Pα−1)α,Pα−1 ◦ T gr1 ...grq = (U
r−(Pα−1)α,Pα−1 )−1 · T gr1 ...grq ,
1 ≤ α ≤ t, 2 ≤ τ ≤ Pα − 1
(2.46)
Since Eq. (2.46) is a set of linear equations in T , they can be numerically solved
efficiently. For all the models we have explicitly checked (e.g. Zq−1XZq−1 with
2 ≤ q ≤ 6), Eq. (2.46) has one non-zero solution up to an overall scaling.
34
Chapter 3
Restricted Boltamann Machine
State for Stabilizer Code in 1D
In this chapter, we develop a criteria judging when the ground state of a 1D stabilizer
code can be exactly written as a RBM state. Once the criteria is satisfied, we find
the RBM state with the minimal number of hidden spins.
3.1 (Restricted) Boltzmann Machine
In this section, we introduce the notion of Boltzmann machine (BM) states, restricted
Boltzmann machine (RBM) states and their connection to MPS.
3.1.1 Definitions
A BM state is a state defined by a classical Ising model on a graph. Each vertex of
the graph carries a classical Ising spin sr = 0, 1 where r is the index of the vertex.
Each edge of the graph carries a weight Wrr′ ∈ C that mimics the Ising “interaction”
between sr and sr′, and each vertex also carries a bias αr ∈ C that mimics “an
35
external magnetic field”. The “energy” for such an Ising model is:
EBM({sr}) =∑
r,r′
Wrr′srsr′+∑
r
αrsr, (3.1)
where the summation runs over all spins. In turn, a BM can be efficiently represented
by a graph: (1) the vertices of the graph represent the spins {sr}; (2) the nonzero
weight of sr and sr′
is represented by the link connecting sr and sr′. The set of spins
is divided into two disjoint subsets: the visible spins whose set is denoted by V and
the hidden spins denoted by H. We denote gr the visible spins and hs the hidden
spins. Using these notations, the BM state is defined as:
|BM〉 = C∑
{gr}r∈V
∑
{hs}s∈H
exp
(− EBM({hs}, {gr})
)|{gr}〉, (3.2)
where C is a normalization constant that we will drop for simplicity. The states
|{gr}〉 are the basis states over the visible spins, i.e., a given |{gr}〉 is the direct
product of Pauli Z eigenstates with eigenvalues {(−1)gr}. The “energy” terms in
EBM({hs}, {gr}) can be split into
EBM({hs}, {gr}) =∑
r,r′∈V
Rrr′grgr
′+∑
s,s′∈H
Sss′hshs
′+∑
r∈Vs∈H
Wrsgrhs +
∑
r∈V
βrgr +
∑
s∈H
αshs,
(3.3)
where Wrs, Rrr′ , Sss′ ∈ C are the weights between visible and hidden, visible and
visible, hidden and hidden spins respectively. βr ∈ C is the bias of the visible spin
gr, and αs ∈ C is the bias of the hidden spin hs.
A restricted Boltzmann machine (RBM) state is a special BM state satisfying
Rrr′ = 0, ∀ r, r′ ∈ V ; Sss′ = 0, ∀ s, s′ ∈ H. (3.4)
36
Thus an RBM state reads
|RBM〉 =∑
{gr}r∈V
∑
{hs}s∈H
exp
(− ERBM({hs}, {gr})
)|{gr}〉
(3.5)
with
ERBM({hs}, {gr}) =∑
r∈Vs∈H
Wrsgrhs +
∑
r∈V
βrgr +
∑
s∈H
αshs. (3.6)
In this article, we will consider RBM states for 1D translational invariant systems.
For this reason, we use r, s to label the unit cells, and i, a to label the visible spin
and hidden spins within a unit cell (which are dubbed “orbitals”) respectively. We
further require the RBM to be finitely connected, and by properly enlarging the unit
cell, we can always choose the RBM to be nearest unit cell connected. Due to the
requirement of translational invariance and nearest neighbor connectivity, we label
the visible spins, the hidden spins, the weights and the biases as follows:
1. The visible spins within the unit cell at r are labeled by gri where i = 1, . . . , q
labels the orbitals within the unit cell. q is the number of visible spins within
each unit cell.
2. The hidden spins are divided into two categories:
(a) hra, a ∈ {1, . . . ,M}, labels the hidden spins connecting to the visible spins
from the unit cell at r−1 and those from the unit cell at r, i.e., hra connects
to both {gr−1i } and {gri }. M is the total number of such hidden spins within
the unit cell. Since we assume that the RBM is nearest unit cell connected,
hra does not connect to the visible spins of another unit cell. We will dub
such hidden spins as type-h hidden spins.
(b) hrb, b ∈ {1, . . . , M}, labels the hidden spins connecting to the visible spins
within the unit cell at r, i.e., hrb only connects to {gri }. M is the total
37
number of such hidden spins within the unit cell. We will dub such hidden
spins as type-h hidden spins.
3. The weight connecting hra and gri is labeled by Aia, i ∈ {1, . . . , q}, a ∈
{1, . . . ,M}.
4. The weight connecting hra and gr−1i is labeled by Bia, i ∈ {1, . . . , q}, a ∈
{1, . . . ,M}.
5. The weight connecting hrb and gri is labeled by Cib, i ∈ {1, . . . , q}, b ∈
{1, . . . , M}.
6. The bias of the visible spin gri is βi, i ∈ {1, . . . , q}.
7. The bias of the hidden spin hra is αa, a ∈ {1, . . . ,M}.
8. The bias of the hidden spin hrb is αb, b ∈ {1, . . . , M}.
Due to translational invariance, the weights Aia, Bia, Cib and the biases βi, αa and
αb are all independent of the position of the unit cell r. We have distinguished the
hidden spins into type-h and type-h because, as will be explained in Sec. 3.1.2, the
hidden spins of type-h contribute to the entanglement, while those of type-h do not.
Correspondingly, we distinguish the weights Aia which connect the visible spin gri to
the hidden spins of type-h, i.e., hra, and Cib which connect the visible spin gri to the
hidden spins of type-h, i.e., hrb. In Fig. 3.1, we show an example of such an RBM
state with q = 3,M = 2 and M = 2. The visible spins (i.e., gri ) are represented by
red circles. The hidden spins connecting to the visible spins from the neighboring
unit cells (i.e., hra) are represented by the rectangles and the hidden spins connecting
to the visible spins from a single unit cell (i.e., hrb) are represented by triangles.
38
Figure 3.1: An example of RBM state corresponding to q = 3,M = 2, M = 2. The redcircles represent visible spins. The black rectangles are the hidden spins connectingvisible spin belonging to different unit cells, which are linked to the purple and orangelines representing the weights Aia and Bia respectively. The black triangles are thehidden spins connecting visible spins within the same unit cell, which are linked tothe green lines representing the weights Cib. The blue region represents a unit cell.Notice that the nonzero weights are only between the hidden spins and the visiblespins.
With the notations introduced above, a translational invariant and nearest neigh-
bor connected RBM state is
|RBM〉 =∑
{gri }
∑
{hra,hrb}
exp
(− ERBM({hra, hrb}, {gri })
)|{gri }〉, (3.7)
with
ERBM =∑
r
q∑
i=1
M∑
a=1
(Aiagri h
ra +Biag
ri h
r+1a ) +
M∑
b=1
Cibgri h
rb
+∑
r
q∑
i=1
βigri +
M∑
a=1
αahra +
M∑
b=1
αbhrb
.
39
3.1.2 Relation to MPS
The RBM state defined by Eq. (3.7) can be cast into an MPS by mapping the hidden
spins of the RBM to the virtual indices of the MPS. We name such MPS an RBM-
MPS. Specifically, Eq. (3.7) can be rewritten as
|RBM〉 =∑
{gri }
Tr
(∏
r
T gr1 ...g
rq
)|{gri }〉, (3.8)
where
Tgr1 ...g
rq
hr1...hrM ,h
r+11 ...hr+1
M
= e−∑q,Mi,a=1(Aiagri hra+Biag
ri hr+1a )−
∑qi=1 βig
ri−
∑Ma=1 αah
ra
∑
{hrb}
e−∑q,Mi,b=1 Cibg
ri hrb−
∑Mb=1 αbh
rb .
(3.9)
The bond dimension of the RBM-MPS is determined by the number of type-h hidden
spins, i.e., M . Hence only the hidden spins of type-h contribute to the entanglement,
while those of type-h do not. The optimal M will be determined in Sec. 2. For
instance, if each hra ∈ {0, 1} is Z2 valued, the bond dimension is 2M . The tensor T
satisfies two useful properties:
Theorem 3.1.1. (a) T gr1 ...g
rq in Eq. (3.9) is either strictly zero or all its matrix el-
ements are non-vanishing. (b) If T gr1 ...g
rq is non-vanishing, it is of rank 1. If T g
r1 ...g
rq
vanishes, it is of rank 0.
Proof. To prove (a), we notice that each matrix element of T gr1 ...g
rq is a com-
mon multiplicative factor∑{hrb}
e−∑q,Mi,b=1 Cibg
ri hrb−
∑Mb=1 αbh
rb independent of the hid-
den spins {hra, hr+1a } for all a, times a strictly nonzero expression of {hra, hr+1
a }:
e−∑q,Mi,a=1(Aiagri hra+Biag
ri hr+1a )−
∑qi=1 βig
ri−
∑Ma=1 αah
ra . If the common multiplicative factor is
zero then T gr1 ...g
rq vanishes. If the common multiplicative factor is nonzero, all matrix
elements are non-vanishing.
40
To prove (b), we observe that, when the matrix elements of T gr1 ...g
rq are non-
vanishing, the ratio
Tgr1 ...g
rq
hr1...hrM ,h
r+11 ...hr+1
M
Tgr1 ...g
rq
hr1...hrM ,h
′r+11 ...h′r+1
M
(3.10)
is independent of hr1 . . . hrM , for any hr+1
1 . . . hr+1M and h′r+1
1 . . . h′r+1M . Hence any two
rows of the matrix T gr1 ...g
rN are proportional to each other, and thus the matrix is of
rank 1. When T gr1 ...g
rq vanishes, by definition, it is of rank 0.
Since the non-vanishing matrices of the RBM-MPS are of rank 1, it is natural to
ask if the reverse statement also holds true, i.e., whether an MPS can be expressed
as an RBM-MPS if the non-vanishing MPS matrices are of rank 1. In the rest of this
article, we study this problem in the context of stabilizer codes. We conjecture that
if the non-vanishing MPS matrices of the ground state of a translational invariant
stabilizer code are of rank 1, such a ground state can also be found as an RBM state.
In Sec. 2, we first determine the condition for the non-vanishing MPS matrices of a
stabilizer code to be of rank 1. In Sec. 3.4, we give an algorithm to generate the RBM
state for a large class of models (the cocycle models) whose MPS matrices are of rank
1.
3.2 More on ZZXZZ Model
In section 2.2, we derived the MPS for the ZZXZZ model, as shown in Eq. (2.32).
These matrices are of rank 1 and all the tensor elements are nonzero. We emphasize
that they match the two properties (a) and (b) in Theorem 3.1.1, and this match
depends on the proper choice of the matrices for U operators. Indeed, if there is a U
operator containing Pauli Z matrix, for instance U r1,1 = X ⊗X,U r
2,1 = X ⊗ I, U r−12,2 =
I⊗Z,U r3,1 = Z⊗I, then the MPS matrix elements can have both zeros and nonzeros.
The appearance of zero matrix elements makes it difficult to match the MPS to the
41
RBM element-wise, because the matrix elements of RBM-MPS are all non-vanishing
as shown in Theorem 3.1.1.
In fact, we do not have to solve the matrices and then find their ranks. We
can immediately find the rank of the matrices from the Hamiltonian terms. From
Eq. (2.31) used only for T 000,
T 000h1h2,h3h4
= T 000h1h2,(1−h3)(1−h4)
T 000h1h2,h3h4
= T 000h1h2,h3(1−h4)
T 000h1h2,h3h4
= −iT 000(1−h1)h2,h3h4
(−1)h1
T 000h1h2,h3h4
= −iT 000h1(1−h2),h3h4
(−1)h2 .
(3.11)
The physical indices are unchanged on both sides of Eq. (3.11) simply because the
four equations are coming from acting with the physical operators on the LHS of
Eq. (3.11),
Lr1,1 = Ir1 ⊗ Zr2 ⊗ Zr
3 ,
Lr2,1 = Ir1 ⊗ Ir2 ⊗ Zr3 ,
Rr2,2 = Zr
1 ⊗ Ir2 ⊗ Ir3 ,
Rr3,1 = Zr
1 ⊗ Zr2 ⊗ Ir3 ,
(3.12)
which contain only Pauli Z operators and identities. Hence, the left indices h1 and h2
of the matrix T 000 obey two independent constraints, and the right indices h3 and h4
obey two independent constraints. Therefore, the rank of the matrix T 000 can be at
most 1, since each constraint for the left (or right) indices eliminates half of the total
rank. Acting with Eq. (3.12) on the T -matrices with other physical indices, we also
get two independent constraints on the left and right indices respectively. Hence the
T matrices with any physical indices are of rank 1. For other general models obeying
the assumptions (2.0.1), (2.0.2) and (2.0.3), we can similarly find the constraints on
42
the rank of the matrices by counting the independent L or R operators with only
Pauli Z matrices without solving explicitly the matrices by brute-force. We elaborate
this idea in Sec. 3.3.
We finally comment that for the ZZXZZ model with one spin per unit cell, i.e.,
H1−siteZZXZZ =
r−1∑
i=0
Zr−2Zr−1XrZr+1Zr+2 (3.13)
Using the same calculation in this section, the ground state of H1−siteZZXZZ can be ex-
pressed as an MPS,
|GS〉1−site =∑
{gr}
Tr
( L−1∏
r=0
T gr
)|{gr}〉 (3.14)
where the MPS matrices are
T 0 =
1 0 1 0
−i 0 −i 0
0 1 0 1
0 −i 0 −i
, T 1 =
0 −1 0 1
0 −i 0 i
−1 0 1 0
−i 0 i 0
(3.15)
Notice that both T 0 and T 1 are of rank 2. By theorem 3.1.1, it is impossible to
express the MPS Eq. (3.14) as an RBM state.
3.3 An Inequality for Rank of MPS
As discussed in Sec. 3.1, a necessary condition for the existence of a finitely connected
RBM of a stabilizer code ground state is Theorem 3.1.1. In this section, we propose an
inequality which allows us to directly constrain the rank of the MPS without solving
for the MPS matrices.
43
Before we state and prove our theorem, it is convenient to introduce two notations.
Denote a set of operators:
L =
{Lr1,1,Lr2,1, . . . ,Lrt,1
}. (3.16)
In particular, L contains a special subset dubbed as LZ such that the operators in
LZ are only the tensors products of Pauli Z and the identity I matrices. Denote
NLZ as the number of independent operators in LZ . Notice that due to translational
invariance, NLZ is independent of r.
Theorem 3.3.1. For the matrices T gr1 ...g
rq satisfying Eq. (2.46) where the U matrices
are tensor product of Pauli matrices, the rank of T gr1 ...g
rq is upper bounded:
rank(T gr1 ...g
rq ) ≤ D
2NLZ
= 2rank(t)
2−NLZ , ∀{gri }. (3.17)
where NLZ is the number of independent operators in LZ.
Proof. To constrain rank(T gr1 ...g
rq ), we only focus on a subset of Eq. (2.46) satisfying:
(1) the physical operator on LHS of Eq. (2.46) only involves the operators in LZ ;
(2) the virtual operator on RHS of Eq. (2.46) only acts on the right virtual index.
Explicitly, this subset of equations are all included in the following equations:
Lrα,1 ◦ T gr1 ...g
rq = T g
r1 ...g
rq · U r
α,1, ∀ Lrα,1 ∈ LZ . (3.18)
This subset is useful to constrain rank(T gr1 ...g
rq ) because
(1) both LHS and RHS of this subset of equations only involve the same matrix
T gr1 ...g
rq . Indeed, since Lrα,1 belongs to LZ , the LHS is proportional to the matrix
T gr1 ...g
rq ;
44
(2) only the columns of T gr1 ...g
rq are constrained.
Using Theorem A.6.2 of App. A.6, the number of independent equations among
Eq. (3.18)(i.e., the number of independent constraints for the columns ) is given
by the number of independent operators in LZ , i.e., NLZ . We know that U operators
form a generalized Clifford algebra, and as proven in App. A.7, their matrices are
tensor products of the Pauli matrices. More precisely, each virtual U operator either
swaps and/or multiplies by some factors (±i or ±1) on half of the columns. Hence,
each independent constraint eliminates half of the rank. Therefore, the rank of the
MPS T matrix is upper bounded:
rank(T gr1 ...g
rq ) ≤ D
2NLZ
. (3.19)
This completes proving Theorem 3.3.1.
In the 1D stabilizer codes we have studied, the upper bound in Eq. (3.17) always
saturates.
3.4 Restricted Boltzmann Machine State of a Sta-
bilizer Code
In this section, we discuss how to express the ground states of a class of stabilizer
codes, which we dub as cocycle models, as RBM states. They are a special class
of Hamiltonians describing 1D symmetry protected topological phases. We first use
Theorem 3.3.1 to prove that the rank of the ground state MPS is 1. Then we use the
ZZXZZ model as an example to illustrate the construction of the RBM state with
the RBM-MPS bond dimension 4. We further present a general and explicit algorithm
to construct the RBM states for an arbitrary cocycle model, with the minimal RBM-
MPS bond dimension. We finally conjecture that for any stabilizer code which satisfies
45
Assumptions 2.0.1, 2.0.2 and 2.0.3 and also the necessary condition 3.1.1, it is possible
to express its ground state as an RBM state with the minimal RBM-MPS bond
dimension.
3.4.1 MPS Matrix Rank For Cocycle SPT Models
In this section, we apply Theorem 3.3.1 to a particular family of stabilizer codes —
the cocycle Hamiltonians for symmetry protected topological phases — and show that
their MPS matrices are of rank 1. In App. B.1, we provide some backgrounds about
the cocycle Hamiltonians, including the projective representations of the global sym-
metry G, cocycles ω2, cohomology group H2(G,U(1)) and 1D SPT phases. The cocy-
cle ω2 ∈ H2(G,U(1)) classifies the 1D SPT phases with the discrete onsite symmetry
G. In this paper, we restrict G to be (Z2)q. The group elements are parametrized
by g = (g1, g2, . . . , gq) with gi ∈ Z2 = {0, 1}, and the generic form of the cocycle is
[111, 112]:
ω2(g, g′) = exp
(−iπ
∑
1≤i<j≤q
Pijgjg′i
), g, g′ ∈ G, (3.20)
where Pij can be either 0 or 1. The cocycles can also be used to construct representa-
tive SPT wave functions and representative parent Hamiltonians which are stabilizer
codes. For simplicity, we dub the representative states and representative Hamiltoni-
ans as cocycle states and cocycle Hamiltonians respectively. See App. B.1 for a brief
overview.
The cocycle Hamiltonian for a (Z2)q SPT phase (with q spin-12’s per unit cell)
with a given generic cocycle ω2 Eq. (3.20) is
H(Z2)q ,ω2 = −L−1∑
r=0
q∑
α=1
Orα, (3.21)
46
with
Orα =
∏
1<l≤q
(Zr+1l Zr
l )P1lXr+1
1 α = 1
∏α<l≤q(Z
r+1l Zr
l )PαlXr+1
α
∏1≤k<α(Zr+2
k Zr+1k )Pkα 1 < α < q
Xrq
∏
1≤k<q
(Zr+1k Zr
k)Pkq α = q.
(3.22)
For 1 < α < q, Orα are supported on 3 unit cells; while for α = 1, q, Or1 and Orqare supported on 2 unit cells. The Hamiltonian H(Z2)q ,ω2 has the ground state (see
App. B.1 for details)
|GS〉(Z2)q ,ω2 =∑
{gri }
exp
(iπ
L−1∑
r=0
∑
1≤i<j≤q
Pij(grj − gr−1
j )gri
)|{gri }〉. (3.23)
When
P =
0 1 1
0 0 1
0 0 0
, (3.24)
the Hamiltonian Eq. (3.21) reduces to the Hamiltonian of the ZZXZZ model, i.e.,
Eq. (2.3).
Theorem 3.4.1. For the stabilizer codes of Eq. (3.21), if T gr1 ...g
rq is not null, then
rank(T gr1 ...g
rq ) = 1. (3.25)
Proof. To calculate rank(T gr1 ...g
rq ), we apply Theorem 3.3.1, where the upper bound
of the rank of T gr1 ···grq is given by 2
rank(t)2−NLZ . We will first compute rank(t) and NLZ
respectively, and show that the upper bound is 1. We further show that the upper
bound is saturated, which completes the proof of the theorem.
47
We first compute rank(t). To calculate the t-matrix, we enumerate all possible
Lr−τ+1α,τ with all possible (α, τ) and fixed r. For 1 < α < q, τ = 1, 2; for α = 1 or q,
τ = 1. Hence there are 2(q − 1) L operators:
Lr1,1 ≡ (Zr2)P12 ⊗ (Zr
3)P13 ⊗ · · · ⊗ (Zrq−1)P1(q−1) ⊗ (Zr
q )P1q
...
Lrq−2,1 ≡ (Zrq−1)P(q−2)(q−1) ⊗ (Zr
q )P(q−2)q
Lrq−1,1 ≡ (Zrq )P(q−1)q
Lr−12,2 ≡ (Zr−1
3 )P23 ⊗ · · · ⊗ (Zr−1q )P2q ⊗ (Zr
1)P12 ⊗Xr2 ⊗ (Zr
3)P23 ⊗ (Zr4)P24 ⊗ · · · ⊗ (Zr
q )P2q
Lr−13,2 ≡ (Zr−1
4 )P34 ⊗ · · · ⊗ (Zr−1q )P3q ⊗ (Zr
1)P13 ⊗ (Zr2)P23 ⊗Xr
3 ⊗ (Zr4)P34 ⊗ (Zr
5)P35 ⊗ · · · ⊗ (Zrq )P3q
...
Lr−1q−1,2 ≡ (Zr−1
q )P(q−1)q ⊗ (Zr1)P1(q−1) ⊗ (Zr
2)P2(q−1) ⊗ · · · ⊗ (Zrq−2)P(q−2)(q−1) ⊗Xr
q−1 ⊗ (Zrq )P(q−1)q
Lrq,1 ≡ (Zr1)P1q ⊗ (Zr
2)P2q ⊗ · · · ⊗ (Zrq−1)P(q−1)q ⊗Xr
q .
(3.26)
We have suppressed the identity operators for simplicity. Among all the op-
erators in Eq. (3.26), the first q − 1 and the last one act only on the r-th
unit cell, while the remaining act both on the r − 1-th and r-th unit cells.
It is straightforward to compute the commutation relation and determine the
t matrix. In the basis where the operators are listed as in Eq. (3.26), i.e.,
{Lr1,1, · · · ,Lrq−2,1,Lrq−1,1,Lr−12,2 ,Lr−1
3,2 , · · · ,Lr−1q−1,2,Lrq,1}, the t matrix reads
t =
0 Λ
−ΛT 0
, (3.27)
48
where 0 is a (q − 1) × (q − 1) dimensional zero matrix, and Λ is a (q − 1) × (q − 1)
upper triangular matrix:
Λ =
P12 · · · P1(q−1) P1q
. . ....
...
P(q−2)(q−1) P(q−2)q
P(q−1)q
. (3.28)
Therefore, by Eq. (3.27), we have:
rank(t) = 2rank(Λ). (3.29)
Counting rank(Λ) is simply counting the number of independent rows in Λ.
We proceed to evaluate NLZ . Recall that NLZ is defined to be the number of
independent operators among LZ . In this case, we have:
LZ = {Lr1,1,Lr2,1, . . . ,Lrq−1,1}. (3.30)
A crucial observation is that the powers of the Z’s among the operators in Eq. (3.30)
are in one-to-one correspondence with the rows of the Λ matrix in Eq. (3.28). Hence,
the number of independent operators among Eq. (3.30) coincides with the number of
independent rows of the Λ matrix Eq. (3.28), i.e.,
NLZ = rank(Λ). (3.31)
Using Theorem 3.3.1 and Eqs. (3.29) and (3.31), we obtain
rank(T g1...gq) ≤ 2rank(t)
2−NLZ = 2
2rank(Λ)2
−rank(Λ) = 1. (3.32)
49
We have assumed that T g1...gq is not null. rank(T g1...gq) is thus assumed to be positive.
Constrained by Eq. (3.32), we conclude that
rank(T g1...gq) = 1. (3.33)
Since in the ground state Eq. (3.23) for any spin configuration {gri } the coefficient
of the basis |{gri }〉 is a non-vanishing number, the MPS matrices are non-vanishing
for any physical indices gr1 . . . grq . This shows that the matrices T g
r1 ...g
rq are indeed not
null. Hence the MPS matrix rank is 1 for the ground state MPS of an arbitrary
cocycle Hamiltonian in Eq. (3.21) with the global symmetry (Z2)q.
3.4.2 An Example: ZZXZZ Model Revisited
In this section, we derive the RBM for the ZZXZZ model with the RBM-MPS bond
dimension 4.
We start with the ground state |GS〉 of the ZZXZZ model Eq. (2.5). Concretely,
by restricting Eq. (3.23) to q = 3, and using P12 = P23 = P13 = 1, we obtain the
ground state
|GS〉ZZXZZ =∑
{gri }
exp
(iπ
L−1∑
r=0
∑
1≤i<j≤3
(grj − gr−1j )gri
)|{gri }〉. (3.34)
The coefficient of the configuration |{gri }〉 is an exponent of a quadratic function of
the physical spins. The idea to write Eq. (3.34) in the form of an RBM state is to
introduce hidden spins and to transform the quadratic terms in g to linear terms.
This is achieved by applying a series of identities proved in App. B.2. The identities
50
can be summarized as
exp
(iπSym(g1, · · · , gn)
)
=1√2
1∑
h=0
exp
(iπ
2(1− 2h)
n∑
i=1
gi − iπ
4(1− 2h)
),
(3.35)
where gi ∈ {0, 1}, and Sym(g1, · · · , gn) is a symmetric summation of quadratic ex-
pressions in gi, i.e.,
Sym(g1, · · · , gn) ≡∑
1≤i<j≤n
gjgi. (3.36)
We introduce the following definitions to simplify the discussion below:
1. The on-site terms : the quadratic terms involving only the visible spins from a
single unit cell. For example: grjgri , g
r−1j gr−1
i , etc.
2. The inter-site terms : the quadratic terms involving the visible spins from dif-
ferent unit cells. For example: gr−1j gri , g
rjgr−1i , etc.
3. The on-site symmetric expressions : the symmetric expressions involving only
visible spins from a single unit cell. For example: Sym(gri , grj , g
rk), etc.
4. The inter-site symmetric expressions : the symmetric expressions involving vis-
ible spins from different unit cells. For example: Sym(gr−1i , grj , g
rk), etc.
To convert Eq. (3.34) into an RBM state, our strategy is as follows. We group all
the quadratic terms in the exponent of Eq. (3.34) into a sum of symmetric expressions,
and apply the identity Eq. (3.35) to each symmetric expression. For the inter-site
symmetric expression, applying Eq. (3.35) introduces a hidden spin of type-h; for the
on-site symmetric expression, applying Eq. (3.35) introduces a hidden spin of type-
h. As discussed in Sec. 3.1, each hidden spin of type-h doubles the bond dimension
once we write the RBM state as an MPS (i.e., RBM-MPS), while the hidden spin of
type-h does not contribute to the bond dimension. Hence, to obtain the RBM state
51
whose RBM-MPS bond dimension is as small as possible, we are aiming to group
the quadratic expressions in Eq. (3.34) to as few inter-site symmetric expressions as
possible, together with some additional on-site symmetric expressions.
We first discuss the inter-site terms in Eq. (3.34), i.e.,∑
1≤i<j≤3 gr−1j gri , because
on-site terms do not contribute to the inter-site symmetric expressions. There are
different ways to decompose the inter-site terms in the exponent of Eq. (3.34) as
a summation of symmetric expressions. Superficially, there are 3 inter-site terms,∑
1≤i<j≤3 gr−1j gri = gr−1
2 gr1 + gr−13 gr1 + gr−1
3 gr2, and it seems that one has to introduce 3
hidden variables by applying Eq. (3.35) to the three terms separately. However, it is
possible to organize the three inter-site terms into the sum of two inter-site symmetric
expressions and one on-site symmetric expression. Concretely,
∑
1≤i<j≤3
gr−1j gri =
Sym(gr−12 , gr1) + Sym(gr−1
3 , gr1, gr2)− Sym(gr1, g
r2).
(3.37)
Under the decomposition Eq. (3.37) and applying Eq. (3.35), we need to introduce
2 hidden spins of type-h, which we denote as hr1 and hr2. From the discussion in
the last paragraph, the bond dimension of the RBM-MPS is 22 = 4, which precisely
matches the minimal bond dimension of the ZZXZZ model derived in Sec. 2.2. This
shows that there is no way to decompose the quadratic expression∑
1≤i<j≤3 gr−1j gri
in Eq. (3.34) as a sum of at most one inter-site symmetric expression, together
with some additional on-site symmetric expressions. Different decompositions of∑
1≤i<j≤3 gr−1j gri should include at least two inter-site symmetric expressions. We
will provide a general recipe of grouping the inter-site terms in Sec. 3.4.3 for all the
1D cocycle models and show that the grouping is optimal.
52
We further consider the on-site terms∑
1≤i<j≤3 grjgri . We use the same decompo-
sition as Eq. (3.37) by replacing gr−1j with grj , and obtain
∑
1≤i<j≤3
grjgri = Sym(gr2, g
r1) + Sym(gr3, g
r1, g
r2)− Sym(gr1, g
r2). (3.38)
Applying Eq. (3.35) for all the symmetric expressions, the ground state |GS〉ZZXZZcan be rewritten as an RBM state
|GS〉ZZXZZ
=∑
{gri }
exp
(iπ
L−1∑
r=0
−Sym(gr−12 , gr1)− Sym(gr−1
3 , gr1, gr2) + Sym(gr2, g
r1) + Sym(gr3, g
r1, g
r2)
)|{gri }〉
=∑
{gri }
∑
{hr1,hr2}{hr1,hr2}
L−1∏
r=0
exp
(− iπ
2(1− 2hr1)(gr−1
2 + gr1) + iπ
4(1− 2hr1)
− iπ2
(1− 2hr2)(gr−13 + gr1 + gr2) + i
π
4(1− 2hr2) + i
π
2(1− 2hr1)(gr1 + gr2)
− iπ4
(1− 2hr1) + iπ
2(1− 2hr2)(gr1 + gr2 + gr3)− iπ
4(1− 2hr2)
)|{gri }〉
(3.39)
We have suppressed the overall normalization constant. From the discussion in
Sec. 3.1, the RBM state Eq. (3.39) can further be written as an MPS with the MPS
matrix:
Tgr1g
r2gr3
hr1hr2,h
r+11 hr+1
2
=
∑
{hr1,hr2}
exp
(− iπ
2(1− 2hr1)gr1 − i
π
2(1− 2hr+1
1 )gr2 + iπ
4(1− 2hr1)− iπ
2(1− 2hr2)(gr1 + gr2)
− iπ2
(1− 2hr+12 )gr3 + i
π
4(1− 2hr2) + i
π
2(1− 2hr1)(gr1 + gr2)− iπ
4(1− 2hr1)
+ iπ
2(1− 2hr2)(gr1 + gr2 + gr3)− iπ
4(1− 2hr2)
).
(3.40)
53
The bond dimension of the RBM-MPS Eq. (3.40) is indeed 4, which matches the
bond dimension derived from the RBM state Eq. (3.39). Since we have shown in
Sec. 2.2 that the minimal bond dimension of the ZZXZZ MPS is 4, there can not
be an RBM state with the number of hidden spin of type-h per unit cell less than 2.
This implies that our RBM state is the most optimal, in the sense that the number
of hidden spins of type-h is minimal.
Fig. 3.2 is a graphical representation of the RBM state Eq. (3.39). In fact, the
RBM-MPS matrices Eq. (3.40) are the same as the MPS matrices Eq. (2.32) in derived
in Sec. 2.2. As we will see in the next subsection, for more general models Zq−1XZq−1,
each unit cell contains q visible spins. Our construction yields the RBM-MPS bond
dimension 2q−1, and we need to introduce 2(q − 1) hidden spins on average for each
unit cell. Among them, (q − 1) are of the type-h while the remaining (q − 1) are of
the type-h.
3.4.3 RBM States of Cocycle Hamiltonians
In Sec. 3.4.1, we have shown that the MPS matrices of the (Z2)q cocycle Hamiltonians
(with q spin-12’s per unit cell) are all of rank 1. Then it is natural to ask if the ground
state of the cocycle Hamiltonians can always be expressed as an RBM state, whose
Figure 3.2: Graphical representation of the RBM state of the ZZXZZ model.
54
RBM-MPS bond dimension being D defined in Eq. (2.45). In this subsection, we
describe a procedure to obtain the RBM states with minimal number of hidden spins.
In particular, we generalize and apply the procedures of Sec. 3.4.2, and we present
explicit RBM states for Zq−1XZq−1 cocycle Hamiltonians with arbitrary q.
The cocycle Hamiltonian in Eq. (3.21) has the ground state |GS〉(Z2)q ,ω2 in
Eq. (3.23). To convert it to an RBM state, we follow the same procedures in Sec. 3.4.2.
The core idea is that we need to group the inter-site terms∑
1≤i<j≤q Pijgr−1j gri as a
sum of the rank(Λ) inter-site symmetric expressions together with some on-site terms.
Since each inter-site symmetric expression contributes a hidden spin of type-h which
doubles the bond dimension of the RBM-MPS, the bond dimension of the RBM-MPS
is thus 2rank(Λ) ≡ 2rank(t)
2 . This is precisely the minimal bond dimension derived in
Sec. (2.4), which in turn implies that the decomposition of the inter-site terms is
optimal, i.e, the number of type-h hidden spins is minimal in our construction.
Lemma 3.4.2. For an inter-site quadratic term,
(gr−1
)T · Γ · gr =
q∑
i,j=1
Γijgr−1i grj , Γij ∈ {0, 1}, (3.41)
there exists a unimodular transformation G such that Γ transforms to
Γ→ Γ = (G)T · Γ ·G =
γ
0
mod 2, (3.42)
where the integer matrix γ of size rank(Γ)× q has full row rank:
rank(γ) = rank(Γ). (3.43)
55
The vectors gr−1 and gr transform as
gr−1i → gr−1
i =
q∑
j=1
G−1ij g
r−1j , gri → gri =
q∑
j=1
G−1ij g
rj . (3.44)
and
(gr−1
)T · Γ · gr =(gr−1
)T · Γ · gr (3.45)
Proof. Our proof is based on the Gaussian elimination algorithm. For simplicity, we
first introduce the matrix notations: I represents the identity q×q matrix, and E(i, j)
represents a q × q matrix whose elements are
(E(i, j))m,n = δm,iδn,j, ∀ m,n = 1, 2, . . . , q. (3.46)
In other words, the only nonzero value of E(i, j) is 1 located at the i-th row and j-th
column. Moreover, we use the following two types of matrix row transformations:
G1(i, j) = I + E(i, j) + E(j, i)− E(i, i)− E(j, j)
G2(i, j) = I + E(j, i), i 6= j.
(3.47)
It is obvious that both G1 and G2 are unimodular, i.e.,
| det(G1(i, j))| = 1, | det(G2(i, j))| = 1. (3.48)
The products of G1’s and G2’s are also unimodular.
The first transformation G1(i, j) interchanges the i-th row and the j-th row of
Γ, and the second transformation G2(i, j) adds the i-th row to the j-th row.1 There
1Notice that the matrix determinant det(G2(i, i)) = det(I +E(i, i)) = 0. Since we only consideruni-modular transformation, we do not allow i = j in G2(i, j).
56
exists a sequence of G1(i, j) and G2(i, j) such that:
∏
m
Gkm(im, jm) · Γ =
γ′
0
mod 2, (3.49)
where the matrix γ′ of size rank(Γ)× q has full row rank, and its elements are either
0 and 1. Denote:
G =
(∏
m
Gkm(im, jm)
)T
. (3.50)
Using Eq. (3.49), we have:
Γ = GT · Γ ·G =
γ′
0
·G =
γ
0
mod 2, (3.51)
where
γ = γ′ ·G, (3.52)
and γ of size rank(Γ)× q has full row rank.
Lemma 3.4.3. The inter-site term in the ground state |GS〉(Z2)q ,ω2 Eq. (3.23)∑
1≤i<j≤q Pijgr−1j gri can be grouped into rank(Λ) number of inter-site symmetric
expressions and rank(Λ) on-site symmetric expressions, where Λ is defined in
Eq. (3.28).
Proof. We first define the Γ matrix:
Γ ≡
0 0 · · · 0 0
P12 0 · · · 0 0
P13 P23 · · · 0 0
......
. . ....
...
P1q P2q · · · P(q−1)q 0
=
0 0 · · · 0 0
0
0
ΛT ...
0
. (3.53)
57
The matrix Γ is a q × q matrix, whose each element is defined modulo 2. There are
0s in the first row and last column because gr−11 and grq do not appear in the sum
∑1≤i<j≤q Pijg
r−1j gri . The bottom-left (q − 1) × (q − 1) block of Γ is ΛT where Λ is
defined in Eq. (3.28). In particular,
rank(Γ) = rank(Λ). (3.54)
Using this notation, we have:
∑
1≤i<j≤q
Pijgr−1j gri = (gr−1)T · Γ · gr. (3.55)
Using Lemma 3.4.2, Eq. (3.55) can be simplified:
∑
1≤i<j≤q
Pijgr−1j gri =
rank(Λ)∑
i=1
gr−1i
q∑
j=1
Γij grj . (3.56)
It can be decomposed by the symmetric expressions:
∑
1≤i<j≤q
Pijgr−1j gri =
rank(Λ)∑
i=1
Sym(gr−1i , Γi1g
r1, . . . , Γiqg
rq)−
rank(Λ)∑
i=1
Sym(Γi1gr1, . . . , Γiqg
rq).
(3.57)
The first rank(Λ) terms are inter-site symmetric expressions, and the remaining
rank(Λ) terms are the on-site terms. This completes the proof.
Theorem 3.4.4. There exists an RBM for the state Eq. (3.23) whose RBM-MPS has
the minimal bond dimension 2rank(Λ) where Λ is defined in Eq. (3.28).
58
Proof. Using Lemma 3.4.2 and 3.4.3, we obtain
exp
(− iπ
∑
1≤i<j≤q
Pijgr−1j gri
)= exp
(− iπ(gr−1)T · Γ · gr
)
= exp
(− iπ
rank(Λ)∑
i=1
Sym(gr−1i , Γi1g
r1, . . . , Γiqg
rq) + iπ
rank(Λ)∑
i=1
Sym(Γi1gr1, . . . , Γiqg
rq)
).
(3.58)
Applying Eq. (3.35) to the inter-site symmetric expressions leads to:
exp
(− iπ
∑
1≤i<j≤q
Pijgr−1j gri
)
=
rank(Λ)∏
i=1
[1√2
1∑
hri=0
exp
(− iπ
2(1− 2hri )(g
r−1i +
q∑
j=1
Γij grj ) + i
π
4(1− 2hri )
)
× exp
(− iπSym(Γi1g
r1, · · · , Γiqgrq)
)].
(3.59)
Notice that further introducing the hidden spins by linearizing the on-site terms on
RHS of Eq. (3.58) does not increase the bond dimension of the RBM-MPS. Hence we
have shown that the RBM-MPS derived via the above algorithm has rank(Λ) hidden
spins of type h, which corresponds to the RBM-MPS bond dimension D = 2rank(Γ) =
2rank(Λ). This matches the bond dimension Eq. (2.45) associated with the irreducible
representation in Sec. 2.
We use the rest of this section to express the state Eq. (3.23) as an RBM explicitly.
exp
(iπ
∑
1≤i<j≤q
Pij(grj − gr−1
j )gri
)
= exp
(− iπ
rank(Λ)∑
i=1
Sym(gr−1i , Γi1g
r1, . . . , Γiqg
rq) + iπ
rank(Λ)∑
i=1
Sym(gri , Γi1gr1, . . . , Γiqg
rq)
).
(3.60)
59
Applying Eq. (3.35) to Eq. (3.60), we can write the ground state |GS〉(Z2)q ,ω2 as an
RBM state
|GS〉(Z2)q ,ω2
=∑
{gri },{hri },{hri }
rank(Λ)∏
i=1
exp
(− iπ
2(1− 2hri )(g
r−1i +
q∑
j=1
Γij grj )
+ iπ
4(1− 2hri ) + i
π
2(1− 2hri )(g
ri +
q∑
j=1
Γij grj )− i
π
4(1− 2hri )
)|{gri }〉.
(3.61)
We find that in the particular construction Eq. (3.61), the number of inter-site hid-
den spin is the same as the number of on-site hidden spin, for an arbitrary cocycle
Hamiltonian. The relation between {gri } and {gri } depends on the cocycle parameters
Pij, as per Eq. (3.44).
3.4.4 RBM Construction for Zq−1XZq−1 Model
To exemplify our RBM construction, we apply the above algorithm to the stabilizer
code Zq−1XZq−1 for an arbitrary cocycle. Another example is discussed in App. B.3.
The Zq−1XZq−1 model corresponds to the cocycle Hamiltonian with Pij = 1 for any
1 ≤ i < j ≤ q.
The Hamiltonian of the Zq−1XZq−1 model is
HZq−1XZq−1 =−L−1∑
r=0
(q−1∏
i=1
ZriX
rq
q−1∏
i=1
Zr+1i +
q∏
i=2
ZriX
r+11
q∏
i=2
Zr+1i
+
q∑
s=3
( q∏
i=s
Zri
s−2∏
j=1
Zr+1j Xr+1
s−1
q∏
k=s
Zr+1k
s−2∏
l=1
Zr+2l
) ).
(3.62)
Its ground state is
|GS〉Zq−1XZq−1 =∑
{gri }
L−1∏
r=0
exp
(iπ
∑
1≤j<i≤q
(gri − gr−1i )grj
)|{gri }〉. (3.63)
60
The the q × q Γ matrix and the (q − 1)× (q − 1) Λ matrix are
Γ =
0 0 · · · 0
1 0
1 1 0
......
. . ....
1 1 · · · 1 0
, Λ =
1 1 · · · 1
1 · · · 1
. . ....
1
. (3.64)
To transform the Γ matrix to the form as in Eq. (3.42), we switch the rows using
GT = G1(q − 1, q) · · ·G1(1, 2). (3.65)
The visible spins transform as
gr1
gr2...
grq−1
grq
→
gr1
gr2...
grq−1
grq
= G−1 ·
gr1
gr2...
grq−1
grq
=
gr2
gr3...
grq
gr1
. (3.66)
The Γ matrix transforms as
Γ→ Γ = GT · Γ ·G =
1 0
1 1 0
......
. . ....
1 1 · · · 1 0
0 0 · · · 0 0
. (3.67)
61
All the q − 1 rows in the top (q − 1)× q block of Γ are independent,
rank(Γ) = rank(Γ) = rank(Λ) = q − 1. (3.68)
As a result, the exponents in Eq. (3.63) can be written as
∑
1≤j<i≤q
(gri − gr−1i )grj =−
q−1∑
i=1
Sym(gr−1i+1 , g
ri , g
ri−1, . . . , g
r1)
+
q−1∑
i=1
Sym(gri+1, gri , g
ri−1, . . . , g
r1).
(3.69)
On RHS of the equality, the first q−1 symmetric functions are inter-site terms. Using
Eq. (3.35) we introduce q − 1 hidden spins of type-h contributing to 2rank(Γ) = 2q−1
bond dimension of the RBM-MPS. The remaining q − 1 symmetric functions only
contain on-site quadratic terms. Using Eq. (3.35), we introduce q− 1 hidden spins of
type-h. Combining these two operations, we have:
|GS〉Zq−1XZq−1
=∑
{gri }
∑
{hr1}...{hrq−1}
L−1∏
r=0
exp
(− iπ
2
q−1∑
i=1
(1− 2hri )(gr−1i+1 +
i∑
j=1
grj ) + iπ
4
q−1∑
i=1
(1− 2hri )
)
×∑
{hri }
exp
(iπ
2
q−1∑
i=1
(1− 2hri )i+1∑
j=1
grj − iπ
4
q−1∑
i=1
(1− 2hri )
)|{gri }〉.
(3.70)
62
Figure 3.3: Graphical representation of the RBM state of the ZXZ model. The redcircles represent visible spins, the black rectangles represent the hidden spins con-necting visible spin belonging to different unit cells, and the black triangles representthe hidden spins connecting visible spins within the same unit cell.
This RBM can be casted into an rank-1 MPS, and the matrix elements of the RBM-
MPS are:
Tgr1 ,...,g
rq
hr1...hrq−1,h
r+11 ...hr+1
q−1
= exp
(− iπ
2
q−1∑
i=1
(1− 2hri )(i∑
j=1
grj )− iπ
2
q−1∑
i=1
(1− 2hr+1i )gri+1 + i
π
4
q−1∑
i=1
(1− 2hri )
)
×∑
{hri }
exp
(iπ
2
q−1∑
i=1
(1− 2hri )i+1∑
j=1
grj − iπ
4
q−1∑
i=1
(1− 2hri )
).
(3.71)
We discuss two particular cases. When q = 2, the model corresponds to the ZXZ
model. A graphical representation of the ZXZ model is shown in Fig. 3.3. We notice
that the corresponding RBM-MPS has bond dimension 2. In the RBM derived in
Ref. [113], the corresponding bond dimension is 4, which is not minimal. When q = 3
which corresponds to the ZZXZZ model, we find that the RBM-MPS matrices in
Eq. (3.71) precisely agrees with the MPS matrices in Eq. (2.32).
In summary, we have shown that for cocycle Hamiltonians, the ground state can
be expressed as an RBM state with the minimal RBM-MPS bond dimension. We
63
further conjecture, that for an arbitrary translational invariant stabilizer code with
non-degenerate ground state with PBC, if its ground state MPS matrix is of rank
1, then it is possible to express its ground state as an RBM state with the minimal
RBM-MPS bond dimension matching Eq. (2.45). We leave the proof of this conjecture
for future work.
64
Chapter 4
Tensor Network States,
Entanglement Entropy of CSS
Stabilizer Codes and Fracton
Models in 3D
4.1 Stabilizer Code Tensor Network States
In this section, we provide an overview of the stabilizer codes and the tensor network
state description of their ground states. In this article, we focus on a few ”main”
stabilizer codes in three dimensions : the toric code[114] and the Haah code[71].
(In [47], we also discussed another stabilizer code: the X-cube model[115]. ) The
TNS for these models have similarities in their derivation and they share several (but
importantly not all!) common features. Both aspects are presented in this section.
4.1.1 Notations
We first fix the notations, to which we will refer throughout the chapter:
65
1. We introduce a g tensor, which denotes the projector from a physical index to
virtual indices. g tensors are essentially the same (up to the number of indices)
for all stabilizer codes. g tensors have two virtual indices and one physical index
for the 3D toric code model and the X-cube model, while g tensors for Haah
code have 4 virtual indices and 1 physical index. They are depicted in Eq. (4.4),
(4.52) and (4.53).
2. We introduce the T tensor, which denotes the local tensor for each model. It
has only virtual indices and no physical index. The specific tensor elements
differ for different models.
3. Since we consider mostly models on cubic lattices, the indices of T tensors will
be denoted as x, x, y, y, z and z in the 3 directions (forward and backward)
respectively. The indices will be collectively denoted using curly brackets. For
instance, the physical indices are collectively denoted as {s}, while the virtual
indices are denoted as {t}. The virtual indices which are not contracted over
are called “open indices”. Both the physical indices and the virtual indices are
non-negative integer values.
4. Graphically, the physical indices are denoted by arrows, while the virtual indices
are not associated with any arrows.
5. The contraction of a network of tensors over the virtual indices is denoted as
CM ( ) whereM is the spatial manifold that the TNS lives on. The correspond-
ing wave function that arises from the contraction is denoted as |TNS〉M. When
evaluating the TNS wave function norms or any other physical quantities, we
contract over the virtual indices from both the bra and the ket layer. This
contraction is still denoted by CM ( ).
66
U-1U
A1 A2
A1 A2
(a)
(b)
U-1UA1 A2
A1 A2
(c)
(d)
Figure 4.1: TNS gauge in MPS. (a) A part of an MPS. A1 and A2 are two localtensors contracted together. (b) We insert the identity operator I = UU−1 at thevirtual level - it acts on the virtual bonds. The tensor contraction of A1 and A2 doesnot change. (c) We further multiply U with A1 and U−1 with A2, resulting in A1 andA2 respectively in Panel (d). The tensor contraction of A1 and A2 is the same as thetensor contraction of A1 and A2. The TNS wave function does not change as well.Similar TNS gauges also appear in other TNS such as PEPS.
6. Lx, Ly and Lz refer to the system sizes in the three directions (the bound-
ary conditions will be specified), while lx, ly and lz refer to the sizes of the
entanglement cut. Both are measured in units of vertices.
7. TNS gauge is defined as the gauge degrees of freedom of TNS such that the
wave function stays invariant while the local tensors change. One can insert
identity operators I = UU−1 on the virtual bonds, where U is any invertible
matrix acting on the virtual index, multiplying U and U−1 to nearby tensors
respectively. The local tensors then change but the wave function stays invari-
ant. We refer to this gauge degree of freedom as TNS gauge. TNS gauge exists
in MPS, PEPS etc. See Fig. 4.1 for an illustration. In our calculations, we only
fix the tensor elements up to a choice of TNS gauge.
67
4.1.2 CSS Stabilizer Code and TNS Construction
We now summarize the general idea of constructing TNS for stabilizer codes. In the
following, we assume that the physical spins are defined on the bonds of the cubic
lattice (such as the 3D toric code). The cases where the physical spins are defined on
vertices can be analyzed similarly. The generic philosophy of any CSS stabilizer code
model (named after Robert Calderbank, Peter Shor and Andrew Steane) is captured
by the following exactly solvable Hamiltonian:
H = −∑
v
Av −∑
p
Bp (4.1)
where the Hamiltonian is the sum of the Av terms composed of only Pauli Z operators
and the Bp terms composed of only Pauli X operators, and v and p denotes the
positions of the lattice. (There are more general non-CSS stabilizer codes that contain
the terms with both X and Z operators. We considered non-CSS models in 1D (see
chapter 2 and 3), but we will not discuss them in higher dimensions.) In the 3D toric
code, v is the vertex of the cubic lattice, while p is the plaquette. In the Haah code,
both v and p are cubes. See Sec. 4.3.1 and 4.4.1 for the definition of Hamiltonians of
these three models. All these local operators commute with each other:
[Av, Av′ ] = 0, ∀ v, v′
[Bp, Bp′ ] = 0, ∀ p, p′
[Av, Bp] = 0, ∀ v, p
(4.2)
The Hamiltonian eigenstates are the common eigenstates of these local terms indi-
vidually. In particular, any ground state |GS〉 should satisfy:
Av|GS〉 = |GS〉, ∀ v
Bp|GS〉 = |GS〉, ∀ p(4.3)
68
for all positions labeled by v and p.
The ground states for the stabilizer codes with Hamiltonian as in Eq. (4.1) can be
written exactly in terms of TNS. Our construction, when restricted to the 2D toric
code model, is the same as in the literature[116, 117]. In the following, we provide one
possible general construction for such TNS wave functions. We introduce a projector
g tensor with one physical index s and two virtual indices i, j:
gsij =
i j
s=
1 s = i = j
0 otherwise
(4.4)
where the line with an arrow represents the physical index, and the lines without
arrows correspond to the virtual indices. The physical index s = 0, 1 represents
the Z-eigenstates of |↑〉, |↓〉 respectively where Z|↑〉 = |↑〉, and Z|↓〉 = −|↓〉. The
projector g tensor maps the physical spin into the virtual spins exactly. As a result,
the virtual index has a bond dimension 2. 1 When a Pauli operator acts on the
physical index of a projector g tensor, its action transfers to the virtual indices of g.
For instance, a Pauli operator X acting on the physical index of a g tensor amounts
to two Pauli operators X acting on both virtual indices of the same g tensor, and
a Pauli operator Z acting on the physical index of a g tensor amounts to a Pauli
operator Z acting on either virtual index of the same g tensor.
1This construction is limited to the models where the entanglement entropy of the ground statescales as S / Area ln 2. If the entanglement area law is higher than Area ln 2, for example 2Area ln 2,than this construction does not work. However, for the stabilizer codes to be discussed in this chapter,the entanglement entropy of the ground states indeed satisfy the above scaling property.
69
Tg
T
g
(a) (b)
xzy
xzy
Figure 4.2: (a) A plane of TNS on a cubic lattice. (b) TNS on a cube. The lines
with arrows are the physical indices. The connected lines are the contracted virtual
indices, while the open lines are not contracted. On each vertex, there lives a T
tensor, and on each bond, we have a projector g tensor.
To each vertex, we associate a local tensor T which only has virtual indices. To
each bond, we associate a projector g tensor. The TNS is obtained by contracting
the g and T tensors as depicted in Fig. 4.2 (a) and (b). We define the TNS wave
function as:
|TNS〉 =∑
{s}
CR3
(gs1gs2gs3 . . . TTT . . .) |{s}〉 (4.5)
where CR3denotes the contraction over all virtual indices on R3 as illustrated in
Fig. 4.2 (b); |{s}〉 is a wave function basis for spin configurations on the cubic lattice
in Pauli Z basis. The TNS can be put on other spatial manifolds such as T 3 and
T 2 ×R. In our notation, they are denoted by changing CR3to CT 3
and CT 2×R. The
TNS for the ground states satisfies:
Av|TNS〉 = |TNS〉, ∀ v
Bp|TNS〉 = |TNS〉, ∀ p(4.6)
for all positions labeled by v and p.
Since we have projector g tensors contracted with all virtual indices of a T tensor,
the actions of Av and Bp operators on the TNS can be transferred to the virtual
70
indices, using the definition of the g tensor. Then the actions of Av and Bp on the
physical indices will be transferred to actions on the local tensors T . By enforcing
the local tensors T to be invariant under Av and Bp actions, we obtain Eq. (4.6), and
|TNS〉 belongs to the ground state manifold. For the three models analyzed in this
paper, we have found that up to TNS gauge, the elements of the local tensor T can
be reduced to two values, either 1 or 0. The first equation of Eq. (4.6) restricts the
local T tensor to be:
Txxy...
6= 0 if the indices xxy . . . satisfy
some constraints
= 0 otherwise
(4.7)
Applying the second equation of Eq. (4.6) will further restrict the local T tensor to
be:
Txxy... =
1 if the indices xxy . . . satisfy
some constraints
0 otherwise
(4.8)
For simplicity, we calculate the entanglement entropies of the wave function on R3.
We emphasize that in this chapter, we are only concerned about the bulk wave
functions and their entanglement entropies. In principle, the TNS of Eq. (4.5) re-
quires boundary conditions, i.e. the virtual indices at infinity on R3. The boundary
conditions are assumed not to make a difference to the reduced density matrices in
the bulk. (Note that this is true as long as the region considered for the reduced
density matrices does not contain any boundary virtual index.) Hence, we do not
need to specify the boundary conditions for the TNS in the following calculations of
entanglement entropies.
71
4.2 Entanglement properties of the stabilizer code
TNS
The specific structure of the TNS discussed in the previous section allows to derive
its entanglement properties. In this section, we show that for a large class of en-
tanglement cuts the TNS is already in Schmidt form i.e. is exactly a singular value
decomposition.
4.2.1 TNS as an exact SVD
We propose a general sufficient condition that the TNS is SVD with respect to partic-
ular entanglement cuts. Suppose we denote the TNS wave function with open virtual
indices {t} as:
|{t}〉 =∑
{s}
CM (TTT . . . gs1gs2gs3 . . .) |{s}〉 (4.9)
whereM is an open manifold which the TNS lives on, CM stands for the contraction
over the virtual indices inside M, but not over the open ones {t} that straddle the
boundary of M. In Eq. (4.9), the T tensors and g tensors are the tensors inside M
such that the nodes of the local T tensors and the projector g tensors are inside M.
For example, when M is a cube, we have a TNS figure:
|{t}〉 =
{t}
(4.10)
where inside the cube is a network of contracted tensors which are not explicitly
drawn, and the red lines denote the open virtual indices {t}. With this notation of
72
|{t}〉, the TNS wave function can be written as:
|TNS〉 =∑
{t}
|{t}〉A ⊗ |{t}〉A (4.11)
with respect to a region A and its complement A. |{t}〉A is the TNS wave function
in region A with open indices {t}, while |{t}〉A is the TNS wave function in region A
with the same open indices {t} due to tensor contraction. In other words, the TNS
naturally induces a bipartition of the wave functions. However, the two partitions do
not need to each form orthonormal sets.
We now propose a simple sufficient (but not generally necessary) condition to
determine when Eq. (4.11) is an exact SVD for the TNS constructed in this paper.
We first have to make an assumption, satisfied by all our TNS:
Local T tensor assumption: We assume that the indices of the nonzero elements
of the local T tensor are constrained: if all the indices of the element T...t... except for
t are fixed, then there is only one choice of t such that T...t... is nonzero.
We are now ready to express our SVD condition:
SVD condition: If there are no two open virtual indices in {t} (see Eq. (4.10))
of the region A that connect to the same T tensor in the region A, i.e. if every
open virtual index in {t} belongs to different T tensors, then the non-vanishing states
|{t}〉A span an orthogonal basis. Similarly, if there are no two open virtual indices in
{t} of the region A that connect to the same T tensor in the region A, i.e. if every
open virtual index in {t} belong to different T tensors, then the non-vanishing states
|{t}〉A form an orthogonal basis. Therefore, Eq. (4.11) is an exact SVD.
Proof :
We first prove the statement for region A. Suppose that |{t}〉A and |{t′}〉A are
two non-vanishing TNS wave functions in the region A. Any open index in {t}
of the region A must connect to either a projector g tensor or a local tensor T . We
73
discuss the two situations respectively, and examine the overlap of two different states
A〈{t′}|{t}〉A as a function of the two indices configurations {t′} and {t}.
(1) If the open virtual index m in the ket layer (i.e. |{t}〉A) connects to a projector
g tensor, then the open virtual index m′ in the bra layer (i.e. A〈{t′}|), at the same
place as the index m, also connects to a projector g tensor. If we “zoom in” on the
local area of A〈{t′}|{t}〉A near the index m and m′, we have the following diagram:
A
m'
m
A
Entanglement Cut
ket layer
bra layer
(4.12)
By using the projection property of the g-tensor Eq. (4.4), we can conclude that
m = m′, otherwise A〈{t′}|{t}〉A = 0.
(2) If the open virtual index m0 in the ket layer connects to a local T tensor, we
require by the SVD condition that there are no other open virtual indices connecting
to this T tensor. Then the other indices of this T tensor are all inside the region A.
Similarly for the index m′0 in the bra layer. In terms of a diagram, A〈{t′}|{t}〉A near
the area of the index m0 and m′0 can be represented as:
m'
T
T
A
ket layer
bra layer
A
mi
m'i
m0
0
Entanglement Cut
(4.13)
where mi and m′i with i = 1, 2, 3 . . . denote the other virtual indices of the T tensor in
the bra and ket layer respectively, except m0 and m′0. Note that in the ket layer, the
74
virtual indices mi (i = 1, 2, . . .) of the T tensor (all indices except the index m0) are
all connected with contracted projector g tensors inside region A. Correspondingly,
in the bra layer, the virtual indices m′i (i = 1, 2, . . .) are also all connected with the
same contracted projector g tensors. Hence, due to these projector g tensors, all
the indices except m0 of the T tensor in the ket layer are equal to their respective
analogues in the bra layer:
mi = m′i, i = 1, 2, . . . (4.14)
otherwise the overlap would be A〈{t′}|{t}〉A = 0. The only remaining question is
whether the open indices m0 and m′0 should be identified in order to have a non-
vanishing overlap A〈{t′}|{t}〉A.
Using the local T tensor assumption:, mi (i = 1, 2, . . .) will uniquely determine
m0 in order to have nonzero element of the T tensor in the ket layer. Similarly,
m′i (i = 1, 2, . . .) will uniquely determine m′0 in order for the T tensor in the bra layer
to give a nonzero element. Therefore, Eq. (4.14) implies that:
m0 = m′0 (4.15)
such that the overlap A〈{t′}|{t}〉A is nonzero.
Therefore, both situations (1) and (2) lead to the conclusion that the open indices
{t} and {t′} should be identical in order to have a nonzero overlap A〈{t′}|{t}〉A. The
non-vanishing states |{t}〉A are orthogonal basis. A similar proof can be derived for
the region A. The orthogonality of each set |{t}〉A and |{t}〉A implies that Eq. (4.11)
is indeed an SVD. However, the singular values are not clear at this stage since the
basis may not be orthonormal (i.e., the states might not be normalized). 2
In the following specific discussions of the 3D toric code model and the Haah code,
we will show that we can select a region A and a cut on the TNS such that |{t}〉A
75
and |{t}〉A are not only orthogonal, but also normalized. In particular for the 3D
toric code model, we can just select the region A to be a cube which satisfies the SVD
condition directly. See respectively Sec. 4.3.4 for detailed discussions. However, the
Haah code is different: a cubic region A does not fulfill the SVD condition However,
in Sec. 4.4.3 we can generalize the SVD condition to the Generalized SVD Condition
and apply Bc operators to make the TNS wave function an SVD.
4.2.2 Summary of the results
We now summarize the major results derived in this paper for the three stabilizer
codes. Fundamentally, our calculations come down to the fact that the indices of the
nonzero elements of the local tensor T and g are constrained. More specifically, when
we calculate the entanglement entropies with a TNS which is an exact SVD, the only
task is to count the number of independent Schmidt states |{t}〉A. The number of
independent Schmidt states |{t}〉A is determined by the Concatenation lemma,
i.e., when a network of T tensors and g tensors are concatenated, the open indices of
the nonzero elements of the resulting tensors are constrained as well.
1. The TNS is the exact SVD for the ground states with respect to particular
entanglement cuts. The entanglement spectra are flat for models studied in
this paper.
2. The entanglement of TNS is bounded by the area law:
S ≤ Area× log(D),
where D is the virtual index dimension and Area is measured in the units of
vertices. For the models studied in this paper, the entanglement entropies are
strictly smaller than the area law when one is computing in terms of vertices. For
the toric code, the correction is a negative constant, − log(2). For Haah code,
76
the correction includes a negative term linear with the system size, presented
in Sec. 4.4.4.
4.3 3D Toric Code
In this section, we construct the TNS for 3D toric code model and then calculate
the entanglement entropy and GSD of the toric code model, both deriving from the
Concatenation lemma. The results are the immediate generalizations of those in
2D toric code model. We find a topological entanglement entropy in accordance to
that obtained by Ref. [118] using field theoretic approach.
This section is organized as follows: In Sec. 4.3.1, we briefly review the toric code
model in a cubic lattice. In Sec. 4.3.2, we construct the TNS for the toric code model.
In Sec. 4.3.3, we prove a Concatenation lemma for toric code TNS, which is useful
in the following calculations. In Sec. 4.3.4, we calculate the entanglement entropies
on R3.
4.3.1 Hamiltonian of 3D Toric Code Model
The 3D toric code model can be defined on any random lattice. However, for sim-
plicity, we only work on the cubic lattice. On a cubic lattice, the physical spins are
defined on the bonds of the lattice, and the Hamiltonian is built from two types of
terms:
H = −∑
v
Av −∑
p
Bp (4.16)
Av is defined around a vertex v, Bp is defined on a plaquette p.
Av =∏
i∈v
Zi, Bp =∏
i∈p
Xi (4.17)
77
Z ZZ
Z
x
x
xx
x
z
y
Z
Z
(a) (b)
Figure 4.3: The Hamiltonian terms of the 3D toric code model. Panel (a) is Av whichis a product of 6 Z operators, and Panel (b) is Bp which is a product of 4 X operators.The circled X and Z represent the Pauli matrices acting on the spin-1/2’s. The toriccode Hamiltonian includes Av terms on all vertices v and Bp terms on all plaquettesp.
where Zi and Xi are Pauli matrices for the i-th spin. On a cubic lattice, Av is
composed of 6 Pauli Z operators while Bp is composed of 4 Pauli X operators. These
two terms are depicted in Fig. 4.3. In the 2D toric code, Av is composed of 4 Pauli
Z operators on a square lattice. The Hamiltonian is the sum of Av operators on all
vertices v and Bp operators on all plaquettes p.
It is easy to verify that all the Hamiltonian terms commute:
[Av, Av′ ] = 0, ∀ v, v′
[Bp, Bp′ ] = 0, ∀ p, p′
[Av, Bp] = 0, ∀ v, p
(4.18)
and their eigenvalues are ±1:
A2v = 1, B2
p = 1. (4.19)
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The ground states |GS〉 should satisfy:
Av|GS〉 = |GS〉, ∀ v
Bp|GS〉 = |GS〉. ∀ p(4.20)
These two sets of equations are enough to derive the local T tensor and to construct
TNS for the toric code model. In particular, one of the ground states on the torus
that we will find is
|ψ〉 =∏
v
(1v + Av)|0x〉 (4.21)
where |0x〉 is the tensor product of all X = 1 eigenstates defined on each link.
4.3.2 TNS for 3D Toric Code
We first introduce a projector g tensor Eq. (4.4) on each bond of the lattice. The
range of the virtual index is only from 0 to 1. The local physical indices are |0〉 ≡ |↑〉
and |1〉 ≡ |↓〉. The projector g tensor satisfies:
Z = =
x x=x
(4.22)
79
In terms of algebraic equations, these diagrams correspond to:
gsi,j(−1)s = gsi,j(−1)i = gsi,j(−1)j
g1−si,j = gs1−i,1−j
(4.23)
These two sets of equations are true, because (1) the indices s, i and j are identified for
nonzero gsi,j, (2) the nonzero gsi,j are always 1 Eq. (4.4). We can use these conditions
to transfer the action of the physical operators to the virtual operators. Now we
introduce additional T tensors on each vertex of the cubic lattice, these T tensors
have six virtual indices. Graphically, we represent this T tensor as:
Txy
z
z
y
x (4.24)
Next we need to fix the elements of the T tensor, up to the TNS gauge freedom. The
method to fix the T tensor is to make it invariant under the actions of Av and Bp
operators, in order to implement the local conditions for ground states Eq. (4.20).
80
The actions of Av and Bp operators on local tensors are:
TZ
ZZ
Z
Z
Z
Z
Z Z
ZZ
Z
g
gg
g
g
g
=
Tg
gg g
x
x
x x
x xx
xx x
x
x=
(4.25)
where we have used Eq. (4.22) to transfer the physical operators to the virtual ones.
We require a strong version of the solution to of the above equations. We want the
tensors in the dashed red rectangles to be invariant under the actions of any of the
Av and Bp (this is a sufficient constraint that guarantees that the tensors form the
81
ground state) , which leads to the following equations:
T TZ Z
Z
Z=
T T
x x
T
x
xT
x
x
T
x
xT
xx
Tx
x
Tx
Tx
xx
= = =
=
=
==
Z
Z
=
Tx
x T
xx=== =
Tx
x
Tx
T
x
x
x
===T
xxTx
x
(4.26)
In the second set of equations, the first 12 equalities are obvious from the red dashed
squares, and the last 3 equalities can be derived from the first 12 ones. Expanding
the first set of conditions by using Zij = δij(−1)i, we have:
Txx,yy,zz = (−1)x+x+y+y+z+zTxx,yy,zz
⇔
Txx,yy,zz
= 0, if x+ x+ y + y + z + z = 1 mod 2
6= 0, if x+ x+ y + y + z + z = 0 mod 2
(4.27)
82
where x, x, y, y, z, z are the six indices of T in the three directions respectively. We
emphasize for notation clarity that x is not−x, they are notations for different indices.
The second set of conditions in Eq. (4.26) further enforces that an even number of
index flipping of the virtual indices of a tensor does not change the value of the tensor
elements. For instance, in terms of components, we have:
Txx,yy,zz =T(1−x)(1−x),yy,zz
=T(1−x)x,(1−y)y,zz
=Txx,yy,(1−z)(1−z)
= . . .
(4.28)
Hence, the elements of the T tensor are all equal. Up to an overall normalization, we
have the unique solution:
Txx,yy,zz =
0, if x+ x+ y + y + z + z = 1 mod 2
1, if x+ x+ y + y + z + z = 0 mod 2
(4.29)
The ground state TNS wave function is then Eq. (4.5) with the local T to be Eq. (4.29).
The local T tensors are the same on other spatial manifolds, such as T 3.
A similar set of conditions as the first equality in Eq. (4.26) have been intro-
duced by several other names in tensor network literature: Z2-injectivity[119], MPO-
injectivity[120], Z2 gauge symmetry[24] etc. The previous studies were in 2D, and
our condition is the 3D generalization. Note that the first equation in Eq. (4.26)
alone will not necessarily lead to topological order. It only implies that the ground
state is Z2 symmetric. The state which only satisfies the first condition in Eq. (4.26)
could also be a topological trivial state by tuning the relative strength of the nonzero
elements of T tensor. This can be interpreted as a condensation transition from topo-
83
logical phases to trivial phases. See Refs. [24, 121, 122, 123, 124] for explanations and
examples in the case of 2D TNS.
4.3.3 Concatenation Lemma
In this section, we consider some contraction of a network of local T tensors with
open virtual indices. Since the elements of a local T tensor are 0 for the odd sector
and 1 for the even sector (see Eq. (4.29)), we will show that, in general, a network of
contracted T tensors obeys a similar rule: some elements are zeros while the others
are identical and nonzero. A Concatenation lemma is proposed to define the rule
for the contraction of several tensors in general and will be frequently used in the
following discussions.
Concatenation Lemma: For a network of contracted T tensors Eq. (4.29)
with open indices, the open indices need to sum to 0 mod 2, otherwise the element
of the network tensor is zero. Moreover, if nonzero, the elements of the network
tensor are constants, independent of open indices.
This lemma can be easily proved by using Z2 symmetry Eq. (4.29) and induction.
The proof is in App. C.1. We explain this lemma by a simple example. Suppose we
have two T tensors contracted over a pair of indices:
Tx1,x1,y1,y1,z1,x2,x2,y2,y2,z2
=∑
z1,z2
Tx1x1,y1y1,z1z1Tx2x2,y2y2,z2z2δz1z2(4.30)
84
T
T
x
z
y
Figure 4.4: Contraction of two local T tensors in the z-direction.
Graphically, the tensor T is represented by Fig. 4.4. The open indices of the
tensor T need to sum to an even number in order for the elements of the T tensor to
be nonzero. This comes out of writing the constraints of each of the T tensors:
x1 + x1 + y1 + y1 + z1 + z1 = 0, mod 2
x2 + x2 + y2 + y2 + z2 + z2 = 0, mod 2
z1 = z2
⇒ x1 + x1 + y1 + y1 + z1 + x2 + x2 + y2 + y2 + z2
=0, mod 2
(4.31)
Otherwise, the tensor element of T is zero. Moreover, the elements of the contracted
tensor are 1, if nonzero:
Tx1,x1,y1,y1,z1,x2,x2,y2,y2,z2 =
0 if x1 + x1 + y1 + y1 + z1 + x2 + x2 + y2 + y2 + z2 = 1, mod 2
1 if x1 + x1 + y1 + y1 + z1 + x2 + x2 + y2 + y2 + z2 = 0, mod 2
(4.32)
85
For a more complicated contraction of T tensors, we have:
T{t} =
0 if∑
i ti = 1, mod 2
Const if∑
i ti = 0, mod 2
(4.33)
where {t} denotes all the indices of the tensor T. Note that the nonzero constant
does not depend on {t} .
4.3.4 Entanglement
We now show that Eq. (4.5) is exactly the SVD for the wave function with respect
to the entanglement cut illustrated in Fig. 4.5. For simplicity, suppose that the TNS
is on infinite R3. As we have emphasized at the end of Sec. 4.1.2, we do not specify
the boundary conditions of TNS, since we are only concerned with the bulk wave
functions whose reduced density matrices are assumed not to be influenced by the
boundary conditions. If we put the wave function on a large but finite R3, we have
to specify the boundary conditions of the TNS by fixing the indices on the boundary.
Suppose the open indices on the boundary are denoted as {tb}. The norm of the
TNS on open R3, which can be expressed as a network of contracted T tensors with
open virtual indices {tb}, is zero when∑
i tbi = 1 mod 2 and nonzero when
∑i tbi = 0
mod 2, according to the Concatenation lemma of the 3D toric code model. Hence,
we can only fix the boundary indices {tb} to be∑
i tbi = 0 mod 2. Calculating
the entanglement on a nontrivial manifold is ambiguous since multiple degenerate
ground states, which cannot be distinguished locally, appear. Their superpositions
have different entanglement entropies.
We rewrite Eq. (4.5) by separating the tensor contractions to a spatial region A
and its complement region A. Region A contains the g tensors near the entanglement
86
cut as illustrated in Fig. 4.5:
|TNS〉R3 =∑
{t}
|{t}〉A ⊗ |{t}〉A (4.34)
where
|{t}〉A =∑
{s}∈A
∑
{i}∈A
CA(gs1t1i1gs2t2i2
. . . gs3i3i4gs4i5i6Ti7...Ti8... . . .)|{s}〉 (4.35)
Indices denoted by s are the physical indices; indices denoted by t are the open virtual
indices going out of the entanglement cut from the region A; indices denoted by i are
the contracted virtual indices inside the region A. The tensors gs1t1i1 and gs2t1i2 etc are
the projector g tensors near the entanglement cut on the region A side as illustrated
in Fig. 4.5; gs3i3i4 and gs4i5i6 are the projector g tensors inside the region A; for this cut,
all the T tensors are inside the region A. The summation is over all physical indices
{s} inside the region A.
Tg
A Acut
Figure 4.5: The splitting of tensors near the entanglement cut.
Thereby, |{t}〉 is the TNS wave function for region A with open virtual indices {t}.
We choose a convention of splitting tensors whereby g tensors near the entanglement
cut belong to the region A, as illustrated in Fig. 4.5. For instance, when the region A
87
is a cube, we can graphically denote the basis |{t}〉 as Eq. (4.10), where in the bulk
of this cube is a TNS, and the red lines are the outgoing virtual indices {t}. The g
tensors connecting with these red lines are inside the cube. Similarly for the region
A:
|{t}〉A =∑
{s}∈A
∑
{i}∈A
CA(gs1i1i2gs2i3i4Tt1i5...Tt2i6... . . .)|{s}〉 (4.36)
Since the TNS for region A and A share the same boundary virtual indices {t}, then
in Eq. (4.34) the two basis for region A and A have the same label {t}. For the TNS
wave function of Eq. (4.5), the boundary virtual indices {t} of the regions A and A
are contracted over, and thus in Eq. (4.34) {t} are summed over.
We now show that |{t}〉A and |{t}〉A are an orthonormal basis (normalized up to
constant) for the region A and the region A respectively. Therefore, Eq. (4.34) is
exactly the SVD for the ground state wave function, i.e.,
A〈{t′}|{t}〉A ∝ δ{t′},{t}δ(∑
i
ti = 0 mod 2). (4.37)
Proof:
Applying the SVD condition to the toric code TNS, we can immediately conclude
that the |{t}〉A span an orthogonal basis, and the TNS is exactly an SVD. However,
the SVD condition does not tell us whether the basis is orthonormal. In the follow-
ing, we show that |{t}〉A is not only orthogonal, but also orthonormal with a norm
independent on t, which leads to the flat singular values. Following the definition of
88
our basis:
A〈{t′}|{t}〉A =
∑
{s′}∈A
∑
{j}∈A
CA(gs′1?
t′1j1gs′2?
t′2j2. . . g
s′3?j3j4
gs′4?j5j6
T ?j7...T?j8...
. . .)〈{s′}|
∑
{s}∈A
∑
{i}∈A
CA(gs1t1i1gs2t2i2
. . . gs3i3i4gs4i5i6Ti7...Ti8... . . .)|{s}〉
(4.38)
When the open virtual indices {t′} 6= {t}, the overlap is clearly zero, as the spin
configurations on the boundary are different due to the projector g tensors. Hence,
the basis |{t}〉A are orthogonal.
Next we show that A〈{t}|{t}〉A is zero when(∑
ti∈{t} ti
)is odd. Using the defi-
nition of the g-tensor, we have:
A〈{t}|{t}〉A = CA (. . . TTT . . .) (4.39)
with the open virtual indices {t}. The contraction CA is over the T tensors in the
region A. Applying the Concatenation lemma, A〈{t}|{t}〉A is zero if the open
indices {t} are summed to be 1 mod 2:
∑
i
ti = 1 mod 2 ⇒ A〈{t}|{t}〉A = 0 (4.40)
Moreover,
A〈{t}|{t}〉A = Const, when∑
i
ti = 0 mod 2 (4.41)
Hence |{t}〉 is orthonormal basis up to an overall normalization factor that can be
obtained by the normalization of |TNS〉. 2
The same proof works for the region A and |{t}〉A. Therefore, we can conclude
that Eq. (4.34) is indeed an SVD, and the singular values are all identical. Hence,
89
for a entanglement cut, we only need to count the number of singular vectors in
Eq. (4.34). For a connected entanglement surface with N open virtual indices, the
number of singular vectors in Eq. (4.34) is 2N−1, because the open virtual indices
need to sum to be 0 mod 2. Hence, the entanglement entropy for a region whose
entanglement surface is singly connected is:
S = N log(2)− log(2) (4.42)
If the entanglement surface still has N open virtual indices but is separated into n
disconnected surfaces, then the entanglement entropy is:
S = N log(2)− n log(2)
= Area× log(2)− n log(2)
(4.43)
The above is true because the condition that the open indices need to have an even
summation holds true for each component of the entanglement cut. Furthermore, if
we place our TNS ground state on a 3D cylinder T 2xy ×Rz, and the entanglement cut
splits the cylinder into two halves z > 0 and z < 0, then the entanglement entropy of
the either side is also S = Area× log(2)− log(2). The results can be easily generalized
to ZK lattice gauge models on R3:
S = Area× log(K)− n log(K) (4.44)
with the same equation holding on a cylinder T 2xy ×Rz. The entanglement spectrum
is also flat. The area is measured by the number of open virtual indices going out of
the entanglement cut.
Following the same logic, for the toric code in (d + 1) dimensions, all the open
virtual indices of region A, {ti}, have to satisfy a single constraint∑
i ti = 0 mod 2,
90
because they have to obey the Concatenation lemma. If there are N open virtual
indices on the surface of region A, there are N − 1 independent open virtual indices.
Hence the rank of the reduced density matrix is still 2N−1, because each independent
open index can take 2 values. The entanglement entropy is
S = N log(2)− log(2) (4.45)
The topological entanglement entropy Stopo[T d−1] is independent of the dimensional-
ity, and it obeys the conjecture presented in Ref. [118]:
exp(−dStopo[T d−1]) = GSD[T d] (4.46)
where GSD[T d] = 2d.
4.4 Haah Code
In this section, we derive the TNS for Haah code following a similar prescription as
that in Sec. 4.3.2. We then compute the entanglement entropies using the TNS for
several types of entanglement cuts. In Sec. 4.4.1, we review the Haah code and the
Hamiltonian terms. In Sec. 4.4.2, we present the construction of TNS for the Haah
code. In Sec. 4.4.3, we discuss the entanglement cuts for which the tensor network
wave function is an exact SVD. In Sec. 4.4.4, we discuss the cubic entanglement cut,
where the tensor network wave function is not an exact SVD. The calculation for
entanglement entropies proceeds in the same way as that for the Toric code: one
counts the number of constraints for open indices.
91
4.4.1 Hamiltonian of Haah code
The Haah code is defined on a cubic lattice. As opposed to the two previous models,
there are two spin-1/2’s defined on each vertex of a cubic lattice. Each spin-1/2
is associated with a two-dimensional local Hilbert space. Similar to the toric code
and the X-cube model discussed in Sec. 4.3, the Hamiltonian is a sum of commuting
operators where each term is the product of Pauli X or Z operators. Specifically,
there are two types of the Hamiltonian terms:
H = −∑
a,b,c
Aabc −∑
a,b,c
Babc (4.47)
The A and B operators are defined on each cube in the cubic lattice, and the indices
a, b, c represent the vertex coordinates. If we choose the space to be the infinite
without periodic boundary condition, i.e., R3, then a, b, c ∈ Z. If we choose the space
to be a 3D torus of the size Lx × Ly × Lz with periodic boundary condition on each
side, then a ∈ ZLx , b ∈ ZLy and c ∈ ZLz . The operators defined on a = 0, b = 0, c = 0
are
A000 = ZL110Z
L101Z
L011Z
L111Z
R100Z
R010Z
R001Z
R111
B000 = XL000X
L110X
L101X
L011X
R000X
R100X
R010X
R001
(4.48)
The up-indices L/R represent the left or the right spin on a vertex where the Pauli
operators act on. The bottom-indices (ijk) ∈ Z2×Z2×Z2 represent the coordinate of
vertices (on a cube). All other operators Aabc and Babc can be obtained by translation.
92
Pictorially the two types of terms are:
x
z
y
(4.49)
It is straightforward to check that all the Hamiltonian terms commute.
4.4.2 TNS for Haah Code
The ground state |GS〉 is obtained by requiring
Aabc|GS〉 = |GS〉 (4.50)
Babc|GS〉 = |GS〉 (4.51)
for every a, b, c. We can solve these two equations similarly to the toric code model
Sec. 4.3.2 to obtain a TNS representation, although, since the model geometry is
different (spins on sites rather than on bonds), the form of the TNS is also changed.
We now specify the projector g tensor and the local T tensor.
There are 2 types of g tensors gL and gR associated with the left and right physical
spins on each vertex. Each g tensor has 1 physical index s and 4 virtual indices i, j, k, l.
93
The reason for these 4 virtual indices (rather than 2 virtual indices as in the toric
code and the X-cube examples) is that, for each vertex, the virtual indices from T
tensors (to be defined below) in the neighboring 8 octants need to be fully contracted;
this requires the g tensor to have 4 virtual indices. The index assignment of the Left
and Right- spin g tensor gLsijkl and gRsijkl are:
gLsijkl =
s
i
j
k l
I
IIIII
IV
V
V IV II
V III
(4.52)
and
gRsijkl =
l
j
ki
s
I
IIIII
IV
V I
V
V II
V III
(4.53)
94
where s is the physical index in {|0〉 = |↑〉, |1〉 = |↓〉}, and ijkl are virtual indices.
We use a blue dot for the right spin and a red dot for the left spin. The green dots
at the center of each cube represent T tensors (which we define below). Similar to
the toric code model and the X-cube model, we require that the g tensor acts as a
projector from the physics index to the four virtual indices:
gLsijkl =
1 i = j = k = l = s
0 otherwise
, gRsijkl =
1 i = j = k = l = s
0 otherwise
(4.54)
The four virtual indices of gLsijkl extend along the III, VIII, VII, VI octants (as shown
in Eq. (4.52)), and the four virtual indices of gRsijkl extend along the II, VII, IV, V
octants (as shown in Eq. (4.53)).
The tensor T{i} is defined at the center of each cube, and every T tensor has 8
virtual indices. Graphically, the T tensor is:
Ti1i2i3i4i5i6i7i8 = T (4.55)
The T tensor is contracted to 8 of the total 16 (8 vertices times 2 degrees of freedom
per vertex) g tensors located at the cube corners via the virtual indices. The reason
for only 8 virtual indices (instead of 16 virtual indices) in the T tensor is that among
16 spins around the cube (a, b, c) only eight of them are addressed by the Pauli Z
operators in the Aabc term of the Hamiltonian. The elements of the T tensor for
a given set of virtual indices i1i2i3i4i5i6i7i8 are determined by solving Eq. (4.50)
95
and Eq. (4.51). Imposing the condition Eq. (4.50) and transferring the physical Z
operators to the virtual level, we find that:
TZ
Z Z
Z Z
Z
ZZ
= T (4.56)
which amounts to
Ti1i2i3i4i5i6i7i8 = (−1)∑8n=1 inTi1i2i3i4i5i6i7i8 (4.57)
where i1, · · · , i8 are the eight virtual indices of the T tensor defined in Eq. (4.55).
Hence,
Ti1i2i3i4i5i6i7i8 = 0, if8∑
n=1
in = 1 mod 2 (4.58)
96
Imposing the condition Eq. (4.51) and transferring the physical X operators to the
virtual level, we find that
T = Tx x
= T
x
x
= T
x
x
= T
x
x = Tx x
= T
x x
= T
x x
= T
x
x
= T
x
x = T
x
x
= Tx
x
= Tx
x
= Tx
x
= T
x x
x
x
xx
(4.59)
97
In terms of components, Eq. (4.59) means that Ti1i2i3i4i5i6i7i8 = Ti′1i′2i′3i′4i′5i′6i′7i′8 where
i′1i′2i′3i′4i′5i′6i′7i′8 are obtained by flipping arbitrary pairs of indices from i1i2i3i4i5i6i7i8.
For example,
Ti1i2i3i4i5i6i7i8 = T(1−i1)(1−i2)i3i4i5i6i7i8
= Ti1(1−i2)(1−i3)i4i5i6i7i8
= Ti1i2(1−i3)(1−i4)i5i6i7i8
= ...
(4.60)
Combining Eq. (4.57) and (4.60), we find that any configuration of Ti1i2i3i4i5i6i7i8
satisfying the condition∑8
k=1 ik = 0 mod 2 are equal. We can rescale the T tensor
such that Ti1i2i3i4i5i6i7i8 = 1 for∑8
k=1 ik = 0 mod 2, i.e.,
Ti1i2i3i4i5i6i7i8 =
1∑8
n=1 in = 0 mod 2
0∑8
n=1 in = 1 mod 2
(4.61)
For simplicity, we consider the space to be R3. Since there is no non-contractible
spatial cycle, there is only one ground state:
|TNS〉 =∑
{s}
CR3 (gL,s1gR,s2gL,s3gR,s4 . . . TTT . . .
)|{s}〉 (4.62)
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T T
T T
(a)
x
z
y
T T
T T
T T
T T
(b)
Figure 4.6: Tensor contraction for the Haah Code TNS. (a) The lattice size is 2×3×3.
(b) The lattice size is 3× 3× 3
Note that the contraction of the Haah code TNS is quite different from that of the
3D toric code model and the X-cube model. The main difference is that the g tensor
has 4 virtual indices for the Haah code, while it has only 2 virtual indices for the toric
code and the X-cube code. As an example of contraction, we take two blocks of size
2× 2× 1 and 2× 2× 2 in Fig. 4.6. The T tensors with their virtual indices are drawn
99
explicitly. Each red or blue node in the two figures is a projector g tensor, whose
physical index is not drawn; we only draw the virtual legs that are connected to the
T tensors inside the blocks. In the block 2 × 2 × 2, all the 8 virtual indices of the
two g tensors (4 per each g tensor) in the middle of all the cubes are contracted with
T tensors, while other g tensors have open virtual indices (which are not explicitly
drawn).
4.4.3 Entanglement Entropy for SVD Cuts
In this section, we compute the entanglement entropy for two types of cuts for which
the TNS wave function is an SVD.
Two types of SVD Cuts
To compute the entanglement entropy, we use the same convention which was adopted
in the discussion of the toric code (in Sec. 4.3 ): the open virtual indices of the region
A connect directly to the g tensors while the open virtual indices of the region A
connect with T tensors. We further choose a region A such that the TNS is an SVD,
and compute the entanglement entropy. We found two types of entanglement cuts for
which the Haah code TNS is an exact SVD. For more general regions, there exists an
extra step required to make other cuts SVD. This step will be presented in Sec. 4.4.4.
1. Region A only consists of the spins connecting to a set of (l − 1) T tensors
which are contracted along a certain direction. Figure 4.7 shows an example
with l− 1 = 3 contracted along the z direction. (Since in Sec. 4.3, we used l as
the number of vertices along each side of region A, so there are l− 1 bonds (or
cubes) along each side.)
100
T
T
T
Figure 4.7: Region A contains all the spins connecting with l− 1 T tensors which are
contracted along z direction. The figure shows an example with l = 4.
2. Region A contains all the spins connecting with T tensors which are contracted
in a “tripod-like” shape, where three legs extend along x, y, z directions. If there
are lx− 1 cubes in the x leg, ly − 1 cubes in the y leg, and lz − 1 cubes in the z
leg, then there are 1 + (lx − 2) + (ly − 2) + (lz − 2) = lx + ly + lz − 5 cubes (or
T tensors) region A. Figure 4.8 shows an example with lx = ly = lz = 3.
T T
T
T
Figure 4.8: Region A contains all the spins connecting with T tensors which are
contracted in a “tripod-like” shape, where three legs extend along x, y, z directions.
There are three legs extending along x, y, z directions respectively. In general, three
legs can have different length, each with lx−1, ly−1, lz−1 cubes along three directions.
This figure shows an example where lx = ly = lz = 3.
101
In the first case and for l = 2, 3, we used brute-force numerics to find that the
reduced density matrix is diagonal (see App. C.2 for details), which shows that the
wave function is an exact SVD.
In order to show that the above cuts correspond to an SVD, we follow the argu-
ments developed in Sec. 4.2.1. In Sec. 4.2.1, we proposed a SVD Condition. However,
we find that the region A of both types, shown in Fig. 4.7 and 4.8, do not satisfy
the SVD Condition: Two open virtual indices in region A connects with the same T
tensor, which violates the SVD Condition. For instance, the g1 and g2 in Fig. 4.8 con-
nects to the same T tensor in their upper-left cube which is in the region A. Here, we
propose a Generalized SVD Condition which suffice to prove that the entanglement
cut corresponding to Figure 4.7 and 4.8 are SVD.
Generalized SVD condition: Let {t} be the set of open virtual indices. Given a
set of physical indices {s} inside region A, if {t} can be uniquely determined by the
{s} inside region A via the g tensor projection condition Eq. (4.54) and T tensor
constraints Eq. (4.61), then |{t}〉A is orthogonal. Since |{t}〉A is orthogonal because
all the open virtual indices are connected with g tensors, the TNS wave function
|TNS〉 =∑{t} |{t}〉A ⊗ |{t}〉A is SVD.
To prove the Generalized SVD Condition, we notice that if we have two different
sets of open virtual indices {t}A and {t′}A, the physical indices {s}A and {s′}A which
connect (via g tensors) to the T tensors on the boundary of region A cannot be the
same. Otherwise, if {s}A = {s′}A, since the physical indices {s}A and {s′}A in the
region A uniquely determine the open virtual indices {t}A and {t′}A, {t}A = {t′}A,
hence it is in contradiction with our assumption {t}A 6= {t′}A. Therefore, {t}A 6=
{t′}A implies {s}A 6= {s′}A, and hence A〈{t}|{t′}〉A = 0. This is in the same spirit of
the proof in Sec. 4.2.1. The proof of normalization of the wave function is independent
of {t} is also the same as in Sec. 4.2.1. Furthermore, A〈{t}|{t′}〉A = 0 for {t} 6= {t′}
102
is the straightforward because {t}A are connected with g tensors. In summary, if the
entanglement cut satisfies the Generalized SVD Condition, we have
1. A〈{t}|{t′}〉A ∝ δ{t},{t′} when |{t}〉A and |{t′}〉A are not null vectors;
2. A〈{t}|{t′}〉A ∝ δ{t},{t′} when |{t}〉A and |{t′}〉A are not null vectors.
This shows that the TNS wave function is an SVD.
We explain the Generalized SVD Condition in the simplest example, i.e., l = 2 in
case 1. There is only one T tensor, and region A contains 8 physical spins.
T
All other spins apart from the eight connecting with the T tensor belong to region
A. Because the virtual indices and physical indices are related by the g tensor which
is a projector, we use i1 to denote the values of both virtual indices and physical
indices connecting with left g tensor located at (x, y, z) = (0, 0, 1). Here, we use
the coordinate convention where the (x, y, z) = (0, 0, 0) is located at the left down
frontmost corner as in Fig. 4.6. Similarly we use i2, i3, i4, i5, i6, i7, i8 to label the values
of the virtual/physical indices on the remaining seven nodes connecting with the same
T tensor. Hence the set of open indices is effectively {i1, i2, i3, i4, i5, i6, i7, i8} (after
identified by the g tensors). We further consider how the physical indices from the
region A constraint the open indices. Consider the T tensor in the region A (which
we denote by T ′) which shares two spins i7, i8 with region A (The T ′ tensor lives in
103
the lower right corner):
T
T'
(4.63)
Since six among the eight virtual indices of T ′ are contracted with g tensors inside
region A, the remaining two open virtual indices, i.e., i7 and i8 are subject to one
constraint from the T ′ tensor:
i7 + i8 = fixed (4.64)
where “fixed” means that the sum is fixed by the physical indices inside the region
A. We can similarly consider the constraints coming from other T tensors in region
A. The whole set of constraints are listed as follows:
i7 + i8 = fixed, i1 + i2 = fixed, i5 = fixed, i6 = fixed, i6 + i7 = fixed
i2 + i3 = fixed, i8 = fixed, i1 = fixed, i4 = fixed, i3 = fixed, i7 = fixed
(4.65)
The “fixed” on the right hand side of the equations means that the virtual indices
or the sum of the virtual indices are fixed by the physical indices in the region A.
All variables and equations are defined module 2. The above equations uniquely
determine all the open virtual indices i1...i8. Therefore, such a choice of region A of
the entanglement cut satisfies the Generalized SVD Condition.
104
For the first type of region A with general l, and the second type of region A
with general lx, ly, lz, we can similarly check that the TNS wave function satisfies
the Generalized SVD Condition. Numerically, we checked TNS wave function indeed
satisfies the Generalized SVD Condition for 2 ≤ l ≤ 9 for the first type, and 3 ≤ lx ≤
8, 3 ≤ ly ≤ 8, 3 ≤ lz ≤ 8 for the second type.
Entanglement entropy
We now compute the entanglement entropy for the exact SVD TNS wave funtions. We
first consider the case 1 with general l, such as in Fig. 4.7. All the spins connecting
with l − 1 contracted T tensors along the z directions are in region A, and the
remaining belong to region A. The number of open virtual indices, after identified by
the local g tensors, is 8 + 7(l− 2) = 7l− 6. The number of constraints from the local
T tensors is simply the number of T tensors l − 1, because they are all independent.
Hence the number of independent open virtual indices is 7l − 6 − (l − 1) = 6l − 5.
Therefore, the entanglement entropy is
S(A)
log 2= 6l − 5 (4.66)
In appendix. C.2, we numerically brute-force compute the reduced density matrix for
l = 2 and l = 3, and find that the results match the general formula Eq. (4.66).
We further consider the case 2 — region A of tripod shape. The legs in the x, y, z
direction contains lx− 1, ly − 1, lz − 1 T tensors respectively. We first count the total
number of open virtual indices. When lx = ly = lz = 3 as shown in Fig. 4.8, there are
26 physical spins (or g tensors) in total. However, there is one g tensor (at the left spin
of (x, y, z) = (1, 1, 1)) whose four virtual indices are all contracted by the T tensors
within region A. Hence the number of open virtual indices, after identified by the local
g tensor, is 25. Moreover, we notice that adding one T tensor in one of the three legs of
105
region A brings 7 extra spins. Therefore the total number of open virtual indices (after
identified by the g tensor) is (26−1)+7(lx−3)+7(ly−3)+7(lz−3) = 7lx+7ly+7lz−38.
We further numerically count the number of constraints that these open virtual indices
satisfy. We find the number of constraints is the number of cubes minus 1, i.e.,
(lx + ly + lz − 5) − 1 = lx + ly + lz − 6. Therefore the number of independent open
virtual indices is (7lx + 7ly + 7lz − 38)− (lx + ly + lz − 6) = 6lx + 6ly + 6lz − 32. The
entanglement entropy is
S(A)
log 2= 6lx + 6ly + 6lz − 32. (4.67)
4.4.4 Entanglement Entropy for Cubic Cuts
In this section, we consider the case where the region A is a cube of size l × l × l,
where l is the number of vertices in each direction of the cube. The cut is chosen
such that all the open virtual indices coming out of the region A are connected to g
tensors in the region A (i.e., all the physical spins near the boundary belong to the
region A). For example, for l = 2 as shown in (4.55), all 16 physical spins belong to
the region A. For l = 3 as shown in Fig. 4.6 (b), all 54 physical spins belong to the
region A. For the simplicity of notations, in this section, we denote the Hamiltonian
terms as Ac and Bc where the subindex refers to a cube c.
SVD for TNS
For the cubic region A, we find that the TNS for the Haah code is different from that
for the toric code and X-cube models: the TNS for the Haah code is not an exact
SVD. The TNS basis in the region A, |{t}〉A, are orthonormal, since the open virtual
indices are connected with g tensors. However, the TNS basis |{t}〉A in the region A
are not orthogonal. In other words, the basis |{t}〉A is over complete.
106
The subtlety that the TNS bipartition is not an exact SVD manifests as follows:
the singular vectors in the region A for the ground states of the Haah code have to be
the eigenvectors of all Ac and Bc operators that actually lie in the region A, and the
corresponding eigenvalues should all be 1. Note that our TNS basis state |{t}〉A, if
not null, are the eigenvectors of all Ac operators inside the region A with eigenvalues
1, and are also the eigenvectors of Bc operators with eigenvalues 1 when Bc operators
are deep inside the region A, i.e., when they do not act on any spin at the boundary
of A. However, |{t}〉A are not the eigenvectors of Bc operators, when Bc operators
are inside the region A but also adjacent to the region A’s boundary. The reason
is that the Bc operators adjacent to the region A’s boundary, when acting on the
TNS basis |{t}〉A, will flip the physical spins on the boundary, and thus flip the open
virtual indices {t} due to the projector g tensors. Therefore, the basis |{t}〉A is no
longer the singular vectors for the Haah code. This is not an a priori problem, but
a result of the geometry of the Haah code, whose spins cannot be written on bonds
but have to be written on sites. A similar situation would occur if the 2D toric code
model would be re-written to have its spins on sites.
The method to find the correct SVD for the TNS wave function is to use the |{t}〉Ato construct the eigenvectors of Bc operators by projection. We prove the following
statement:
If |{t′}〉A = Bc|{t}〉A when Bc is inside the region A and also adjacent to the
region A’s boundary, then A〈{t′}|{t}〉A = 0 and |{t′}〉A = |{t}〉A.
The proof is as follows. The first part of the statement is a consequence of the
|{t}〉A basis state orthogonality. Indeed, Bc flips physical spins located at the region
A’s boundary. Thus the two sets {t} and {t′} are distinct. The second part of the
statement is more involved. Suppose for simplicity that we consider two nearest
neighbor T tensors for the region A and A in Fig. 4.9. The Bc operator acts on the
right cube Fig. 4.9 (a). The physical spins on the boundary of the region A which
107
T T
Cut
x
x
x
xx
T T
Cut
x
x
(a)
(b)
Figure 4.9: Transferring the Pauli X operators of the Bc operator from the region A(a) to the region A (b).
are flipped by Bc are those covered by circled X in Fig. 4.9 (a). Then these Pauli X
operators can be transferred to the virtual indices due to projector g tensors, and the
virtual indices of the T tensor in the region A obtain two X operators as in Fig. 4.9
(b). Note the T tensor for the Haah code is invariant under this action (see the 12th
cube in Eq. (4.59)). This is also true for other T tensors in the region A that are
affected by Bc. The transfer of X operators from the region A to the region A gives
exactly the same equations in Eq. (4.59) when we solve for the T tensors. Hence, the
X operators transferred to the open virtual indices in the region A do not change the
state at all, i.e., |{t′}〉A = |{t}〉A. As a consequence, we can perform the following
factorization
|{t}〉A ⊗ |{t}〉A + |{t′}〉A ⊗ |{t′}〉A
=[(1 +Bc)|{t}〉A
]⊗ |{t}〉A
(4.68)
108
The left part of the tensor product is an eigenstate of Bc with eigenvalue 1.
Therefore, in the TNS decomposition Eq. (4.11), we can group the basis state |{t}〉Awhich are connected by this Bc operator. This factorization can be extended to
any product of Bc operators inside the region A and also adjacent to the region A’s
boundary. Note that any such product has at least one X operator belonging to
only one Bc and so is different from the identity. When acting with all the possible
products of these Bc operator (including the identity) on a given |{t}〉A will generate
as many unique states as there are Bc’s. The TNS can be brought to the following
form
|TNS〉 =∑
{t}′
[∏
c
(1 +Bc
2
)|{t}〉A
]⊗ |{t}〉A (4.69)
where the product over c only involves the Bc operators inside the region A and also
adjacent to the region A’s boundary and the sum over {t}′ is over the open virtual
index configurations that are not related by the action of these Bc operators.
Counting the number of TNS basis in region A: Notations
To find the upper bound to the entanglement entropy, we need to find the number of
basis states in the region A that are also eigenstates of any Bc operators fully lying
in the region A. This number that we denote as basis(TNS(A)) is
basis(TNS(A)) = 2N−NB (4.70)
where N is the number of independent open virtual indices and NB is the number
of Bc operators inside the region A and also adjacent to the region A’s boundary.
Every open virtual index connected to a g tensor located in A and at the boundary
of this region. Since each g tensor has a unique independent virtual index, we have
N = Ng −Nc where Ng is the number of g tensors in A and at the boundary of this
109
region and Nc are the number of constraints on the open indices coming from the T
tensors within the region A. We this get
log2(basis(TNS(A))) = Ng −Nc −NB (4.71)
and the upper bound on the entanglement entropy reads
S(A) = (Ng −Nc −NB) log 2 (4.72)
Counting Ng and NB
We first count Ng. The number of g tensors can be computed by looking at Fig. 4.6
(b). We consider the region A with size lx × ly × lz (Notice that lx, ly, lz are the
number of vertices in each direction). In eight corners, there are 8× 2 = 16 vertices.
On the four hinges along x direction, there are 2 × 4 × (lx − 2) vertices, where 2
means there are two spins on each vertex, and 4 means four hinges. And similar for
2 × 4 × (ly − 2) and 2 × 4 × (lz − 2) in the y and z directions respectively. For the
xy-plane, there are 2× 2× (lx− 2)(ly− 2), where the first 2 comes from two spins per
vertex, and the second 2 comes from two xy-planes. Similarly 2× 2× (lx− 2)(lz − 2)
and 2 × 2 × (ly − 2)(lz − 2) for xz and yz plane respectively. Therefore, the total
number of g tensors is
Ng =16 + 8(lx − 2) + 8(ly − 2) + 8(lz − 2)
+ 4(lx − 2)(ly − 2) + 4(lx − 2)(lz − 2)
+ 4(ly − 2)(lz − 2)
=4lxly + 4lxlz + 4lylz − 8lx − 8ly − 8lz + 16
(4.73)
We further count NB. As explained in Sec. 4.4.4, NB is the number of Bc operators
inside the region A and adjacent to the boundary of the region A. For a cubic region
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A with size l × l × l (which is the case we consider below), the number of such Bc
operators are
NB = (l − 1)3 − (l − 3)3 = 6l2 − 24l + 26,∀l ≥ 3 (4.74)
For l = 2, we just have 1 Bc operator. Hence we have
NB = 6l2 − 24l + 26− δl,2,∀l ≥ 2 (4.75)
Counting Nc: Contribution from the T tensors
The open indices may be constrained by the T tensor fully inside the region A. In the
following, we will discuss the specific entanglement cuts where lx = ly = lz = l. We
rely on numerical calculations to evaluate Nc. We first consider the examples l = 2
and l = 3 in details, and then we describe our algorithm to search the number of the
linear independent constraints.
For l = 2, no g tensor has all virtual indices contracted. The reader can refer to
Fig. 4.6 (a) as an example. There is only one T tensor. The element of the T tensor
is
Ti1i2i3i4i5i6i7i8 (4.76)
where i1, i2, i3, i4, i5, i6, i7, i8 are all contracted virtual indices. Because they are con-
tracted with g tensors where at least one virtual index is open, all the contracted
virtual indices i1, i2, i3, i4, i5, i6, i7, i8 are equal to some open indices, and we denote
them as
i1 = t1, i2 = t2, i3 = t3, i4 = t4,
i5 = t5, i6 = t6, i7 = t7, i8 = t8
(4.77)
111
The constraints on {i}’s are hence equivalent to the constraints on {t}’s, i.e.,
t1 + t2 + t3 + t4 + t5 + t6 + t7 + t8 = 0 mod 2 (4.78)
There is only one constraint from the T tensor. Hence Nc = 1 for l = 2.
For l = 3, as shown in the Fig. 4.6 (b), we have eight constraints from eight T
tensors which involve the open indices via the g tensors. The eight equations are
8∑
n=1
i(x,y,z)n = 0 mod 2, x, y, z ∈ {0, 1} (4.79)
where the up-index (x, y, z) represents the position of the T tensor, and n counts the
eight indices of each cube in the 2 × 2 × 2 cut. All the i’s are contracted virtual
indices. However, except the virtual indices that are connected with the central two
g tensors (which are defined on the two spins at the vertex (x, y, z) = (1, 1, 1)), all
other indices (which are defined on two spins at vertices (x, y, z), x, y, z ∈ {0, 1, 2}
except (x, y, z) = (1, 1, 1)) are equal to some open indices via g tensors. Specifically,
the virtual indices that are connected with the two center g tensors are
i0004 = i100
3 = i0101 = i001
7 mod 2
i0005 = i110
2 = i1018 = i011
6 mod 2
(4.80)
Since we only count the number of constraints for the open indices, we need to Gauss-
eliminate all these eight virtual indices i0004 , i100
3 , i0101 , i001
7 , i0005 , i110
2 , i1018 , i011
6 from the
above 8 equations. Therefore, we obtain 8− 2 = 6 independent equations in terms of
open indices only. Hence there are 6 constraints for the open indices.
For the general l, we apply the same principle. We first enumerate all possible
constraints from the T tensors, and then we Gauss-eliminate all the virtual indices
that are contracted within region A. Hence we obtain a set of equations purely in
112
terms of the open indices. The number of constraints is the rank of these set of
equations, and we list the number of linear independent constraints for the open
indices as follows:
l(≥ 3) 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Nc 6 12 18 24 30 36 42 48 54 60 66 72 78 84(4.81)
Hence, for l ≥ 3, there are
6l − 12 (4.82)
linearly independent constraints for the open indices. Taking into account the fact
that when l = 2 the number of constraints is 1, we infer that the number of constraints
for a generic l is:
6l − 12 + δl,2 (4.83)
Entanglement entropy
We are ready to collect all the data we have obtained and compute the entanglement
entropy. For the entanglement cut of size l × l × l, the total number of g tensors is
Ng = 12l2 − 24l + 16 (4.84)
The number of of T tensor constraints is
Nc = 6l − 12 + δl,2,∀l ≥ 2 (4.85)
The number of Bc operators is
NB = 6l2 − 24l + 26− δl,2,∀l ≥ 2 (4.86)
113
Therefore entanglement entropy reads
S
log 2=Ng −Nc −NB
=6l2 − 6l + 2,∀l ≥ 2
(4.87)
The entanglement entropies also have negative linear corrections.
If the region A is much larger than the region A, we conjecture that the region A
will not impose any additional constraint. In that case, the upper bound would be
saturated. The numerical calculations in App. C.2 also support this conjecture.
114
Chapter 5
Topological Entanglement Entropy
of (3+1)D Gapped Phases of
Matter
5.1 Reduction formulas for Entanglement Entropy
In this section, we study the general structure of the EE for gapped phases of matter in
(3+1)D. The definitions of the entanglement entropy and the entanglement spectrum
are reviewed in Appendix D.1. We are inspired by the fact that for a (2+1)D system,
the EE of the ground state of a local, gapped Hamiltonian obeys the area law. In
particular, if we partition our system into two subregions, A and Ac, the EE of
subregion A with the rest of the system Ac takes the form
S(A) = αl + γ +O(1/l), (5.1)
where αl is the area term, and l is the length of the boundary of region A. Importantly
the constant term −γ− is topological and thus dubbed “topological entanglement
entropy” [20, 21]. We would like to understand whether an analogous formula holds
115
for gapped phases of matter in (3+1)D. In particular, we ask how the constant part
of the EE depends on the topological properties of both the Hamiltonian and the
entanglement surface.
Our approach to this question relies on the SSA inequality for the entanglement
entropy. We also make certain locality assumptions about the form of the entropy,
detailed in Appendix D.2. This allows us to derive an expression for the constant part
of the EE of a subregion A for a TQFT, STQFTc (A), which depends on the topological
properties (e.g. Betti numbers) of the entanglement surface ∂A ≡ Σ.1
We start by reviewing some general facts about the EE and then use SSA inequal-
ities to determine the formula for the EE across a general surface in Sec. 5.1.1. In
Sec. 5.1.2, we discuss the implications of our EE formula, especially regarding models
away from a renormalization group (RG) fixed point. Our approach is inspired by
Ref. [125].
5.1.1 Strong Sub-Additivity
Structure of the EE of Fixed Point TQFTs
As reviewed in Appendix D.2, for a generic theory with an energy gap, the EE for a
subregion A can be decomposed as
S(A) = F0|Σ|+ Stopo(A)− 4πF2χ(Σ)
+4F ′2
∫
Σ
d2x√hH2 +O(1/|Σ|), (5.2)
where the coefficients F0, F2 and F ′2 are constants that depend on the system under
study. The first term is the area law term, where |Σ| is the area of the entangle-
ment surface, Σ. The second term is the topological entanglement entropy, which
is independent of the details of the entanglement surface and of the details of the
1In this paper, we will denote a generic entanglement surface as Σ.
116
Hamiltonian. The third term is proportional to the Euler characteristic χ(Σ) of the
entanglement surface. Although it only depends on the topology of Σ, it is not uni-
versal, and we expect that the coefficient, F2, will flow under the RG. The fourth
term is proportional to the integral of the mean curvature, H = (k1 +k2)/2, of Σ (see
Appendix D.2 for a derivation of the local contributions). It depends on the geometry
(in contrast to the topology) of Σ, and its coefficient F ′2 also flows under the RG in
general. The remaining terms are subleading in powers of the area |Σ|, and vanish
when we take the size of the entanglement surface to infinity. One of the main goals
of this paper is to understand the structure of the topological entanglement entropy,
Stopo(A), and how it can be isolated from the Euler characteristic term and the mean
curvature term.
In this section, unless otherwise stated, we consider (3+1)D TQFTs describing
the low energy physics of a gapped topologically ordered phase. In this case the
constant part of the EE depends only on the topology of the entanglement surface.
The reason is the following: since a TQFT does not depend on the spacetime metric, it
is invariant under all diffeomorphisms, including dilatations as well as area-preserving
diffeomorphisms. Hence, the term related to the mean curvature (which depends on
the shape of Σ) should not appear. This implies that the coefficient F ′2 flows to
zero at the fixed point. When we regularize the theory on the lattice, we explicitly
break the scaling symmetry while maintaining the invariance under area preserving
diffeomorphisms. Hence the area law term can survive, i.e. F0 can flow to a non-
vanishing value at the fixed point. (We relegate the explanation of this subtlety in
Sec. 5.2.2.) Since the Euler characteristic is topological, F2 can also flow to a non-
vanishing value. In summary, the possible form of the EE for a low energy TQFT
(when regularized on the lattice) is
S(A) = F0|Σ|+ Stopo(A)− 4πF2χ(Σ) +O(1/|Σ|). (5.3)
117
For the sake of clarity, we denote the constant part of the EE for a generic theory
as Sc(A) = Stopo(A) − 4πF2χ(Σ) + 4F ′2∫
Σd2x√hH2, and the constant part of the
EE for a TQFT as STQFTc (A) = Stopo(A) − 4πF2χ(Σ). We point out that the value
of F2 for a general theory and for a TQFT are not the same, since its value flows
under renormalization to the one in the TQFT, which will be specified in Sec. 5.1.2.
Furthermore, the area law part of the EE, F0|Σ|, is denoted as Sarea(A).
For any quantum state, there are several information inequalities relating EEs
between different subsystems that are universally valid[126], such as sub-additivity,
strong sub-additivity, the Araki-Lieb inequality[127] and weak monotonicity[128].
Special quantum states, such as quantum error correcting codes[129] and holographic
codes[130, 126, 131], obey further independent information inequalities. The major
constraint on the EE utilized in this paper is the strong sub-additivity inequality,
which is typically used in quantum information theory. Explicitly, the SSA inequality
is
S(AB) + S(BC) ≥ S(ABC) + S(B), (5.4)
where the space is divided into four regions A,B,C, and (ABC)c. Here, (ABC)c is
the complement of ABC ≡ A ∪ B ∪ C. SSA strongly constrains the structure of the
constant part of S(A), i.e., Sc(A), as we will see below.
Reduction to the Constant Part of the EE
The SSA is universal, and hence it is valid for any choice of the regions A, B and
C. Here we will only need to consider the special cases with A ∩ C = ∅. This
configuration is chosen precisely to cancel the area law part of the EE on both sides
of the SSA inequality, thus giving us information about the constant part Sc(A).
Explicitly, when A ∩ C = ∅, we have
Sarea(AB) + Sarea(BC) = Sarea(ABC) + Sarea(B). (5.5)
118
Equation (5.4) then implies
Sc(AB) + Sc(BC) ≥ Sc(ABC) + Sc(B) . (5.6)
When restricted to a TQFT, we have
STQFTc (AB) + STQFT
c (BC) ≥ STQFTc (ABC) + STQFT
c (B) . (5.7)
Structure of Sc(A)
We need to parametrize STQFTc (A) in order to proceed. For a TQFT (where F ′2 = 0),
we see that STQFTc (A) = Stopo(A) − 4πF2χ(Σ) only depends on the topology of the
entanglement surface Σ through its Euler characteristic. Two-dimensional orientable
surfaces are classified by a set of numbers {n0, n1, n2, . . .}, where ng is the number of
disconnected components (parts) with genus g.2 We will show that this is an over-
complete labeling for STQFTc (A), and that STQFT
c (A) only depends on the zeroth and
first Betti number[132] of Σ defined below in terms of {n0, n1, n2, · · · }.
For the time being, we use the (over-)complete labeling scheme for STQFTc (A)
STQFTc [(0, n0), (1, n1), · · · , (g, ng), · · · ], (5.8)
where in each bracket, the first number denotes the genus, and the second number
denotes the number of disconnected boundary components ∂A with the corresponding
genus. The list ends precisely when ng∗ 6= 0 and ng = 0 for any g > g∗. In other words,
STQFTc [(0, n0), (1, n1), . . . , (g∗, ng∗)] is the constant part of the EE of the region with
n0 genus 0 boundaries, n1 genus 1 boundaries, · · · and ng∗ genus g∗ boundaries. We
emphasize that the region A can have multiple disconnected boundary components.
The set {ng} is related to the Betti numbers bi and the Euler characteristic χ through
2In this paper, the entanglement surfaces do not wrap around non-contractible cycles of the space.
119
g∗∑
g=0
ng = b0,
g∗∑
g=0
ng(2− 2g) = 2b0 − b1 = χ. (5.9)
These numbers will be useful in the following calculations.
By applying the SSA inequality to a series of entanglement surfaces, we derive an
expression for STQFTc in terms of the Betti numbers b0 and b1, as well as the entropies
STQFTc [T 2] and STQFT
c [S2] across the torus and sphere, respectively. Relegating the
details of the derivation to Appendix D.3, we find:
STQFTc [(0, n0), (1, n1), · · · , (g, ng)]
= b0STQFTc [T 2] +
χ
2
(STQFT
c [S2]− STQFTc [T 2]
). (5.10)
Notice that Eq. (5.10) is consistent with the expectation that disconnected parts of
the entanglement surface result in additive contributions due to the local nature of
the mutual information.
5.1.2 Topological Entanglement Entropy
Our first main result is Eq. (5.10), which clarifies two points. First, as we mentioned in
the introduction (and as was also discussed in Ref. [125]), given a general entanglement
surface [(0, n0), (1, n1), ..., (g∗, ng∗)], we can reduce the computation of the constant
part of the EE of a TQFT, STQFTc [(0, n0), (1, n1), ..., (g∗, ng∗)], to that of STQFT
c [S2]
and STQFTc [T 2]. Second, using Eq. (5.10), we can identify the topological and universal
part of Sc(A) for a generic theory beyond the TQFT fixed point. We now elaborate
on these points.
STQFTc [S2] and STQFT
c [T 2]
For a TQFT, Eq. (5.10) proves that the constant part of the EE across a general
surface can be reduced to a linear combination of the constant part of the EE across
120
S2 and T 2. Whether STQFTc [S2] and STQFT
c [T 2] are independent of each other depends
on the type of TQFT. As we show in Sec. 5.2, for a BF theory [see Eq. (5.22)] in
(3+1)D, STQFTc [S2] = STQFT
c [T 2]. For the GWW models [see Eq. (5.19)] in (3+1)D,
we show in Sec. 5.2 that STQFTc [S2] and STQFT
c [T 2] are different in general. Thus,
Eq. (5.10) is the simplest expression that is universally valid for any TQFT.
Away from the Fixed Point
In Sec. 5.1.1 and Appendix D.2, we revisited the arguments presented in Ref. [125]
that the constant part of the EE for a theory away from the fixed point is generically
not topological. The structure of the EE of a generic theory was shown in Eq. (5.2).
Combining Eq. (5.2) and Eq. (5.10), we now extract more information about the
structure of the EE.
First, we argued in Sec. 5.1.1 that
F ′2 → 0, (5.11)
when the theory is renormalized to a TQFT fixed point.
Second, by setting F ′2 = 0 in Eq. (5.2) and comparing the TEE and the coefficient
of the Euler characteristic χ in Eq. (5.2) and Eq. (5.10), we find that
Stopo[(0, n0), · · · , (g∗, ng∗)]
= b0STQFTc [T 2] =
( g∗∑
i=0
ni
)STQFT
c [T 2],(5.12)
and
F2 → −1
8π
(STQFT
c [S2]− STQFTc [T 2]
). (5.13)
121
Equation (5.12) suggests that the TEE across an arbitrary entanglement surface (for
a generic theory) is proportional to STQFTc [T 2]; in particular, the TEE across T 2 (for
a generic theory) equals STQFTc [T 2], i.e., Stopo[T 2] = STQFT
c [T 2]. Equation (5.13)
shows that while F2 can flow when the theory is renormalized, it converges to a
nontrivial value − 18π
(STQFT
c [S2] − STQFTc [T 2]
)at the RG fixed point. Our iden-
tification of the TEE Eq. (5.12) further elaborates on the result from Ref. [125],
which showed that the TEE across a genus g entanglement surface Σg is Stopo[Σg] =
gStopo[T 2]− (g−1)Stopo[S2]. Our result Eq. (5.12) suggests that Stopo[S2] = Stopo[T 2]
and therefore further simplifies the result of Ref. [125] to Stopo[Σg] = Stopo[T 2] for any
g. Our identification of the TEE also works for entanglement surfaces with multiple
disconnected components.
Extracting the TEE
Equation (5.12) suggests an “algorithm” to compute the TEE for a generic theory:
1) take a ground state wavefunction |ψ〉 for a generic system; 2) renormalize |ψ〉 to
the fixed point; 3) compute the entanglement entropy for an entanglement surface
T 2, STQFT[T 2]. The constant part STQFTc [T 2] is the TEE across T 2. Notice that this
is consistent with our definition STQFTc [T 2] = Stopo[T 2]− 4πF2χ(T 2) since χ(T 2) = 0.
The TEE across an arbitrary surface immediately follows from Eq. (5.12).
In this section, we will explain a more practical algorithm for extracting the TEE
(across T 2) which is applicable to the groundstate wavefunction of any generic the-
ory, and does not require renormalization to the TQFT fixed point. Our algorithm
(which is termed the KPLW prescription) builds upon the study of the topological
entanglement entropy in (2+1)D systems initiated by Kitaev, Preskill, Levin and
Wen[20, 21](KPLW) and the proposal in Ref. [125] in (3+1)D. We compute a par-
ticular combination of the EE of different regions, which we call SKPLW[T 2], and
demonstrate that this combination equals Stopo[T 2]. The same KPLW prescription
122
was studied in Ref. [125], but here we provide a rigorous proof of the equivalence
between the entanglement entropy from the KPLW prescription Eq. (5.14) and the
TEE Stopo[T 2], as we derive in Eq. (5.17). Via Eq. (5.12), we can then obtain the
TEE across a general surface.
A CB
Figure 5.1: KPLW prescription of entanglement surface T 2. The space inside the two
torus is divided into three regions, A, B and C, each being a solid torus.
We generalize the KPLW prescription to (3+1)D by considering the configuration
of the entanglement regions shown in Fig. 5.1 and computing the combination of EEs
SKPLW[T 2] ≡ S(A) + S(B) + S(C)− S(AB)
−S(AC)− S(BC) + S(ABC). (5.14)
Following similar arguments in Ref. [20], it can be shown that SKPLW[T 2] satisfies two
properties:
1. SKPLW[T 2] is insensitive to local deformations of the entanglement surface.
2. SKPLW[T 2] is insensitive to local perturbations of the Hamiltonian.
We first argue that the property 1 holds. If we locally deform the common bound-
ary of region A and B (but away from the common boundary of region A, B and C,
123
which is a line), the deformation of SKPLW[T 2] is
∆SKPLW[T 2] = [∆S(A)−∆S(AC)]
+ [∆S(B)−∆S(BC)].
(5.15)
Because the deformation is far away from region C (farther than the correlation
length ξ ' 1/m, where m is the energy gap), ∆S(A) − ∆S(AC) = 0, and similarly
∆S(B) − ∆S(BC) = 0. Hence SKPLW[T 2] is unchanged under the deformation of
common boundary of A and B, away from the line which represents the common
boundary of A, B and C. If we now locally deform the common boundary of regions
A, B and C 3 (the line A ∩ B ∩ C),
∆SKPLW[T 2] ≡∆S(A) + ∆S(B) + ∆S(C)−∆S(AB)
−∆S(AC)−∆S(BC)
= [∆S(DBC)−∆S(BC)] + [∆S(DAC)
−∆S(AC)] + [∆S(DAB)−∆S(AB)],
(5.16)
where region D is the complement of the region ABC, i.e., D = (ABC)c, and we
have used Ac = DBC and S(A) = S(Ac). Since the deformation is far from region D
(farther than the correlation length ξ) as it is acting only on the line A∩B∩C, each of
three square brackets vanishes separately. Hence SKPLW[T 2] is unchanged under the
deformation of the common boundary line of A, B and C. In summary ∆SKPLW[T 2] =
0 under an arbitrary deformation of the entanglement surface. Therefore property 1
holds.
We now argue that property 2 holds. As suggested in Refs. [20, 21], when we
locally perturb the Hamiltonian far inside one region4, for instance region A, the
3We should distinguish between the common boundary of A, B and C, which is a line A∩B∩C,and the boundary of region ABC, which is a surface
4Quantitatively, the shortest distance d between the position of the local deformation and theentanglement surface should be much longer than the correlation length ξ ' 1/m, i.e., d� ξ.
124
finiteness of the correlation length ξ guarantees that the perturbation does not affect
the reduced density matrix for the region Ac. Therefore the entanglement entropy
S(A) = S(Ac) is unchanged. If a perturbation of the Hamiltonian occurs on the
common boundary of multiple regions, for example region A and B, one can deform
the entanglement surface using property 1 such that the perturbation is non-vanishing
in one region only. This shows that SKPLW[T 2] is invariant under local deformations
of the Hamiltonian which does not close the gap (i.e., those which leave ξ <∞), and
property 2 holds. In summary SKPLW[T 2] is a topological and universal quantity.
Lastly we show that the combination SKPLW[T 2] equals the TEE, Stopo[T 2], i.e.,
SKPLW[T 2] = Stopo[T 2], (5.17)
where Stopo[T 2] is defined in Eq. (5.12). We insert the expansion of the EE (5.2) in the
definition of SKPLW[T 2]. First, it is straightforward to check that the KPLW combina-
tion of the area law terms cancel. Second, the KPLW combination of the Euler char-
acteristic terms vanish since each region in the KPLW combination is topologically a
T 2, and χ(T 2) = 0. Third, as we prove in Appendix D.4, the KPLW combination of
the mean curvature terms vanishes as well, i.e,
4F ′2
∫∂A+∂B+∂C−∂AB−∂AC−∂BC+∂ABC
d2x√hH2 = 0. (5.18)
This was assumed implicitly in Ref. [125], but we demonstrate it explicitly here so
as to close the loop in the argument.
Finally, the KPLW combination simplifies to Stopo[T 2]: it is given by the sum
of the TEE across the four tori ∂A, ∂B, ∂C and ∂ABC, minus the TEE across the
three tori ∂AB, ∂AC and ∂BC. Therefore, Eq. (5.17) holds. In summary, we have
demonstrated that the KPLW prescription, Eq. (5.14), gives a concrete method to
extract the TEE for a generic (non-fixed-point) theory.
125
5.2 Application: Entanglement Entropy of Gener-
alized Walker-Wang Theories
In this section, we construct lattice ground state wave functions for a class of TQFTs
known as the generalized Walker-Wang (GWW) models, whose actions are given by
Eq. (5.19) below. We then compute the EE across various two dimensional entangle-
ment surfaces. The calculations in this section are independent of the SSA inequality
used in Sec. 5.1. The calculations in this section provide support for our assumptions
about the entanglement entropy for fixed-point models, and suggest a conjecture
about higher dimensional topological phases.
The GWW models are described by a TQFT with the action[133, 134, 135]
SGWW =
∫n
2πB ∧ dA+
np
4πB ∧B, n, p ∈ Z. (5.19)
The Walker-Wang models correspond to the special cases p = 0 and p = 1. In
Eq. (5.19) B is a 2-form U(1) gauge field and A is a 1-form U(1) gauge field. (When
we formulate the theory on a lattice, they will be Zn valued. See Appendix D.5 for
details.) The gauge transformations of the gauge fields are
A→ A+ dg − pλ,
B → B + dλ,
(5.20)
where λ is a u(1) valued 1-form gauge field (where u(1) is the Lie algebra of U(1))
with gauge transformation λ → λ + df (where f is a scalar satisfying f ' f + 2π),
and g is a compact scalar (i.e., g ' g + 2π). The gauge invariant surface and line
126
operators are respectively
exp(ik
∮
Σ1
B), k ∈ {0, 1, ..., n− 1},
exp(il
∮
γ
A+ ilp
∫
Σ2
B), l ∈ {0, 1, ..., n− 1},
(5.21)
where Σ1 is a closed two dimensional surface, γ is a closed one dimensional loop and Σ2
is an open two dimensional surface whose boundary is γ. The gauge invariance follows
from the compactification of the scalar g and the standard Dirac flux quantization
condition of U(1) gauge field λ:∮γdg ∈ 2πZ and
∮Σ1dλ ∈ 2πZ.5 We will use
canonical quantization to explain that exp(in∮
Σ1B) and exp(in
∮γA+ inp
∫Σ2B) are
trivial operators in App. D.5.
5.2.1 Wave Function of GWW Models
BF Theory: (n, 0)
For simplicity, we first discuss the special case when p = 0, which is referred to as a
BF theory. The action is
SBF =
∫
M4
n
2πB ∧ dA, (5.22)
where A is a 1-form gauge field and B is a 2-form gauge field. The theory is defined
on a spacetime which is topologically a four ball, M4 ' B4, whose boundary S3 is
a spatial slice, as shown in Fig. 5.2. In the following, we formulate the theory on a
triangulated spacetime lattice. The 1-form gauge field A corresponds to 1-cochains
A(ij) ∈ 2πnZn living on 1-simplices (ij). The 2-form gauge field B corresponds to
5The Dirac flux quantization of the U(1) gauge field λ can be derived as follows:∮
Σ1dλ =∫
Σ+1dλ+ −
∫Σ−1
dλ− =∫∂Σ+
1λ+ −
∫∂Σ−1
λ−, where Σ+1 ∪ Σ−1 = Σ1 and the minus sign of the Σ−1
term is due to orientation. We use λ+ and λ− to emphasis that the gauge field are evaluated in Σ+1
and Σ−1 respectively. The U(1) gauge symmetry implies that λ+ − λ− on the common boundary∂Σ+
1 = ∂Σ−1 = Σ+1 ∩ Σ−1 does not have to vanish, but it can be a pure gauge df . Therefore,∮
Σ1dλ =
∮Σ+
1 ∩Σ−1df ∈ 2πZ. This proves the Dirac flux quantization.
127
M4
S3
l
S
S0
Figure 5.2: A schematic figure of the topology of spacetimeM4 and space S3. InsideS3, we schematically draw a loop l representing the loop configurations C of the Bfield in the dual lattice. The dashed surface S bounding the loop l extends into thespacetime bulkM4, representing the B field configuration in the dual lattice of space-time. S ′ represents the B field configurations that form closed surfaces away fromthe boundary of the spacetime ∂M4. The boundary condition in the path integralEq. (5.23) is specified by a fixed B configuration C on S3. The path integral shouldintegrate over all the configurations in the spacetime bulk M4 with the boundaryconfiguration C on S3 fixed.
2-cochains B(ijk) ∈ 2πnZn living on 2-simplices (ijk)6. We define the Hilbert space
to be H = ⊗(ijk)H(ijk), where H(ijk) is a local Hilbert space on the 2-simplex (ijk)
spanned by the basis |B(ijk)〉 = |2πq/n〉, q ∈ Zn.7 More details about the lattice
formulation of the TQFT are given in Appendix D.5.
We now discuss the ground state wave function for this theory. The ground state
wave function is defined on the boundary of the open spacetime manifold S3 = ∂M4
as[136, 137]
|ψ〉 = C∑
C,C′
∫
C′|∂M4
DA∫
C|∂M4
DB exp(in
2π
∫
M4
B ∧ dA)|C〉, (5.23)
6We use i, j, k to label vertices, and (ij), (ijk) to label 1-simplices and 2-simplices with thespecified vertices.
7Note that the Hilbert space on each 1-simplex is defined independently, and does not have tosatisfy the closed loop (Gauss law) constraint Eq. (5.25).
128
where C ′ and C indicate the boundary configurations for the A and B fields respec-
tively, i.e., the value of A and B fields on ∂M4. We integrate over all A and B
subject to the boundary conditions C ′ and C. C is a normalization factor. Because
A and B are canonically conjugate, the states are specified by the configuration of B
only; |C〉 is a specific state corresponding to the particular B field configuration C on
∂M4. The summation over C ranges over all possible configurations of B-cochain with
weights determined by the path integral. C|∂M4 means the path integral is subject
to the fixed boundary conditions C on ∂M4, and similarly for C ′|∂M4 . If we take the
spacetime M4 to be a closed manifold, Eq. (5.23) reduces to the partition function
overM4. Because the spacetime is topologically a 4-ball B4, there is only one ground
state associated with the boundary S3.8
We first work out the wavefunction for the BF theory with n = 2 explicitly as a
generalizable example. We use B field values as a basis to express |C〉. Integrating
out A (notice that we both integrate over the configurations of the A-field with
fixed boundary configurations and also sum over the boundary configurations, i.e.,∑C′∫C′|∂M4
DA, which is tantamount to integrating over all configurations of A), we
get the constraint δ(dB),
|ψ〉 = C∑
C
∫
C|∂M4
DBδ(dB)|C〉. (5.24)
where the delta function δ(dB) constrains dB(ijkl) = 0 mod 2π on each tetrahedron
(ijkl) in M4. Concretely,
dB(ijkl) = B(jkl)−B(ikl) +B(ijl)−B(ijk)
= 0 mod 2π.
(5.25)
8Topologically degenerate ground states are the representation of line and surface operators whichwrap around the nontrivial spatial cycles. Since there are no nontrivial 1-cycles and 2-cycles in thespatial manifold S3 that line and surface operators can wrap around, the ground state is topologicallynon-degenerate.
129
Any B configuration satisfying this constraint is said to be flat (see Appendix D.5
for details). Since B(ijk) ∈ {0, π},∀i, j, k for the n = 2 theory, Eq. (5.25) means
that for each tetrahedron, there are an even number of 2-simplices where B(ijk) = π
mod 2π, and an even number of 2-simplices with B(ijk) = 0 mod 2π. We refer to
the π 2-simplices as occupied and to the 0 2-simplices as unoccupied.
i
j l
k
d
a
e
b
c
Figure 5.3: A tetrahedron is drawn with solid lines, and its dual is drawn in dash
and gray lines. The 2-simplex (ijk) in the original lattice is dual to the 1-simplex
(ab) in the dual lattice. Similarily, (ikl) is dual to (ad), (ijl) is dual to (ca) and
(jkl) is dual to (ea). The colored dash arrows indicate the orientations of the four
2-simplices, where (ijk) and (ikl) share the same orientation, and (ijl) and (jkl)
share the opposite orientation. The orientations of the dual-lattice 1-simplices are
also indicated by the arrows on the grey/dashed lines.
It is more transparent to consider the configurations in the dual lattice of the
spatial slice S3. (In the next paragraph, we will discuss the dual lattice configurations
in the spacetime M4.) As an example, the dual lattice of a tetrahedron is shown in
Fig. 5.3. The 2-simplices in the original lattice are mapped to 1-simplices in the dual
lattice.9 A 2-cochain B(ijk) defined on a 2-simplex in the original lattice is mapped to
9The dual lattice of a triangulation is not necessarily a triangulation. For example, the dual latticeof a triangular lattice in two dimensions is a honeycomb lattice. Therefore, it is inappropriate totalk about cochains and simplices in the dual lattice of a triangulation. However, we will still use
130
a 1-cochain B(ab) defined on an 1-simplex in the dual lattice. If B(ijk) = π, then we
define the corresponding B(ab) = π in the dual lattice. In the dual lattice, Eq. (5.25)
means that there are an even number of occupied bonds (1-simplices) associated with
each vertex, as well as an even number of unoccupied bonds. If we glue different
tetrahedra together, we find that the occupied bonds in the dual lattice form loops.
Pictorially, this is reminiscent of the wave function of the toric code model in one
lower dimension[138, 22, 139].
In the (3 + 1)D spacetime M4 [rather than the 3D space S3], 2-simplices are
dual to the (4 − 2) = 2-simplices [rather than the 1-simplices] in the dual lattice.
Equation (5.25) means the occupied 2-simplices form continuous surfaces in the dual
spacetime lattice. (Continuous means that the simplices in the dual lattice connect via
edges, rather than via vertices. We discuss the continuity of the dual lattice surfaces
in Appendix D.6.) If these surfaces are inside the bulk of the spacetime and do not
touch ∂M4 (such as S ′ in Fig. 5.2), they are continuous and closed surfaces; if the
surfaces intersect with the spatial slice ∂M4 (such as S in Fig. 5.2), the intersections
are closed loops in ∂M4.
For the BF theory with a general coefficient n, the wavefunction is also a su-
perposition of loop configurations. The only difference is that the loops are formed
by 1-simplices in the dual lattice with B = 2πn
. When there is a loop formed by
1-simplices with B = 2πln
in the dual lattice, we regard the loop as composed of l
overlapping loops formed by the same 1-simplices with B = 2πn
. We emphasize that
the loop configuration is enforced by the flatness condition Eq. (5.25). For n > 2,
we need to specify the orientations of the simplices and keep tract of the signs in
Eq. (5.25). The orientation of each simplex is specified in Fig. 5.3, where the ori-
entations of (jkl) and (ijl) are pointing into the tetrahedron, while the orientation
of (ikl) and (ijk) are pointing out of the tetrahedron. For example, if the values
such notions for simplicity as long as the context is clear. In the dual lattice, we use “1-simplex” todenote a link, and “1-cochain” to denote a discretized 1-form on the link.
131
of the B-cochains are B = 2πq1/n, 2πq2/n, 2πq3/n, 0 with q1 − q2 + q3 = 0 on the
2-simplices (jkl), (ikl), (ijl), (ijk) respectively, the dual of (jkl) and (ikl) (i.e., (ea)
and (ad)) belong to one loop in the dual lattice, while the dual of (ijl) and (ikl)
(i.e., (ca) and (ad)) belong to another loop in the dual lattice. Note that the two
loops share the same dual lattice bond (ad) where the value of the B-cochain is the
sum of the B values from the two loops B(ad) = 2π(q1 + q3)/n = 2πq2/n. The gauge
transformation, B(ijk)→ B(ijk)+λ(jk)−λ(ik)+λ(ij), preserves Eq. (5.25). Hence,
although it deforms the position of loops, it never turns closed loops into open lines.
Open lines in the dual lattice violate the flatness condition Eq. (5.25), and so do
not contribute to the wave function Eq. (5.24). Summing over the configurations C
ensures gauge invariance of the wave function. Notice that Eq. (5.24) implies that
the weights associated with different loop configurations C are equal, similar to the
toric code. Thus we see that Eq. (5.24) reduces to
|ψ〉 = C∑
C∈L
|C〉, (5.26)
where the sum is taken over the set L of all possible loop configurations C at the
spatial slice S3 = ∂M4. This is termed “loop condensation”, since the wave function
is the equal weight superposition of all loop configurations in the dual lattice.
General Case: (n, p)
In this section, we consider GWW models with nontrivial p described by the action
in Eq. (5.19), where A is still a 1-form and B a 2-form. Canonical quantization of
the GWW theories implies that B ∈ 2πnZn on the lattice (see Appendix D.5 for more
details).
In order to find the ground state wave function, we still use B as the basis to label
the configurations C and the corresponding states |C〉 on the spatial slice. The wave
132
function is formally given by
|ψ〉 =C∑
C,C′
∫
C′|∂M4
DA∫
C|∂M4
DB exp(in
2π
∫
M4
B ∧ dA+ inp
4π
∫
M4
B ∧B)|C〉. (5.27)
For simplicity, we consider the case n = 2, p = 1 in the following. As in the BF
theory, we first integrate out the A fields, yielding
|ψ〉 = C∑
C
∫
C|∂M4
DB δ(dB)
exp(i
2
4π
∫
M4
B ∧B)|C〉. (5.28)
The difference between this wave function and that of the BF theory, Eq. (5.24), is that
when the flatness condition δ(dB) is satisfied, the states with different configurations
C are associated with different weights. The weights are determined by the integral
exp(i
2
4π
∫
M4
B ∧B), (5.29)
where B must satisfy the flatness condition dB = 0 with the boundary condition
labeled by C.
We proceed to evaluate the integral in Eq. (5.29). Notice that the flatness condi-
tion, Eq. (5.25), implies that the 2-simplices at which B = π form two-dimensional
spacetime surfaces in the dual lattice of M4 whose boundaries on the spatial slice
S3 are closed loops belonging to C. Relegating the details of the derivation to Ap-
pendix D.7, we show that when B = π only at two dual lattice surfaces S1, S2, whose
boundaries are dual lattice loops l1 = ∂S1, l2 = ∂S2 in C, it follows that
exp(i
2
4π
∫
M4
B ∧B)
= exp(iπlink(l1, l2) + i
π
2link(l1, l1) + i
π
2link(l2, l2)
). (5.30)
The first term is associated with the mutual linking number, link(l1, l2), between dif-
ferent loops, while the second and the third terms are associated with the self-linking
133
number, link(li, li), of one loop, li, with itself, defined in Appendix D.7. Equation
(5.30) can be generalized to configurations with many loops, and the weights of differ-
ent configurations are determined by the linking numbers of the loops. In summary,
the ground state wave function for the (n, p) = (2, 1) theory is:
|ψ〉 = C∑
C∈L
(−1)#(Mutual links)i#(Self links)|C〉. (5.31)
For general (n, p), a similar argument can be made. B can now take n different
values 2πkn, k = 0, 1, · · · , n−1 on each 2-simplex in the lattice, or on each 1-simplex in
the dual lattice. Due to the constraint of Eq. (5.25), the 1-simplices where B = 2π/n
form loops in the dual lattice. Similar to the discussion of the case p = 0 and
general n, two dual-lattice loops can touch in one tetrahedron. We also regard a
loop with B = 2πq/n to be q overlapping loops with B = 2π/n. If there are q1
loops with B = 2π/n that are overlapping on l1 (which is equivalent to one loop with
B = 2πq1/n on l1) and q2 loops with B = 2π/n that are overlapping on l2 (which is
equivalent to one loop with B = 2πq2/n on l2), then
exp(inp
4π
∫
M4
B ∧B)
= exp[2inp(2π)2q1q2
4πn2link(l1, l2) + i
np(2π)2q21
4πn2link(l1, l1) + i
np(2π)2q22
4πn2link(l2, l2)
]
= exp[i2πpq1q2
nlink(l1, l2) + i
πpq21
nlink(l1, l1) + i
πpq22
nlink(l2, l2)
].
(5.32)
Therefore after evaluating these weights, the wave function Eq. (5.27) reduces to
|ψ〉 = C∑
C∈L
ei2πpn
#(Mutual links)eiπpn
#(Self links)|C〉, (5.33)
134
where the mutual-linking and self-linking numbers are counted with multiplicities q1
and q2 as given in Eq. (5.32). The sum over C ∈ L contains configurations with all
possible q1 and q2.
5.2.2 Entanglement Entropy of GWW Models
In this section, we show that the constant part of the EE of GWW theories depends on
the topology of the entanglement surface in a nontrivial way. In particular, Sc[S2] 6=
Sc[T2] in general. Hence, Sc[S
2] and Sc[T2] are truly independent quantities.
This section is divided into two parts: In Sec. 5.2.2, we calculate the EE for GWW
models with arbitrary (n, p) across the entanglement surface T 2. In Sec. 5.2.2, we
compute the EE for GWW models across closed surfaces with arbitrary genus and
an arbitrary number of disconnected components. These independent calculations
confirm Eq. (5.10).
EE for the Torus, n = 2, p = 1
In this subsection, we compute the EE of GWW models across Σ = T 2. For simplicity,
we first consider the case n = 2, p = 1, and then generalize to models with arbitrary
n and p.
We start with the wave function obtained in the last section, Eq. (5.31):
|ψ〉 = C∑
C
(−1)#(Mutual links)i#(Self links)|C〉. (5.34)
We choose the subregion A to be a solid torus whose surface is T 2, and Ac to be the
complement of A. We illustrate the microscopic structure of the spatial partitioning
in Fig. 5.4 via a lower-dimensional example. The entanglement surface Σ is chosen to
be a smooth surface in the real spatial lattice (green simplices in Fig. 5.4). The real
space simplices that form the entanglement surface Σ are counted as part of region
135
A
⌃
B=π simplices on the entanglement surface ∑
Ac
Figure 5.4: An example of the lattice structure of an entanglement cut in (2 + 1)D.The green simplices form the entanglement cut Σ, which partitions the lattice intoregion A and region Ac. We include Σ as part of region A. B = π on the red simplices,while B = 0 elsewhere. The dotted loop is the dual lattice configuration of the redsimplices. In this example, the configuration CE contains two B = π 1-simplices atthe entanglement cut Σ, which are the fourth and eighth 1-simplices of Σ (countingfrom the left side) as shown in the figure.
A. 10 We will find the Schmidt decomposition of the wavefunction corresponding to
this spatial partitioning in order to calculate the EE. To do so, we first parametrize
the configurations C appearing in Eq. (5.34) as:
C 7→ {CE, (a, α), (b, β)}, (5.35)
which we now explain. CE labels the real space B-cochain configuration at the entan-
glement surface Σ. (In Fig. 5.4, the fourth and the eighth green 1-simplices (counting
from the left side) are occupied on the entanglement surface Σ, which also belong
to region A according to our partition.) We denote by NA(CE) the number of con-
figurations in the region A (but not including Σ) consistent with the choice of CE.
We label such configurations by (a, α), where α is the parity (even e or odd o) of
the number of occupied loops winding around the nontrivial spatial cycle inside the
10There are other choices of spatial partitioning. For example, we can count the real simplices thatform the entanglement surface as part of region Ac. We will consider only consider the partitioningmentioned in the main text for definiteness.
136
region A in the dual lattice, and the configurations of either parity are enumerated by
a = 1, . . . , NA(CE)/2.11 Similarly, (b, β) labels the NAc(CE) configurations in region
Ac. Figure 5.5 presents a particular configuration where, besides two contractible
dual-lattice loops, there is one dual lattice loop wrapping the non-contractible cycle
in the dual lattice of region A and one dual lattice loop wrapping the non-contractible
cycle in the dual lattice of region Ac, which corresponds to α = o and β = o. Note
that two non-contractible cycles are in different regions A and Ac. To be illustrative,
we also draw 2-simplices in the real lattice where B = π whose dual configurations
form loops in the space. Hence the summation over C splits as:
∑
C
=∑
CE
NA(CE)/2∑
a=1
NAc (CE)/2∑
b=1
∑
α=e,o
∑
β=e,o
. (5.36)
11We can establish a one-to-one correspondence between the configurations of loops in the evenparity sector and the odd parity sector. If we start with a configuration in the even parity sectorin which k dual lattice loops wrap around the non-contractible cycle in region A, we can obtain aconfiguration in the odd parity sector by adding a single loop wrapping the non-contractible cycle sothat there are (k+1) non-contractible dual lattice loops in total. Similarly, we can start with the oddparity sector and obtain the even parity sector. This demonstrates that the number of configurationsin the even parity sector is equal to that of the odd parity sector. Therefore, we denote the numberof configurations in both sectors by NA(CE)/2. This argument can be generalized to the case ofgeneral n.
137
A
Ac
�4
�3 �2
�1
Figure 5.5: A particular spatial configuration with one loop γ1 (dashed line) threading
through the hole (the hole itself belongs to region Ac) inside the region A and one
loop γ2 (grey line) threading through the hole inside the region Ac. γ3 and γ4 are
two linked contractible loops, where γ3 locates inside region A, and γ4 locates both in
region A and Ac. The two blue points are the intersection of l4 with Σ. The simplices
(gray triangles) are living in the real lattice where B = π. The lines perpendicular to
the simplices are living in the dual lattice where B = π and they form loops in the
dual lattice. This configuration corresponds to α = o, β = o.
For convenience we also introduce the notation
lCEa,e =(−1)#(Mutual links with fixed CE configuration of region A in even sector),
lCEa,o =(−1)#(Mutual links with fixed CE configuration of region A in odd sector),
sCEa,e =i#(Self links with fixed CE configuration of region A in even sector),
sCEa,o =i#(Self links with fixed CE configuration of region A in odd sector),
(5.37)
where even/odd sector refers to the set of states with an even/odd number of loops in
the dual lattice threading the non-contractible cycle in region A. Similar definitions
apply to region Ac. See Fig. 5.5 for an illustration. We further define |ACEa 〉α to
be a state associated with one particular configuration in region A, which is labeled
138
by {CE, a, α}, and define |AcCEb 〉β likewise in region Ac. There is a subtlety: we
also need to specify the mutual-linking/self-linking number of loops which cross the
entanglement surface. We specify that when two loops (among which at least one of
them crosses the entanglement surface) are linked, such as γ3 and γ4 in Fig. 5.5, the
mutual-linking number is counted as part of the A side, i.e., lCEa,e and lCEa,o. Additionally,
when a loop crosses the entanglement surface, the self-linking number of the loop
is counted as part of the A side, i.e., sCEa,e and sCEa,o. We are able to make such a
choice because there is a phase ambiguity in the Schmidt decomposition, and phases
can be shuffled between A and Ac by redefining the basis |ACEa 〉e/o and |AcCEb 〉e/o.
(For example, we can define another set of states via |ACEa 〉e/o = sCE−1a,e/o |ACEa 〉e/o, and
|AcCEb 〉e/o = sCEa,e/o|Ac
CEb 〉e/o.) As we will see, the reduced density matrix Eq. (5.39)
does not depend on the choice of phase assignment. Combining the above, we get
|ψ〉 =C∑
CE
NA(CE)/2∑
a=1
NAc (CE)/2∑
b=1
∑
α=e/o
∑
β=e/o
(−1)αβlCEa,αlCEb,βs
CEa,αs
CEb,β|ACEa 〉α|Ac
CEb 〉β. (5.38)
The factor (−1)αβ, which equals −1 when α = β = o and 1 otherwise, reflects the
mutual-linking between the non-contractible loops in region A (such as γ1 in Fig. 5.5)
and the non-contractible loops in region Ac (such as γ2 in Fig. 5.5). Figure 5.5 shows
a special configuration where there is one non-contractible loop in region A and one
non-contractible loop in region Ac.
From this we easily obtain the reduced density matrix for region A by tracing over
the Hilbert space in region Ac,
ρA =|C|2∑
CE
NAc(CE)
2
NA(CE)/2∑
a,a=1
∑
α,α,γ=e,o
(−1)(α−α)γ|ACEa 〉α〈ACEa |α
=|C|2∑
CE
NAc(CE)
NA(CE)/2∑
a,a=1
(|ACEa 〉e〈ACEa |e + |ACEa 〉o〈ACEa |o
),
(5.39)
139
where we have performed unitary transformations on the bases |ACEa 〉e/o and |AcCEb 〉e/o
to absorb the mutual-linking and self-linking factors within region A and region
Ac respectively. The transformed bases are denoted |ACEa 〉α = lCEa,αsCEa,α|ACEa 〉α and
|AcCEb 〉β = lCEb,βs
CEb,β|Ac
CEb 〉β.
Furthermore, the constraint
TrHA(ρA) = |C|2
∑
CE
NAc(CE)NA(CE) = 1 (5.40)
fixes the normalization constant C. For each fixed configuration CE on the entangle-
ment surface, the product of the number of configurations in the region A and the
number of configurations in region Ac, i.e., NAc(CE)NA(CE), is independent of CE (see
Appendix D.8 for details). Thus, to compute C we need only to count the number
of different choices of CE. There are in total 2|Σ|−1 different boundary configurations,
where the 1 comes from the constraint that closed dual lattice loops always inter-
sect the entanglement surface twice (hence the number of occupied 1-simplices on Σ
is even), and |Σ| is the number of 2-simplices on the entanglement surface. Since
|C|2NAc(CE)NA(CE) is independent of CE, and there are 2|Σ|−1 choices of CE,
|C|2NAc(CE)NA(CE) =1
2|Σ|−1. (5.41)
We give a more detailed derivation of this formula in Appendix D.8.
From the reduced density matrix ρA, we can calculate the entanglement entropy
of the ground state |ψ〉 associated with the torus entanglement surface by the replica
trick,
S(A) = −TrHAρA log ρA = − d
dN
(TrHA
ρNA(TrHA
ρA)N
)∣∣∣∣N=1
(5.42)
140
Using Eq. (5.39),
TrHAρNA
= |C|2N∑
CE0
NA0/2∑
a0=1
∑
α0=e,o
〈ACE0a0 |α0
N∏
I=1
(∑
CEI
NAI/2∑
aI ,aI=1
∑
αI=e,o
NAc(CEI )|ACEIaI 〉αI 〈A
CEIaI|αI)|ACE0
a0 〉α0
= |C|2N∑
CE0,a0,α0
N∏
I=1
( ∑
CEI ,aI ,aI ,αI
NAc(CEI )
)δCE0
CE1δCE1
CE2· · · δCEN CE0
× δa0a1δa1a2δa2a3 · · · δaN−1aN δaNa0
× δα0α1δα1α2 · · · δαNα0
= |C|2N∑
CE0
NAc(CE0)N∑
α0=o,e
NA(CE0/2)∑
a1=1
· · ·NA(CE0
/2)∑
aN=1
1
= |C|2N∑
CE0
2NAc(CE0)N(NA(CE0)
2
)N= 2−|Σ|(N−1).
(5.43)
In the first equation, we expand the trace over the Hilbert space in region A. In the
second equation, we use the orthogonal condition 〈ACEa |α|AC′Ea′ 〉α′ = δCEC′Eδaa′δαα′ . In
the third equation, we simplify the formula using the delta functions CE0 = CE1 =
· · · = CEN , α0 = α1 = · · · = αN , and eliminate {a0, aI} by {aI}. In the last equation,
we used Eq. (5.41). Moreover, notice that TrHAρA = 1, we obtain the entanglement
entropy
S(A) = − d
dN2−|Σ|(N−1)|N=1 = |Σ| log 2. (5.44)
Since |Σ| is the number of 2-simplices on Σ, which is proportional to the area of
Σ, hence it is the area law term. Since there is no constant term, the topological
entanglement entropy is trivial, reflecting the absence of topological order in this
model.
141
EE for the Torus: general (n, p)
We carry out the analogous calculations for a general GWW theory with arbitrary
coefficients n and p. We start by writing down the ground state wave function,
|ψ〉 =C∑
CE
NA(CE)/n∑
a=1
NAc (CE)/n∑
b=1
n−1∑
α,β=0
ei2πpαβn lCEa,αl
CEb,βs
CEa,αs
CEb,β|ACEa 〉α|Ac
CEb 〉β, (5.45)
where lCEa,α, lCEb,β, s
CEa,α, s
CEb,β are straightforward generalizations of Eq. (5.37) to the cases
with arbitrary coefficients p and n, c.f. Eq. (5.33). The reduced density matrix is
ρA = |C|2∑
CE
NAc(CE)
n
NA(CE)/n∑
a,a=1
n−1∑
α,α,γ=0
ei2πp(α−α)γ
n |ACEa 〉α〈ACEa |α, (5.46)
where we again performed the unitary transformations to absorb the self-linking and
mutual-linking factors, and denote the resulting new basis as |ACEa 〉α and |AcCEb 〉β.
For the same reason as in Eq. (5.41),
|C|2NAc(CE)NA(CE) =1
n|Σ|−1, (5.47)
where |Σ| is the number of 2-simplices on the entanglement surface.
In order to compute the entanglement entropy
SA = −TrHAρA log ρA, (5.48)
we first calculate the entanglement spectrum, i.e., we diagonalize ρA. As a first step,
we carry out the sum over γ in Eq. (5.46). We note that the sum is nonvanishing
only if p(α− α)/n is an integer, in which case the sum takes the value n. Thus,
n−1∑
γ=0
ei2πp(α−α)γ
n = n δ
(α− α = 0 mod
n
gcd(n, p)
). (5.49)
142
We find
ρA = |C|2∑
CE
NAc(CE)
NA(CE)/n∑
a,a=1
n−1∑
α,α
δ
(α− α = 0 mod
n
gcd(n, p)
)|ACEa 〉α〈ACEa |α(5.50a)
=∑
CE,a,α,a,α
[ρCEA
]a,α;aα
|ACEa 〉α〈ACEa |α, (5.50b)
where[ρCEA
]a,α;aα
are matrix elements given by
[ρCEA
]a,α;aα
= |C|2NAc(CE)[1 n
gcd(n,p)⊗ Jgcd(n,p)
]αα⊗[JNA(CE)
n
]aa. (5.50c)
Here, 1m is the m ×m identity matrix, and Jl is an l × l matrix of ones (which has
one nonzero eigenvalue equal to l). The first term in this expression originates from
the periodic delta function in Eq. (5.50a), and the second term comes from the sum
over a, a in the outer product. Noting that each Jm is a rank one matrix with nonzero
eigenvalue m, we see immediately that ρCEA can be put in diagonal form
ρCEA = |C|2NAc(CE)NA(CE)
ngcd(n, p)(1 n
gcd(n,p)⊕ 0NA(CE)−n/gcd(n,p)). (5.51)
The matrix in Eq. (5.51) is
1
1
. . .
1
0
0
. . .
0
0
0
ngcd(n,p) 1’s
NA (C
E )−ngcd(n,p) 0’s
(5.52)
143
Finally, using Eq. (5.47), we find that the nonzero entanglement eigenvalues are
given by
e−ξCE,r =gcd(n, p)
n|Σ|, (5.53)
where r = 1, · · · , n|Σ|/gcd(n, p). With this spectrum, it is straightforward to evaluate
Eq. (5.48) to obtain the entanglement entropy as
S(A) = |Σ| log n− log gcd(n, p). (5.54)
The first term is proportional to the area of the entanglement surface. The second
constant term is the TEE 12:
STQFTc (A) = Stopo(A) = − log gcd(n, p). (5.55)
We see that the TEE depends nontrivially on the parameters n and p. If n and p
are coprime, i.e., gcd(n, p) = 1, the TEE vanishes. If p = 0, using the definition
gcd(n, 0) = n, the constant part of the EE reduces to − log n. Alternatively, we can
also compute the EE of the BF theory using the wave function Eq. (5.26), and we
find the constant part to be − log n.
Note that this result is consistent with Refs. [134] and [140] where the ground
state degeneracy (GSD) on T 3 was computed to be gcd(n, p)3. The ground state
degeneracy suggests that the GWW models can be topologically ordered, which, in
our context, is reflected by the nonzero TEE, − log gcd(n, p). When gcd(n, p) = 1,
the ground state on T 3 is non-degenerate, and the TEE vanishes. In particular, for
the case of the Walker-Wang model n = 2, p = 1, we obtain
S(A) = |Σ| log 2, (5.56)
12The constant part of the EE is STQFTc (A) = − log gcd(n, p). According to the discussion in
Sec. 5.1, because the entanglement surface is T 2, whose Euler characteristic vanishes, Stopo(A) ≡Stopo[T 2] = STQFT
c (A) = − log gcd(n, p).
144
and there is no topological order. We notice the relation between the GSD on T 3 and
the TEE across the torus T 2,
exp(−3Stopo[T 2]) = GSD[T 3], (5.57)
which should be compared to the similar relation, exp(−2Stopo[T 1]) = GSD[T 2], for
the (2+1)D Abelian theories.
For an Abelian theory in (d+ 1)D, our computation leads us to conjecture that
exp(−dStopo[T d−1]) = GSD[T d]. (5.58)
For (d + 1)D BF theory with level n, we have computed both the TEE and the
GSD[T d], and we found Stopo[T d−1] = − log n and GSD = nd. This is consistent
with our conjecture. (See Appendix D.9 for details.) We conjecture that this re-
lationship is true for more general theories such as Dijkgraaf-Witten models, and
higher dimensional Chern-Simons theories as well. For a generic (2 + 1) dimensional
nonabelian Chern-Simons theory, Eq. (5.58) may not hold. For example, the TEE
of the SU(2)3 Chern-Simons theory is Stopo[T 1] = − log(√
5/(2 sin(π/5)))[141], and
exp(−2Stopo[T 1]) is not an integer. Hence Eq. (5.58) can not hold because the GSD
should be an integer. However, we note that for some nonabelian theories, the con-
jecture still holds. For example, for the bosonic Moore-Read quantum Hall state in
(2 + 1)D, GSD[T 2] = 4 (which consists of 3 states from the even parity sector and 1
state from the odd parity sector), and Stopo[T 1] = − log 2, hence Eq. (5.58) holds in
this case.
EE for Arbitrary Genus
Following the same procedure used for the torus, we calculate the EE across a general
entanglement surface with genus g. (The results are summarized in Table 5.1.) For
145
S2 T 2 [(0, n0), · · · , (g∗, ng∗)]n2πBF
STQFTc − log n − log n −b0 log nStopo − log n − log n −b0 log n
n2πBF + np
4πBB
STQFTc − log n − log gcd(n, p) (−b0 + χ
2) log gcd(n, p)− χ
2log n
Stopo − log gcd(n, p) − log gcd(n, p) −b0 log gcd(n, p)
Table 5.1: Constant part and topological part of the entanglement entropy for gen-eralized Walker-Wang models. STQFT
c is the constant part of the EE for the TQFT,while Stopo is the TEE for a general theory which belongs to the same phase of
the TQFT. b0 is the zeroth Betti number of entanglement surface b0 =∑g∗
g=0 ng.
χ =∑g∗
g=0(2 − 2g)ng is the Euler characteristic of the entanglement surface. In
particular, we have Stopo(S2) = Stopo(T 2).
each hole i (i = 1, · · · , g) of the entanglement surface, we introduce a pair of additional
indices αi and βi that count the number of loops (modulo n) winding around the non-
contractible cycles around the hole in region A and region Ac, respectively. Then the
wavefunction is
|ψ〉 =C∑
CE
NA(CE)
ng∑
a=1
NAc(CE)
ng∑
b=1
n−1∑
α1···αg=0
n−1∑
β1···βg=0
g∏
i=1
ei2πpαiβi
n |ACEa 〉α|AcCEb 〉β.
(5.59)
We collect the set of indices α1, · · · , αg into a index vector α. We first consider the
configurations in region A. Since each hole is associated with an index αi, which can
take n different values, the complete set of indices α can take ng different values.
Hence, the NA(CE) configurations are partitioned into ng classes, where each class
contains NA(CE)/ng configurations. For this reason the summation in Eq. (5.59)
reaches only up to NA(CE)/ng. For region Ac, similar arguments hold. Then the
146
reduced density matrix on a genus g surface takes the form
ρA =|C|2∑
CE
NAc(CE)
ng
n−1∑
α1,··· ,αg=0
n−1∑
α1,··· ,αg=0
n−1∑
γ1,··· ,γg=0
NA(CE)/ng∑
a,a=1
g∏
i=1
ei2πp(αi−αi)γi
n |ACEa 〉α〈ACEa |α
=|C|2∑
CE
NAc(CE)n−1∑
α1,··· ,αg=0
n−1∑
α1,··· ,αg=0
NA(CE)/ng∑
a,a=1
g∏
i=1
δ
(αi − αi = 0 mod
n
gcd(n, p)
)|ACEa 〉α〈ACEa |α
=∑
CE
∑
α,α
NA(CE)/ng∑
a,a=1
[ρCEA
]
aα,aα
|ACEa 〉α〈ACEa |α,
(5.60)
where
[ρCEA
]
aα,aα
=|C|2NAc(CE)
g⊗
i=1
[1 n
gcd(n,p)⊗ Jgcd(n,p)
]
αiαi
⊗[JNA(CE)
ng
]
aa
=|C|2NAc(CE) gcd(n, p)gNA(CE)
ng
[1 ng
gcd(n,p)g⊕ 0NA(CE)− ng
gcd(n,p)g
]
aα,aα
=gcd(n, p)g
n|Σ|+g−1
[1 ng
gcd(n,p)g⊕ 0NA(CE)− ng
gcd(n,p)g
]
aα,aα
.
(5.61)
In the second line of Eq. (5.60), we summed over γ1, · · · , γg using Eq. (5.49). In the
last line of Eq. (5.60) and the first line of Eq. (5.61), we reorganized the coefficients
|ACEa 〉α〈ACEa |α into a matrix form, where 1 ngcd(n,p)
is the identity matrix due to the
delta function, and Jgcd(n,p) is because all elements of α = ngcd(n,p)
k, α = ngcd(n,p)
k
with k, k = 0, 1, · · · , gcd(n, p) − 1 are enumerated, and similar for JNA(CE)
ng. In the
second line of Eq. (5.61), we expand the tensor product. In the last line, we use the
normalization condition |C|2NAc(CE)NA(CE) = 1n|Σ|−1 . We see that all of the non-zero
eigenvalues of the entanglement spectrum are given by 1/Nn,p,g;|Σ|, where
Nn,p,g;|Σ| ≡n|Σ|−χ/2
gcd(n, p)g, χ = 2− 2g. (5.62)
147
χ is the Euler characteristic of Σ. Thus, the EE across a general surface of genus g
is:
S[(0, 0),(1, 0), . . . , (g − 1, 0), (g, 1)]
=|Σ| log n− g log gcd(n, p)− (1− g) log n
=|Σ| log n− χ
2log
n
gcd(n, p)− log gcd(n, p).
(5.63)
Equation (5.63) is consistent with Eq. (5.10). We summarize Stopo(A) and STQFTc (A)
for various systems and various entanglement surfaces in Table 5.1.
We note that although Eq. (5.63) is the EE for a low energy TQFT, there is still
an area law term. Since the TQFT is independent of the metric of the entanglement
surface, one may naively expect that the area law term should vanish. The reason that
the area law term appears in Eq. (5.63) is that we formulated our theory on a lattice,
which explicitly broke the scaling symmetry (i.e., changing the area of the cut changes
the number of links passing through Σ). However symmetry under area-preserving
diffeomorphisms was unaffected by the lattice regularization (changing the shape of
the cut does not change the number of links passing through Σ). Because of this, we
get terms that scale like the area of the cut (area law term), but no further shape-
dependent terms. Therefore, we expect, and indeed find, that the mean curvature
term vanishes for the TQFT (F ′2 → 0).
148
Chapter 6
Anomaly and Dynamics of (3 + 1)d
SU(2) Yang-Mills Theory
6.1 Introduction
The SU(N) Yang-Mills theory is a non-Abelian gauge theory with a gauge group
SU(N) described by the action
S = − 1
4g2
∫
M4
Tr(F ∧ ?F ) +θ
8π2
∫
M4
Tr(F ∧ F ), (6.1)
which admits a topological term parameterized by a variable θ. Since the second
Chern number
c2(VSU(N)) =1
8π2Tr(F ∧ F ) (6.2)
of the SU(N) vector bundle integrates to be an integer, θ is 2π periodic[142, 143].
The theory has a Z2,[1] one form center symmetry[144, 145, 146, 147]. When θ = 0, π
mod 2π, it is also time reversal symmetric.
149
SU(N) Yang-Mills is the simplest non-Abelian gauge theory in 3 + 1d that ex-
hibits rich dynamics. In contrast to the Abelian U(1) Maxwell gauge theory which
is free, the SU(N) Yang-Mills is strongly coupled due to negative beta function,
and the low energy dynamics is prohibitive via merely perturbative approaches[148].
However, various evidences including ’t Hooft anomalies[149, 143], deformation of
supersymmetric Yang-Mills[150, 151, 152], and holographic calculation in the large
N limit[153, 154] provide various constraints on the low energy dynamics, which we
summarize as the Standard Lore of Yang-Mills.
6.1.1 Standard Lore of SU(N) Yang-Mills
We review the dynamics of SU(N) Yang-Mills as a function of θ ∈ [0, 2π).
• θ = 0: When θ = 0, the only term is the kinetic energy of the gauge field, which
is time reversal symmetric. Various evidences including lattice simulations,
softly broken supersymmetry and large N holographic models suggest that the
ground state is confining with an unbroken center symmetry, and there is a
mass gap[148, 143, 150, 151, 152].
• θ = π: Another instance which is time reversal symmetric is when θ = π. In
this case, there is a mixed anomaly between the time reversal symmetry and the
ZN center symmetry for even N , and a more subtle global inconsistency for odd
N [143, 108, 155]. Both cases are unified from the point of view of anomaly in the
space of coupling constants[156, 157]. For even N , this anomaly immediately
constrains that SU(N) Yang-Mills with θ = π can not flow to a trivial phase.
It is widely believed that at the low energy, the theory confines and the center
symmetry is unbroken. However time reversal is spontaneously broken, leading
to two degenerate ground states[142, 158, 143]. Such spontaneous broken of
150
time reversal has been shown for large N Yang-Mills, where as one tunes from
θ < π to θ > π a first order phase transition has been observed[153, 154].
• 0 < θ < π, π < θ < 2π: The dynamics in this regime is less clear, due to the lack
of time reversal symmetry and consequently the anomaly. It is believed that the
theory confines for all θ. This is also supported by the large N calculation[153,
154]. The phase at θ = 2π, although is believed to be dynamically trivial
as θ = 0, differs from the phase at θ = 0 by a subtle symmetry protected
topological (SPT) phase [144, 145, 152, 147, 159].
Although the Standard Lore is believed to hold for large N , there are less evidences
supporting the standard lore for small N . In particular, for N = 2, the SU(2) Yang-
Mills at θ = π can flow to one of the several possible scenarios at low energy. The
low energy theory should either spontaneously break time reversal, or be deconfined,
or preserve time reversal symmetry and being confined while being gapless. 1 As far
as we know, none of the above scenarios has been excluded for N = 2. Therefore, it
is desirable to study all possible scenarios of SU(2) Yang-Mills in detail.
6.1.2 New Aspects: Lorentz Symmetry Enrichments
For any gauge theory with Z2,[1] one form symmetry, and in particular the SU(2) Yang-
Mills with any theta parameter[108, 155], can be enriched by the SO(3, 1) Lorentz
symmetry,2 via fractionalizing the Lorentz symmetry on the Wilson line operators.
This phenomena has been previously explored in [163, 164, 165] and others, and
has been recently termed in [166] poetically as Lorentz symmetry fractionalization.
1Gapped and confined TQFT that preserve time reversal symmetry has been ruled out in arecent work[160]. In [108, 155], the authors constructed a H-symmetry extended TQFT via theexact sequence 1 → K → H → Z2,[1] → 1, generalizing [161, 162] to higher form symmetries. Bydynamically gauging K, it has been realized that Z2,[1] is spontaneously broken, which is consistentwith [160].
2There are two branches of SO(3, 1), differed by chirality. These are denoted as SO±(3, 1) in theliterature. For our purposes, the choice of chirality will not play a role. In the rest of the paper, wefocus on the positive chirality +, and will suppress the superscript for simplicity.
151
Fractionalization of the Lorentz symmetry on a Wilson line requires that the Wilson
line transforms projectively under SO(3, 1), i.e., the self statistics is shifted by h =
1/2. This is done by shifting the background field B for the center Z2,[1] one-form
symmetry by the second Stiefel-Whitney class of the tangent bundle of the spacetime
manifold, i.e.,
B → B +K2w2, (6.3)
where K2 = 0, 1 represents trivial/nontrivial fractionalization.
For θ = 0, π, the SU(2) Yang-Mills is time reversal symmetric. Thus one can
further enrich the SU(2) Yang-Mills by the time reversal symmetry (or O(3, 1) if
combined with the SO(3, 1) Lorentz symmetry)[108, 155]. In this case, time reversal
symmetry can be fractionalized on the Wilson line. Nontrivial fractionalization of
time reversal means that the Wilson line is a Kramers doublet. Formally, this is done
by shifting the background field B by the square of the first Stiefel-Whitney class of
the tangent bundle of the spacetime manifold, i.e.
B → B +K1w21, (6.4)
where K1 = 0, 1 represents trivial/nontrivial fractionalization of time reversal symme-
try. Of course, one can consider enriching the SU(2) Yang-Mills at θ = 0, π by both
time reversal and SO(3, 1) Lorentz symmetry. In [108, 155], we denote the four differ-
ent O(3, 1) Lorentz symmetry enrichments, labeled by (K1, K2), of SU(2) Yang-Mills
at θ = 0, π as the Four Siblings, which we use throughout the present work.
Keeping the symmetry enrichments in mind, it is natural to revisit the standard
lore and ask a more refined question: How the dynamics of SU(N) Yang-Mills
depends on the O(3, 1) symmetry enrichment, i.e. the four siblings (K1, K2)?
152
In this work, we study the dynamics of SU(2) Yang-Mills at θ = π, and focus on two
low energy scenarios.
In the first scenario (to be discussed in section 6.3), time reversal is spontaneously
broken, and we study the domain wall theory that is constrained by the ’t Hooft
anomalies. We highlight several features of our results:
1. The domain wall theory is not time reversal symmetric, in contrast to the bulk.
Instead, there is a discrete unitary symmetry U .
2. The four siblings in the bulk corresponds to four different enrichments of the
Lorentz symmetry as well as the unitary symmetry U on the wall.
3. Even though the ’t Hooft anomaly of SU(2) Yang-Mills does not depend on the
SO(3, 1) Lorentz symmetry enrichment, the ’t Hooft anomaly on the wall does.
In section 6.4, we also discuss the consequences of symmetry enrichments on the
domain wall theories for SU(2) QCD within the regime of chiral symmetry breaking
Nf < NCFT .
The second scenario will be discussed in section 6.5, where we assume that the
low energy of SU(2) Yang-Mills with θ = π is deconfined. In particular, we only
discuss the case where the low energy theory is described by a U(1) Maxwell theory,
with certain Lorentz symmetry enrichment. The Lorentz symmetry enriched U(1)
Maxwell theories have been studied in [167, 168, 169, 170] where they classify the
phases of time reversal U(1) quantum spin liquids. We will use the ’t Hooft anomaly
to constrain the correspondence between the symmetry enrichments of SU(2) Yang-
Mills at θ = π and the symmetry enrichments of the Maxwell theory. In section 6.6,
we further apply this correspondence to study the phase transitions between different
U(1) spin liquids, as well as the phase transitions between U(1) spin liquids and
trivial paramagnets. Amusingly, we find that SU(2) QCD with Nf fermions (Nf >
NCFT ) in the fundamental representation can be interpreted as the second order phase
153
transition between the the above phases, where the gauge group is enhanced at and
only at the transition point. We denote such transition as gauge enhanced quantum
critical points.
6.2 SU(2) Yang-Mills Theory at θ = π
The 4d SU(2) Yang-Mills gauge theory with an SU(2) gauge group and a theta term
in the Minkowski spacetime M4 is described by an action 3
S = − 1
4g2
∫
M4
Tr(F ∧ ?F ) +θ
8π2
∫
M4
Tr(F ∧ F ), (6.5)
where we denote a as the SU(2) gauge field and F = da − ia2 is the field strength.
Since the second Chern number c2(VSU(2)) =∫M4
Tr(F ∧F )/8π2 is quantized to be an
integer, the θ parameter has periodicity 2π.
6.2.1 Time Reversal Symmetry
We first focus on the discrete time-reversal symmetry ZT2 and its symmetry transfor-
mation T acting on the gauge field aµ ≡ aαµTα, where Tα is the generator of SU(2).
T acts on aµ as:
T : aα0 → −aα0 , aαi → aαi , (t, xi)→ (−t, xi). (6.6)
Tα → Tα, a0 → −a0, ai → ai.
The components of the field strength Fα0i, F
αij transforms under T as
T : Fαij = ∂ia
αj − ∂jaαi + fαβγaβi a
γj → ∂ia
αj − ∂jaαi + fαβγaβi a
γj = Fα
ij(−t, xi),
Fα0i = ∂ta
αi − ∂iaα0 + fαβγaβ0a
γi → −∂−taαi + ∂ia
α0 − fαβγaβ0aγi = −Fα
0i(−t, xi).(6.7)
3For definiteness, the spacetime signature is taken to be (−1, 1, 1, 1).
154
where fαβγ is the structure constant of the SU(2) Lie algebra. Under T , the kinetic
term∫M4 Tr(F ∧ ?F ) is invariant, while the θ term flips the sign:
T :θ
8π2
∫
M4
Tr(F ∧ F )→ − θ
8π2
∫
M4
Tr(F ∧ F ). (6.8)
Because θ is 2π periodic, (6.5) is time reversal invariant only when θ = 0, π mod 2π.
6.2.2 One-form Symmetry
(6.5) also has a Z2,[1] one-form center symmetry which acts on the gauge-invariant
Wilson line
We = TrRP exp
(i
∮a
). (6.9)
where P stands for the path ordering. R can be any possible representation of SU(2).
If R is an irreducible representation and let l be the number of boxes in the Young
diagram of R, then We transforms under Ze2,[1] as
Z2,[1] : We → (−1)lWe. (6.10)
In particular, for the fundamental representation, there is only one box in the Young
diagram, hence the Wilson line flips sign under one form symmetry.
The generator of the symmetry Z2,[1] is a co-dimension two surface operator Ue.
We will find below that
Ue = exp(iπ
∮Λ), (6.11)
where Λ ∈ H2(M4,Z2).
It is useful to couple (6.5) to the Z2,[1] background gauge field B. Following [146?
? ], we first promote the SU(2) gauge field a to a U(2) gauge field a,
a = a+1
2AI2. (6.12)
155
where I2 is a two dimensional identity matrix. The first Chern class of the U(2)
bundle is c1 ≡ c1(VU(2)) ≡ TrF2π≡ dA
2πwhere F = da − iaa is the U(2) field strength.
Then we couple to B by requiring c1 = B mod 2, which can be done via introducing
a Lagrangian multiplier Λ:
∫DΛ . . . exp
(iπ
∫
M4
Λ ∪ (c1 −B)
). (6.13)
The minimal coupling exp(iπ∫
Λ∪B) implies that the generator of ZeN,[1] is precisely
exp(iπ∫
Λ), which explains (6.11). Notice that integrating out the Lagrangian multi-
plier Λ removes the U(1) degree of freedom, hence the gauge group is SO(3)=PSU(2)
(rather than SU(2)),
U(2)
U(1)=
(SU(2)× U(1))/Z2
U(1)=
SU(2)
Z2
= PSU(2) = SO(3). (6.14)
with the gauge bundle constraint
c1(VU(2)) = w2(VSO(3)) = B mod 2. (6.15)
6.2.3 Formulating on Unorientable Manifold and Lorentz
Symmetry Fractionalization
As we are focusing on the time reversal symmetric theory, one should be tempted
to formulate the theory (6.5) on an unorientable manifold. The Lorentz symmetry
associated with an unorientable manifold is O(3,1). In particular, on a generic un-
orientable manifold, both the first and second Stiefel-Whitney classes, w1 and w2, of
the tangent bundle of the spacetime manifold M4 are allowed to be nontrivial. One
can twist the gauge bundle constraint (6.15) as
c1(VU(2)) = B +K1w21 +K2w2 mod 2, K1, K2 = 0, 1. (6.16)
156
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Figure 6.1: Lorentz symmetry fractionalization on the Wilson line. The left panel isthe Wilson line with K1 = K2 = 0. When the background field B for the one-formsymmetry is activated, the Wilson line is attached to a surface Σ bounded by γ.This means that the Wilson line carries charge 1 under Z2,[1]. K1 = K2 = 0 impliesthat W1/2 is the worldline of a boson and a Kramers singlet. The right panel is theWilson line with nontrivial (K1, K2). The quantum number of the Lorentz symmetryis shown in (6.17).
As explained in section 6.1.2, (K1, K2) labels four distinct O(3, 1) Lorentz symmetry
enrichments of SU(2) Yang-Mills theories. In [108, 155], the authors also referred
(K1, K2) as the Four Siblings of O(3, 1) enriched SU(2) Yang-Mills with θ = 0, π.
One can understand (6.16) as follows. When B is nontrivial, the Wilson line
with SU(2) isospin j = 1/2, W1/2(γ), is attached to a surface operator exp(iπ∫
ΣB)
with ∂Σ = γ. See figure 6.1. The twisted gauge bundle constraint modifies the
above surface operator by decorating an additional 2d invertible TQFT of the Lorentz
symmetry: π(K1w21 + K2w2). The physical meanings of these invertible TQFTs are
well known. πw21 is the worldsheet theory of a time reversal symmetric SPT (a.k.a. the
Haldane chain) whose boundary supports a Kramers doublet. πw2 is the worldsheet
theory whose boundary transforms projectively under the Lorentz symmetry SO(3,1),
i.e., the boundary supports a fermion. We further realize that without the twists from
the Lorentz symmetry O(3,1) (i.e. K1 = K2 = 0), the original SU(2) Wilson line
W1/2(γ) transforms under O(3,1) as Kramers singlet and is a boson. Combining the
157
above physical understandings, under the twists using the O(3,1) Lorentz symmetry,
the statistics h(W ) and the Kramers parity T 2W of W1/2(γ) are
h(W ) =K2
2mod 1, T 2
W = (−1)K1+K2 . (6.17)
It is also illuminating to refer twisting the gauge bundle constraint from (6.15) to
(6.16) as Lorentz symmetry fractionalization. See [166] for the related discussions on
3d Chern-Simons (matter) theories and [169, 170] on 4d U(1) gauge theories.
6.2.4 Anomaly on an Unorientable Manifold
The SU(2) Yang-Mills theory with θ = π, coupled to the two-form background field
B, is
S = − 1
4g2
∫
M4
Tr(F − πBI2) ∧ ?(F − πBI2) +π
8π2
∫
M4
Tr(F − πBI2) ∧ (F − πBI2),
(6.18)
subjected to the gauge bundle constraint (6.16). Here F = da− ia2 is the U(2) field
strength. One further attempts to formulate (6.18) on an unorientable and non-spin
manifold M4, which enables one to prove the full quantum anomalies.
On an unorientable manifold, the top differential form is not well-defined, due
to the lack of the volume form whose definition needs an orientation. To make
sense of (6.18) on an unorientable manifold, we reformulate it in terms of the Chern
characteristic classes. We denote the jth Chern class of the U(2) gauge bundle as
cj(VU(2)). Denote the jth Chern class of the U(N) gauge bundle as cj. For j = 1, 2,
we have
c1 =TrF
2π,
c2 = − 1
8π2Tr(F ∧ F ) +
1
8π2(TrF ) ∧ (TrF ).
(6.19)
158
Replacing 18π2 Tr(F ∧ F ) by c1∪c1
2− c2, we rewrite the topological term in (6.18) in
terms of the cocycles and characteristic classes as
π
∫
M4
(− c2 +
c1 ∪ c1
2− 1
2c1 ∪B +
P(B)
4
)(6.20)
where P(B) is the Pontryagin square. On an unorientable manifold, only Z2 char-
acteristic classes can be integrated. Hence, except the first term in (6.20), the other
terms are all ill-defined. To make sense of the last three terms in (6.20), we can
promote these ill-defined terms to a 5d integral using the Stocks rule:
Sanom ≡π∫
M5
δ
(P(B)
4− 1
2c1 ∪B +
c1 ∪ c1
2
)
=π
∫
M5
δP(B)
4− 1
2c1 ∪ δB −
1
2δc1 ∪B +
δc1 ∪ c1
2+c1 ∪ δc1
2
=π
∫
M5
BSq1B + Sq2Sq1B − c1 ∪ Sq1B − Sq1c1 ∪B + Sq1c1 ∪ c1 + c1 ∪ Sq1c1
=π
∫
M5
BSq1B + Sq2Sq1B − (B +K1w21 +K2w2) ∪ Sq1B − (Sq1B +K2Sq1w2) ∪B
+ (Sq1B +K2Sq1w2) ∪ (B +K1w21 +K2w2)
+ (B +K1w21 +K2w2) ∪ (Sq1B +K2Sq1w2)
=π
∫
M5
BSq1B + Sq2Sq1B +K1Sq1B ∪ w21 +K2Sq1(B ∪ w2)
+K2
((K1w
21 +K2w2) ∪ Sq1w2 + Sq1w2 ∪ (K1w
21 +K2w2)
)
=π
∫
M5
BSq1B + Sq2Sq1B +K1Sq1B ∪ w21 +K2Sq1(B ∪ w2).
(6.21)
In the third line, we have used the gauge bundle constraint (6.16), which implies that
c1, when promoted to 5d, can still be valued in Z2 cohomology H2(M5,Z2) (although
c1 is no-longer valued in a Z cohomology class in M5). This implies that on M5, it
makes sense to define Sq1c1. (6.21) has several properties:
159
1. Every term in the last expression of (6.21) is Z2 valued cohomology class, hence
the integration on unorientable M5 is perfectly well-defined.
2. It is straightforward to check that when M5 is closed (without boundary), (6.21)
is invariant under the background gauge transformation
B → B + δλ. (6.22)
Combining with the first point, (6.21) respects all the symmetries of the SU(2)
Yang-Mills.
3. All terms in (6.21) only depend on the background gauge fields, and independent
of the dynamical gauge fields.
4. Because (6.21) is non-vanishing for closed M5, the extension from M4 to M5
depends on the choice of M5, hence (6.21) is really an 5d SPT of the global
symmetry Z2,[1] × ZT2 .
All these properties leads to the conclusion that the SU(2) Yang-Mills theory with
θ = π is anomalous, with the anomaly polynomial
Sanom = π
∫
M5
BSq1B + Sq2Sq1B +K1w21Sq1B +K2Sq1(w2B). (6.23)
To gauge Z2,[1] while preserving time reversal symmetry, we really have to regard the
SU(2) Yang-Mills as a 4d-5d coupled system, where the structures (w1, w2, B) on M4
are extended to M5.
We emphasize that the anomaly polynomial (6.23) depends on K2 only when M5
has a nontrivial boundary M4. This implies that the term K2Sq1(w2B) does not lead
to a distinguished anomaly. Instead, it is a WZW-like counter term. However, we
will show in section 6.3 that, if time reversal is spontaneously broken at θ = π, the
160
WZW-like counter term leads to a nontrivial ’t Hooft anomaly on the time reversal
domain wall.
6.2.5 Low Energy Dynamics: Overview and Questions
The SU(2) Yang-Mills theory is strongly coupled in the infrared, due to negative beta
function. Thus the low energy fate of the SU(2) dynamics is hardly known. It is
famously conjectured [148] that for any N ≥ 2 the SU(N) Yang-Mills with θ = 0 has
a mass gap.4 Moreover, [143] found that the SU(N) Yang-Mills (for even N) has a
nontrivial ’t Hooft anomaly only at θ = π. Since nontrivial ’t Hooft anomaly implies
that the low energy theory can not be trivially gapped, there should be nontrivial
dynamics at θ = π. In particular, the above analysis also apply to SU(2) Yang-Mills.
For the regime within θ ∈ (0, π)∪ (π, 2π), the dynamics is less clear. In fact, [143]
proposed two scenarios for the SU(2) Yang-Mills dynamics at zero temperature. In
one scenario, SU(2) Yang-Mills is confined for every θ. In the other scenario, SU(2)
Yang-Mills is deconfined within a regime θ ∈ [π − x, π + x] for x ∈ [0, π). In the
following discussion, we will not discuss the generic θ and will exclusively focus on
θ = π, where one can infer more on the dynamics based on the ’t Hooft anomaly.
As mentioned above, an immediate consequence of the ’t Hooft anomaly (6.23) for
Yang-Mills theory at θ = π is that the low energy theory can not be trivially gapped.
What should the low energy theory be at the fixed point? [143, 108] discussed several
scenarios, which we enumerate below.
1. The theory confines, and correspondingly the one-form symmetry Z2,[1] is un-
broken. Time reversal symmetry is spontaneously broken. There are two vacua
which are related by the spontaneously broken time reversal transformation.
This scenario is believed to take place for SU(N) Yang-Mills with large N .
4Though the mass gap is supported by numerous evidences, it still remains a conjecture. Insection 6.6, we contemplate another exotic possibility where the low energy of θ = 0 Yang-Mills isgapless, described by a deconfined U(1) Maxwell theory.
161
2. The theory is gapless and deconfined, and correspondingly the one-form sym-
metry Z2,[1] is spontaneously broken. Time reversal is unbroken. The decon-
finement can be realized by a gapless conformal field theory (CFT) (e.g. U(1)
Maxwell theory). See [167, 164] for discussions of different time reversal enriched
gapless CFTs.
3. The theory is gapped and deconfined, and correspondingly the one-form sym-
metry Z2,[1] is spontaneously broken. Time reversal is unbroken. The decon-
finement can be realized by a gapped TQFT (e.g. Z2 gauge theory). In [108],
the authors have proposed the action of Z2 gauge theory in 4d saturating the
anomaly (6.23).
4. Both Z2,[1] and time reversal are preserved by a gapped TQFT. In [155, 108], the
authors constructed a H-symmetry extended TQFT via the exact sequence 1→
K → H → Z2,[1] → 1, generalizing [161, 162] to higher form symmetries. By
dynamically gauging K, it was realized that Z2,[1] is spontaneously broken. This
suggests a possible no go to construct a symmetric TQFT. More systematically,
this scenario is ruled out by a no-go theorem from Cordova and Ohmori[160],
by making use of the quantum surgery constraints on cutting and gluing the
spacetime manifolds[171, 172] and other criteria.
5. Both Z2,[1] and time reversal are preserved by a gapless CFT.
Though the candidate phases have been proposed, it is worthy to discuss in further
detail the following aspects.
1. [143] only discussed one sibling, i.e. K1 = K2 = 0 among the Four Siblings in
[108]. Thus it is worthwhile to explore further the dynamical consequences for
different siblings.
162
2. In the first scenario, time reversal is spontaneously broken, and there are two
vacua related by time reversal symmetry. Hence there can be a domain wall
interpolating between the two vacua. The anomaly of 4d Yang-Mills (6.23) in-
duces an anomaly for the 3d domain wall, hence there must be nontrivial degrees
of freedom supported on the domain wall to saturate the induced anomaly. It
should be interesting to see how the four siblings of the domain wall theory are
related to each other. This will be discussed in section 6.3.
3. The second scenario is particularly interesting. If this scenario takes place in
dynamics, the non-Abelian SU(2) gauge theory with matter can access a direct
second order quantum phase transition between a U(1) spin liquid and the
trivial vacuum, or more exotically between two U(1) spin liquids, depending
on the further details which we discuss in section 6.6. This exotic scenario
tremendously enlarges the range of possible candidates of phase transitions,
and hence the multi-universality class, between the above phases.
Since the Z2 gauge theory has already been studied in [108], we will not study it in
detail in the present work. We also have little to say about the last scenario.
6.3 Domain Wall from Time Reversal Sponta-
neously Broken
We consider the scenario where time reversal symmetry is spontaneously broken in
the low energy. There are two vacua which are time reversal partners. Furthermore,
there exists a domain wall interpolating the two vacua. Since the two time reversal
breaking vacua are separately trivially gapped, the notion of domain wall theory
is well defined. The anomaly (6.23) implies that the domain wall theory itself has
163
nontrivial anomaly, which enforces that the domain wall supports nontrivial degrees
of freedom.
6.3.1 Domain Wall for (K1, K2) = (0, 0): Semion with U2 = 1
In this subsection, we discuss the domain wall theory for the sibling (K1, K2) =
(0, 0). The anomaly (6.23) reduces to π∫M5BSq1B + Sq2Sq1B. Under time reversal
transformation, the above anomaly implies that the partition function transforms as
Z → Z exp(iπ∫M4P(B)/2), hence induces an anomaly for the time reversal domain
wall,
SDWanom =π
2
∫
M4
P(B). (6.24)
The domain wall theory saturating the anomaly (6.24) was proposed in [143] to be a
SU(2)1 Chern Simons (CS) theory with an action
SU(2)1 CS : SCS =1
4π
∫
M3
Tr
(ada− 2i
3a3
), (6.25)
where a is a one-form SU(2) gauge field. The theory (6.25) is a non-spin theory.
There are two lines: an identity line 1, and a line with semionic topological spin s,
i.e. {1, s}. These lines obey the Abelian fusion rule: 1× 1 = 1, 1× s = s, s× s = 1.
Hence the theory is an Abelian semion theory. Coincidentally SU(2)1 is equivalent to
U(1)2 Chern Simons. 5 See Fig. 6.2.
What is the origin of the deconfined topological line s on the domain wall? We
follow the discussions in [159]. Since s is also the SU(2) Wilson line in fundamental
representation, it is natural to identify s with the SU(2) Wilson line in the funda-
mental representation in the 4d Yang-Mills theory, i.e., W1/2 ↔ s. The subscript
5This should be contrasted to the level rank duality SU(2)1 ←→ U(1)−2 which only holds whenboth sides are regarded as spin TQFTs.
164
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Figure 6.2: When time reversal is spontaneously broken, there are two vacua. Weconsider a configuration where each vacuum occupies half of the space, and there isa domain wall in between. Time reversal exchanges the two vacua. The anomaly(6.23) in the bulk induces an anomaly (6.24) on the domain wall, which consequentlyconstrains that there is an Abelian semion TQFT on the wall.
1/2 represents the SU(2) isospin. However, W1/2 in the Yang-Mills obeys area law,
in accordance with the confinement. s on the domain wall has perimeter law, in
accordance with the deconfinement on the wall. The behaviors of the SU(2) line in
the bulk and on the wall can be understood from the different condensates in the two
vacua of the bulk[143, 152, 173, 159]. In one vacua, confinement is due to monopole
condensation. In the other vacua, confinement is due to dyon condensation. Thus
although both vacua are trivially gapped, they differ by a Z2,[1] symmetry protected
topological (SPT) phase which is precisely described by (6.24). When W1/2 tunnels
from one vacuum to the other vacuum, due to the condensate changes, W1/2 has to
deconfine on the wall. The phenomena of deconfinement can also occur on the bound-
ary of a confining (e.g. SPT) or deconfining (e.g. SET) bulk in various dimensions,
see Sec.7 of [162] for further discussions.
s also descends from the Z2,[1] generator U in 4d. The U is a surface operator. In
the vacuum where monopole condenses, U is the spacetime trajectory of the ’t Hooft
line, which does not carry one-form symmetry charge itself. In the vacuum where
165
dyons condense, the Z2,[1] generator is the spacetime trajectory of the dyon line, which
carries one-form symmetry charge. We consider a stretched Z2,[1] generator which
extends to both vacua and crosses the domain wall. U and the domain wall intersects
on a line, which carries Z2,[1] charge which is identified as s. Thus U |DW ↔ s. Notice
that semions in 3d see each other as mutual fermions. Because s ↔ W1/2 ↔ U |DW,
the mutual fermionic statistics between s descends from the mutual semionic statistics
between W1/2 and U : 〈W1/2(γ)U(Σ)〉 = (−1)〈Σ,γ〉.
We further discuss the global symmetries of the domain wall theory SU(2)1. There
is a Z2,[1] one form global symmetry, generated by s. Coupling to the background B
leads to the anomaly (6.24).
What about the time reversal symmetry? In the bulk, time reversal is sponta-
neously broken, hence time reversal exchanges the two vacua on the two sides of the
domain wall. Hence time reversal is not a symmetry of the domain wall theory. In
particular, time reversal T acts as
T [SU(2)1 CS] = SU(2)−1 CS. (6.26)
The reversed sign of the Chern Simons level reflects the reversal of the direction of
the anomaly inflow under T . A useful observation[174, 175] is that T can be modified
to be the symmetry of SU(2)1 by multiplying an unbreakable CP⊥T in 4d. (Analogue
phenomenon and more general relation to the Smith Isomorphism have been discussed
by Hason, Komargodski and Thorngren [174] and independently by Cordova, Ohmori,
Shao and Yan [175]. See also the talk[176] by Thorngren. We apply this general idea
to the special context: the domain wall of SU(2) Yang-Mills.) We define
U = T (CP⊥T ), (6.27)
166
where P⊥ is the reflection along the direction perpendicular to the domain wall. Both
T and CP⊥T are not symmetries of SU(2)1, but their combination U is. Since both
T and CP⊥T are anti-unitary, U is unitary.
How does U act on the line operators in SU(2)1? Because both T and CP⊥T
flip the topological spin of anyons, U preserves the spin. Hence U does not permute
the lines. However, similar to the quantum Hall physics where anyons can trans-
form projectively under U(1) charge conservation symmetry, anyons can transform
projectively under U . The symmetry fractionalization is classified by
H2ρ(Z2, {1, s}) = Z2, (6.28)
where ρ = 1 is the identity because Z2 symmetry generated by U does not permute
the anyons. To determine the action of U , we first compute U2. Using the algebra of
T and CP⊥T in the 4d (K1, K2) = (0, 0) Yang-Mills theory, 6
T 2 = 1, (CP⊥T )2 = 1, T CP⊥T = CP⊥T T . (6.29)
Thus
U2 = T CP⊥T T CP⊥T = T 2(CP⊥T )2 = 1. (6.30)
Hence U generates a Z2 unitary symmetry that acts linearly on W1/2. Since the
Wilson line in the bulk does not transform projectively under T , the Wilson line
on the wall s does not transform projectively under U either.7 Thus the state |s〉
associated with the anyon s carries charge one (rather than the fractional charge)
6T 2 = 1 is because the Wilson line is Kramers singlet. The third equality follows from T (CP⊥) =(CP⊥)T which holds when acting on a bosonic line. If acting on a fermionic line, the third equalityshould be modified to T (CP⊥) = −(CP⊥)T . See section 6.3.5 for further details.
7In the next section, we will see that for the sibling (K1,K2) = (1, 0), the Wilson line transformsprojectively under T and accordingly s transforms projectively under U .
167
under U , i.e., U|s〉 = −|s〉. In summary U is realized linearly on the anyons which
corresponds to the trivial element in (D.67).
How does the domain wall theory couple to the background field of U? Denote the
one-form background field of U as Y , satisfying∮Y ∈ 1 mod 2. The action coupled
to the background field is
2
4πudu− Y du, (6.31)
where u is the U(1) gauge field. Here we have used the equivalence SU(2)1 ≡ U(1)2.
One can check that the Wilson line s = exp(i∮u) indeed has charge one under U .
To see this, one inserts into the path integral a Wilson line along γ, which amounts
to add to the action a term∫u ? j where ?j = δ⊥(γ). To find the U charge of the
Wilson line, we need to find the coefficient of the term πY ? j in the response action
where the dynamical fields are integrated out. This is done by solving the equation
of motion of u and plugging back into the action (6.31).
Further coupling (6.31) to Z2,[1] background field B, the action is
∫
M3
(2
4πudu− uB − Y du+ πY B
)+π
2
∫
M4
P(B), (6.32)
where we suppressed the cup product, e.g. Y B = Y ∪ B. The only anomaly is the
self anomaly of Z2,[1]. There is no anomaly involving U . This is also consistent with
the fact that U is not fractionalized on the anyons {1, s}.
6.3.2 Domain Wall for (K1, K2) = (1, 0): Semion with U2 = −1
We proceed to discuss the domain wall theory for the sibling (K1, K2) = (1, 0). Com-
pared with the anomaly for (K1, K2) = (0, 0), the anomaly for (K1, K2) = (1, 0)
contains an additional term K1π∫w2
1Sq1B = K1π∫w3
1B. Hence one may naively
168
conclude that the anomaly for the domain wall theory is π2
∫M4P(B) + π
∫M4w2
1B.
However, there are several apparent puzzles for the above domain wall anomaly:
1. Since the anomaly involves the background field w1, the domain wall theory
should be time reversal symmetric, and can be formulated on an unorientable
manifold. However, since the 4d theory for (K1, K2) = (1, 0) only differs from
(K1, K2) = (0, 0) by Lorentz symmetry fractionalization, one expects that the
domain wall theory for the sibling (1, 0) should be a modification of SU(2)1 by
modifying the way time reversal acts. But SU(2)1 is not time reversal symmet-
ric in the first place and therefore does not make sense to formulate it on an
unorientable manifold.
2. The anomaly itself, regardless of the details of the domain wall theory, is prob-
lematic. The first term π2
∫M4P(B) is not compatible with unorientable mani-
fold. This is because π2
∫M4P(B) is Z4 valued, while any quantity that can be
integrated on an unorientable manifold has to be Z2 valued.
In this section, we propose a domain wall theory by modifying the U symmetry
realization on the domain wall theory SU(2)1 proposed in section 6.3.1, which resolves
the above puzzles.
For the sibling (K1, K2) = (1, 0), the SU(2) Wilson line in the bulk W1/2 is a
Kramers doublet, hence
T 2 = (−1)2j, (6.33)
where j is the SU(2) isospin. For our purposes, we still regard time reversal symmetry
in 4d as a ZT2 symmetry, and (6.33) is interpreted as the Wilson line transforms in
the projective representation of ZT2 symmetry. The algebra between T , CP⊥T is (see
169
section 6.3.5 for further details)
T 2 = (−1)2j, (CP⊥T )2 = 1, T CP⊥T = CP⊥T T . (6.34)
Hence
U2 = T 2 = (−1)2j. (6.35)
Similar to the discussion below (6.33), we still interpret U as a Z2 unitary symmetry,
and (6.35) implies that the anyon s transforms projectively under U . Such a projective
representation is the nontrivial element in (D.67).
The domain wall theory is thus SU(2)1 with Z2,[1] one-form symmetry and Z2
zero-form symmetry generated by U , satisfying (6.35). How does SU(2)1 couple to
U background field? As in section 6.3.1, we still denote the U background as Y
satisfying∮Y ∈ 1 mod 2. The action coupled to the background field is
2
4πudu− 1
2Y du. (6.36)
Using the method discussed below (6.31), we find that the semion s = exp(i∮u)
carries U charge 1/2, i.e., U|s〉 = i|s〉. U is fractionalized on s as expected.
Is the Z2 symmetry generated by U anomalous? First it does not have anomaly
with itself. To see this, we examine that under the background gauge transforma-
tion Y → Y + δy, (6.36) transforms by −δydu/2, which vanishes modulo 2π. We
further check the mixed anomaly between U and Z2,[1]. The mixed anomaly is most
conveniently seen by activating the Z2,[1] background field B,
∫
M3
(2
4πudu− uB − 1
2Y du
)+
∫
M4
(π2P(B) + πY Y B
). (6.37)
170
Indeed, we find two types of anomaly: π2P(B) is the anomaly already appeared in the
domain wall for the sibling (K1, K2) = (0, 0). πY Y B is the mixed anomaly between
U and Z2,[1], due to nontrivial U symmetry fractionalization in (D.67). Consistently,
πY Y B implies that on the domain wall, the Z2,[1] generator s is attached by a surface
operator exp(iπ∫
ΣY Y ) = exp(iπ/2
∫ΣδY ) = exp(iπ/2
∮∂ΣY ) which precisely reflects
the fact that s carries Y charge 1/2. We make several comments:
1. In (6.37), the domain wall theory SU(2)1 is not time reversal symmetric. Con-
sistently, the anomaly does not involve w1, which resolves the two puzzles men-
tioned in the beginning of this subsection.
2. The time reversal symmetry fractionalization in the 4d induces a unitary Z2
symmetry fractionalization on the domain wall. Correspondingly, the mixed
T − Z2,[1] anomaly πw31B induces a mixed U − Z2,[1] anomaly πY Y B on the
domain wall.
3. Since CP⊥T is always an unbreakable symmetry in 4d Yang-Mills, one can freely
modify the T background field w1 to T (CP⊥T ) background field Y . Hence the
anomaly πw31B for Yang-Mills can be equivalently be written as πw1Y Y B.
This rewriting makes the induced anomaly πY Y B of the domain wall natural,
because under time reversal, 4d Yang-Mills partition function transforms as
Z→ Z exp
(iπ
2
∫
M4
P(B) + iπ
∫
M4
Y Y B
), (6.38)
which naturally provides the anomaly inflow of the 3d domain wall theory (6.37).
We emphasize that in (6.37), one can not replace Y by w1 back, because CP⊥T
is no longer the symmetry of the domain wall.
171
6.3.3 Domain Wall for (K1, K2) = (0, 1): Anti-Semion with
U2 = 1
We proceed to discuss the domain wall theory for the sibling (K1, K2) = (0, 1). We
will find that although the anomaly for 4d Yang-Mills does not depend on K2, the
anomaly of the domain wall does! To see this, we rewrite K2 dependent term in (6.23)
as K2πSq1(w2B) = K2πw1w2B which does not vanish on a manifold with boundary.
This term induces an anomaly on the domain wall K2πw2B. The complete anomaly
for the domain wall is
SDWanom =π
2
∫
M4
P(B) + π
∫
M4
w2B. (6.39)
We look for the domain wall theory that saturates such an anomaly.
We start with SU(2)1 ≡ U(1)2 theory for the sibling (K1, K2) = (0, 0). We have
shown in section 6.3.1 that SU(2)1 saturates the first term in (6.39). One needs to
find a proper fractionalization of the Lorentz symmetry (whose background is w2) to
further match the anomaly πw2B. Denote the topological spin of the Z2,[1] generator
s in SU(2)1 as h(s). The additional anomaly π∫w2B modifies the topological spin
of s by[166]
h(s)→ h(s) +1
2mod 1. (6.40)
Hence after symmetry fractionalization, h(s) shifts from 1/4 to 3/4 mod 1. In other
words, the semion in the domain wall for the sibling (K1, K2) = (0, 0) becomes an
anti-semion for the domain wall in the sibling (K1, K2) = (0, 1). Thus the domain
wall TQFT for (K1, K2) = (0, 1) contains a trivial anyon and an anti-semion, i.e.
172
{1, s}. Such a TQFT is precisely
SU(2)−1 CS. (6.41)
Apart from using Lorentz symmetry fractionalization, the domain wall theory can
further be obtained by rewriting the anomaly (6.39) as
SDWanom =π
2
∫
M4
P(B) + π
∫
M4
P(B) =3π
2
∫
M4
P(B) = −π2
∫
M4
P(B). (6.42)
Comparing with the anomaly (6.24), the anomaly (6.42) simply changes the sign, i.e.,
the direction of the anomaly inflow is reversed. Consistently, the level of the domain
wall Chern Simons theory is also reversed, from SU(2)1 for the sibling (K1, K2) =
(0, 0) to SU(2)−1 for the sibling (K1, K2) = (0, 1).
The Lorentz symmetry fractionalization can also be viewed from the quantum
number of Wilson line W1/2 in the 4d Yang-Mills. For the sibling (K1, K2) =
(0, 1), W1/2 transforms projectively under the SO(3, 1) Lorentz rotation, hence it
is a fermion. As explained in section 6.3.1, the deconfined line s is obtained from
the Wilson line in the bulk. Hence the Lorentz symmetry fractionalization (the shift
of statistics by 1/2) for W1/2 naturally induces a Lorentz symmetry fractionalization
(the shift of statistics by 1/2) for s on the domain wall, which yields s, consistent
with the additional anomaly πw2B for the domain wall.
It is instructive to consider the fractionalization of the unitary Z2 symmetry U
on s. In the 4d Yang-Mills of the sibling (K1, K2) = (0, 1), the Wilson line W1/2
is a Kramers doublet, i.e., T 2 = −1. More generally, T 2 = (−1)2j where j is the
SU(2) isospin. Hence using the algebra of T and CP⊥T , (see section 6.3.5 for further
details)
T 2 = (−1)2j, (CP⊥T )2 = 1, T (CP⊥T ) = (−1)2j(CP⊥T )T , (6.43)
173
we find
U2 = (−1)2jT 2 = (−1)4j = 1. (6.44)
It is ramarkable that although the time reversal symmetry is fractionalized on Wilson
line W1/2 in the bulk, U is not fractionalized on the anyon s ! Hence similar to the
case in section 6.3.1, the anti-semion s transforms linearly under U , i.e., U2(s) = s.
We further couple SU(2)−1 to both Z2,[1] background field B and the U background
field Y ,
∫
M3
(− 2
4πudu+ uB + Y du− πY B
)+
∫
M4
(π2P(B) + πw2B
). (6.45)
The fact that U is not fractionalized on s is in accord with the fact that there is no
anomaly involve U on the wall.
6.3.4 Domain Wall for (K1, K2) = (1, 1): Anti-Semion with
U2 = −1
We finally discuss the domain wall theory for the sibling (K1, K2) = (1, 1). From the
discussion in section 6.3.2 and 6.3.3, we find that the anomaly for the domain wall
theory is
SDWanom =π
2
∫
M4
P(B) + π
∫
M4
(Y Y + w2)B, (6.46)
where Y is the background field for the unitary symmetry U = T (CP⊥T ). The
domain wall theory is SU(2)−1 properly coupled to background fields Y and w2:
∫
M3
(− 2
4πudu+ uB +
1
2Y du
)+
∫
M4
(π2P(B) + πw2B + πY Y B
). (6.47)
174
We emphasize that although time reversal is not fractionalized on the W1/2 in
the bulk, i.e., T 2 = 1, the U unitary symmetry is fractionalized on the anyon s.
Furthermore, we again observe that domain wall carries nontrivial anomaly related
to w2, although the bulk does not.
6.3.5 Remarks On CP⊥ and T , and Summary
We provide some additional remarks on the 4d symmetries CP⊥ and T . The purpose
is to further explain the algebra between T and CP⊥T , i.e. (6.29), (6.34) and (6.43).
As mentioned in section 6.2.3, for the sibling (K1, K2), the Wilson line W1/2 has
spin h(W ) = K2/2, which is explained below (6.16). However, the fact that time
reversal squares to be T 2 = (−1)K1+K2 , rather than T 2 = (−1)K1 , needs further
explanation, which we provide below. (See [169] for similar explanation in 4d Maxwell
theory. ) For K2 = 0, T 2 = (CP⊥)2 = (−1)K1+K2 = (−1)K1 , hence K1 = 0, 1
represents Kramers singlet and doublet respectively. However for K2 = 1, suppose
when we move from a Minkowski spacetime to a Euclidean spacetime, T becomes a
Euclidean reflection R via a Wick rotation. Then T 2 differs by a sign from R2, i.e.
T 2 = −R2. Such a minus sign only occurs when acting on a fermion. Notice that
in Minkowski spacetime, CP⊥ is a still a Euclidean reflection, so T 2 = −(CP⊥)2. To
synthesize, we have
T 2 = (−1)K2(CP⊥)2. (6.48)
If T 2 = (−1)K1+K2 , then (CP⊥)2 = (−1)K1 , hence
T (CP⊥) = CP⊥T CP⊥T T (CP⊥) = (CP⊥T )T 2(CP⊥)2 = CP⊥T (−1)K2 , (6.49)
175
where we used (CP⊥T )2 = 1. This further gives rise to the commutation relation
between T and CP⊥T as
T (CP⊥T ) = (−1)K2(CP⊥T )T . (6.50)
This is precisely the relation in (6.29), (6.34) and (6.43).
We summarize the symmetry properties of the Wilson lines of isospin j in the bulk,
the domain wall theory, their symmetry fractinoalization pattern and the anomalies
in table 6.1.
(K1, K2) (h mod 1, T 2) DW Theory U2 DW Anomaly
(0, 0) (0, 1) SU(2)1 = {1, s} 1 π2
∫M4P(B)
(1, 0) (0, (−1)2j) SU(2)1 = {1, s} (−1)2j π2
∫M4P(B) + π
∫M4Y Y B
(0, 1) (j, (−1)2j) SU(2)−1 = {1, s} 1 π2
∫M4P(B) + π
∫M4w2B
(1, 1) (j, 1) SU(2)−1 = {1, s} (−1)2j π2
∫M4P(B) + π
∫M4
(Y Y + w2)B
Table 6.1: Symmetry fractionalization and anomalies on the domain wall theory for
four siblings of Yang-Mills.
6.4 Application I: Domain Wall Theory Nf < NCFT
We start by considering the domain wall theory for SU(2) QCD with Nf fermions.
The theory depends on the mass and the theta parameter via mNf eiθ. In this section,
we assume m to be real and non-negative, and keep θ in the Lagrangian. (In section
6.6, we will adopt the different assumption.) We exclusively focus on θ = π, which is
time reversal symmetric. We denote Λ as the strong coupling scale.
When m � Λ, one can integrate out the massive fermions, and the low energy
effective theory is the SU(2) Yang-Mills theory with θ = π. Assuming the scenario
176
where the time reversal is spontaneously broken, there are two vacua which are time
reversal partners. Between the two vacua, there is a time reversal domain wall. We
further assume that Nf is below the conformal window, i.e. Nf < NCFT , the domain
wall theory has been conjectured to be[177]
SU(2)1−
Nf2
+Nfψ ←→ U(1)−2 +Nfφ. (6.51)
In the large mass limit, the domain wall theory (6.51) flows to SU(2)1 ≡ U(1)2,
corresponding to the domain wall theory of the pure SU(2) Yang-Mills at θ = π. As
discussed in section 6.3, there are multiple versions of SU(2)1 theories, distinguished
by the enrichments of the unitary symmetry U and the Lorentz symmetry. In this
section, we determine the symmetry enriched versions of SU(2)1−
Nf2
+Nfψ, and how
the symmetry enrichments match across the duality (6.51).
6.4.1 Lorentz Symmetry Fractionalization, K2 = 1
We first show that domain wall theory realized in SU(2) QCD requires K2 = 1. In the
bulk, since the SU(2) gauge field is coupled to fermions, the 2π Lorentz rotation, which
multiplies the fermions by −1, can be compensated by a SU(2) gauge transformation.
More precisely, the gauge-spacetime symmetry is
SU(2)× Spin(3, 1)
Z2
, (6.52)
and the constraint of the symmetry bundle is
w2(VSO(3)) = w2. (6.53)
Comparing with (6.16), we find that the Lorentz symmetry SO(3, 1) is always realized
projectively, hence the effective Yang-Mills corresponds to the sibling K2 = 1. As
177
discussed in section 6.3.3, in the large mass limit on the domain wall SU(2)1, the
SU(2) gauge bundle in the domain wall theory is also twisted by the Lorentz symmetry
SO(2,1), i.e. the gauge-spacetime symmetry on the domain wall, as well as the bundle
constraint are
Domain Wall :SU(2)× Spin(2, 1)
Z2
, w2(VSO(3)) = w2. (6.54)
Thus at large mass limit on the wall, the Chern Simons is the K2 = 1 enrichment
of SU(2)1, i.e. SU(2)−1 Chern Simons theory discussed in section 6.3.3. Notice that
this is precisely the large positive mass limit on the bosonic side of (6.51). Hence
the SO(2, 1) Lorentz symmetry fractionalization is matched across the duality on the
wall. See [166] for more examples.
6.4.2 U Unitary Symmetry Fractionalization
We proceed to discuss the fractionalization of Z2 unitary symmetry generated by
U on the domain wall. We first consider the large positive mass limit in the theory
SU(2)1−Nf/2+Nfψ. There are two options of fractionalization of U on the anti-semion
s,8 labeled by K1. Concretely, there is the correspondence
U2 = (−1)K1 on anti-semion s. (6.55)
When the mass of ψ is finite, the Z2 unitary symmetry acts on the fermion ψ. For
K1 = 0, the fermion carries charge 1, while for K1 = 1, the fermion carries charge
1/2 (i.e. fractionalized).
On the other hand, notice that SU(2)1−Nf/2 + Nfψ naturally has the U(1) sym-
metry associated with Baryon conservation, and we adopt the normalization that the
8Notice that Lorentz symmetry fractionalization of the semion results in an anti-semion.
178
Baryon has U(1) charge 2, while the quark ψ has U(1) charge 1. The symmetry is
U(1)× SU(2)× Spin(2, 1)
Z2 × Z2
, (6.56)
where the constraint between the bundles is w2(VSO(3))+c1(VU(1)/Z2)+w2 = 0 mod 2.
How does the Z2 symmetry generated by U relate to U(1)? For K1 = 0, the
quark ψ carries U charge 1, hence the Z2 is embedded in U(1) in the natural way,
i.e. Z2 ⊂ U(1). For K1 = 1, the quark ψ carries U charge 1/2, hence the Z2 is
embedded into U(1) as Z2 ⊂ U(1)/Z2, or equivalently Z4 ⊂ U(1). We enumerate the
total U -gauge-spacetime symmetry and their gauge bundle constraint as follows:
(K1, K2) = (0, 1) :Z2 × SU(2)× Spin(2, 1)
Z2 × Z2
, w2(VSO(3)) + w2 = 0 mod 2,
(K1, K2) = (1, 1) :Z4 × SU(2)× Spin(2, 1)
Z2 × Z2
, Sq1Y + w2(VSO(3)) + w2 = 0 mod 2.
(6.57)
Notice that the gauge bundle constraints for the domain wall theories (6.57) are nicely
in accord with (6.16) in 4d.
Let us consider the dual theory U(1)−2 + Nfφ, and discuss how the U symmetry
is realized. We first consider the large mass limit, where the theory flows to U(1)−2.
The monopole in the bosonic theory is dual to the Baryon in the fermionic theory. In
the fermionic theory, Baryon carries U(1) charge 2. Using the embedding of Z2 into
U(1), we find that Baryon carries U charge K1 mod 2. Thus the monopole carries U
charge K1 mod 2.
The symmetry breaking quantum phase (described by the nonlinear sigma model)
on the domain wall can be easily seen from the bosonic theory. By turning on the
large negative mass squared of the scalar, we land on the symmetry breaking phase
179
described by the nonlinear sigma model with the target space
G =Sp(2)
Sp(1)× Sp(1)=
Sp(2)
Spin(4). (6.58)
In the sigma model, there exists a configuration of skyrmion which also carries the U
charge K1 mod 2.
6.5 Deconfined Gapless U(1) Gauge Theory
In this section, we discuss the scenario where the low energy theory of SU(2) Yang-
Mills at θ = π is a U(1) gauge theory.9 We attempt to find a U(1) gauge theory that
matches the anomaly (6.23).
We consider the time reversal invariant U(1) gauge theory described by the action
S = − 1
4e2
∫
M4
f ∧ ?f +θ
8π2
∫
M4
f ∧ f, θ = 0, 2π, (6.59)
where f = du and u is the U(1) gauge field. The U(1) theory is time reversal
symmetric, where ZT2 acts on the gauge field as
T (u0(t, ~x)) = −u0(−t, ~x), T (ui(t, ~x)) = ui(−t, ~x). (6.60)
This choice of time reversal flips the U(1) gauge charge, while preserves the U(1)
gauge monopole. Hence one can assign monopole Kramers degeneracy, i.e., T 2 to the
lines with charge (qe, qm) = (0, 1).10 In the present case, T 2 = 1 acting on Wilson
lines. Under Lorentz rotation, the Wilson line transforms with integer spin, while the
9We will also comment on θ = 0.10For θ = 0, the dyonic line with charge (qe, qm) = (0, 1) is denoted the ’t Hooft line. However,
for θ = 2π, due to Witten effect, the ’t Hooft line T is has charge (qe, qm) = (1, 1). The dyonic linewith charge (qe, qm) = (0, 1) is W−1T , i.e. ’t Hooft line attached with an anti-Wilson line.
180
’t Hooft line transforms with half integer spin or integer spin depending on θ = 0, 2π,
due to the statistical Witten effect.
(6.59) also has one form symmetries U(1)e,[1]×U(1)m,[1] where the subscripts e and
m represent electric and magnetic respectively. The electric U(1)e,[1] acts on Wilson
lines, and U(1)m,[1] acts on ’t Hooft lines. To make contact with the SU(2) Yang-
Mills, we will focus on the Z2,[1] subgroup of U(1)e,[1]. To couple (6.59) to two-form
background gauge field B, we replace f by f − πB. The action is
S = − 1
4e2
∫
M4
(f − πB) ∧ ?(f − πB) +θ
8π2
∫
M4
(f − πB) ∧ (f − πB), θ = 0, 2π.(6.61)
We further discuss coupling (6.59) to the Lorentz background fields w1, w2.
6.5.1 U(1) Gauge Theory and Spin Liquids at θ = 0
For θ = 0, one can further couple (6.59) to the Lorentz background fields. Changing
B → B + J2w2 modifies the statistics of the U(1) charge. To modify the Lorentz
symmetries of the U(1) monopole, we add to the action a term
1
2(f − πB − J2πw2)(L1w
21 + L2w2). (6.62)
The Lorentz quantum numbers of the U(1) charge E and the U(1) monopole M are
E : h(E) =J2
2mod 1
M : h(M) =L2
2mod 1, T 2
M= (−1)L1+L2 ,
(6.63)
where we use the tilde to emphasize that the time reversal parities of the U(1) charge
and monopole are the opposite compared with the convention in [167], namely the
time reversal flips the charge E other than the monopole M . We will bridge both
conventions at the end of this section.
181
When coupled to all the background fields B,w1, w2 (i.e. by formulating the
theory on an unorientable and non-spin manifold), the U(1) gauge theory with θ = 0
is11
S = − 1
4e2
∫
M4
(f−πB−J2πw2)∧?(f−πB−J2πw2)+1
2
∫
M4
(f−πB−J2πw2)(L1w21+L2w2).
(6.64)
The last term −12
∫M4
(πB + J2πw2) ∧ (L1w21 + L2w2) ⊂ S is not well-defined on an
unorientable manifold. To make sense of it on an unorientable manifold, we need to
promote it to a 5d action,
−π∫
M5
Sq1((B + J2w2)(L1w
21 + L2w2)
). (6.65)
Among the four terms by expanding (6.65), only two terms represent the ’t Hooft
anomalies,
Sanom = −π∫
M5
(L1w21Sq1B + J2L2w2w3), (6.66)
where w3 ≡ w3(TM5) is the Stiefel-Whitney class for the tangent bundle of M5.
When L1 = 1, there is a mixed anomaly between the time reversal and Z2,[1]. When
J2 = L2 = 1, there is an anomaly for the “all fermion electrodynamics”[178, 167, 179].
11The wedge product of the characteristic classes (e.g. Stiefel-Whitney classes) should be under-stood as the cup product. Below, we suppress the cup product for simplicity.
182
We summarize the U(1) gauge theories at θ = 0 and their ’t Hooft anomalies as
(J2, L2, L1) = (0, 0, 0) EbMb 0,
(J2, L2, L1) = (0, 0, 1) EbMbT − π∫
M5
w21Sq1B,
(J2, L2, L1) = (0, 1, 0) EbMfT 0,
(J2, L2, L1) = (0, 1, 1) EbMf − π∫
M5
w21Sq1B,
(J2, L2, L1) = (1, 0, 0) EfMb 0,
(J2, L2, L1) = (1, 0, 1) EfMbT − π∫
M5
w21Sq1B,
(J2, L2, L1) = (1, 1, 0) EfMfT − π∫
M5
w2w3,
(J2, L2, L1) = (1, 1, 1) EfMf − π∫
M5
w21Sq1B + w2w3,
(6.67)
where we used the Lorentz symmetries of the U(1) charge and U(1) monopoles to
label the spin liquid, similar to [167]. However, we emphasize that E is time reversal
odd and M is time reversal even, in contrast to the conventions of [167] where the
time reversal parities are the opposite to ours.
Comparing with the anomalies of SU(2) Yang-Mills with θ = π (6.23), none of
the U(1) spin liquids in (6.67) can be the potential IR candidate phases of SU(2)
Yang-Mills at θ = π. However, we will see in section 6.6 that some of the U(1) spin
liquids in (6.67) can be obtained by Higgsing SU(2) gauge group to U(1) for the
SU(2) Yang-Mills with θ = 0, although it is very unlikely that the deconfined U(1)
spin liquids are dynamically realized by the RG flow.
It is illuminating to connect our identification of the U(1) spin liquids to those in
[167]. In [167], the convention is that U(1) charge E is time reversal even while the
U(1) monopole M is time reversal odd. For θ = 0, the two conventions are related
by S-duality, i.e. E ↔ M,M ↔ E which can be understood as the π/2 rotation of
183
the charge-monopole lattice. Thus we arrive at the following dictionary:
EbMb = EbMb, EbMbT = EbTMb, EbMfT = EfTMb, EbMf = EfMb,
EfMb = EbMf , EfMbT = EbTMf , EfMfT = EfTMf , EfMf = EfMf .
(6.68)
In the dual theory, only M is charged under Z2,[1].
6.5.2 U(1) Gauge theory and Spin Liquids at θ = 2π
We proceed to discuss the U(1) spin liquids with θ = 2π. Similar to section 6.5.1,
one can still modify the statistics of the U(1) charge by replacing B → B + J2w2.
To modify the Lorentz symmetries of the monopole with charge (qe, qm) = (0, 1), we
realize that due to the Witten effect, the ’t Hooft operator (’t Hooft line) carries
θ/2π = 1 electric charge. Thus to form the pure monopole with vanishing electric
charge, one needs to attach a U(1) charge (i.e. a Wilson line). As noted in [169], for
θ = 2π, a dyon with charge (qe, qm) couples to the U(1)e,[1] and U(1)m,[1] background
fields Be and Bm by attaching a surface operator
exp
(i
∫
Σ
(qe − qm)Be + qmBm + (qe − qm)qmπw2
). (6.69)
Applying (6.69) to our case, Be = π(B + J2w2). We demand that when B = 0, the
surface operator for (qe, qm) = (0, 1) should be L1w21 + L2w2. As we will see below,
to match the mixed anomaly between time reversal and Z2,[1], we need to modify the
above expression to L1w21 +L2w2 +B when B is nonvanishing. This implies that both
E and M are charged under Z2,[1], and the mixed T -Z2,[1] anomaly descends from the
mixed anomaly of Z2,[1] ⊂ U(1)e,[1] and Z2,[1] ⊂ U(1)m,[1]. The Lorentz symmetry of
184
the U(1) charge E and U(1) monopole M is
E : h(E) =J2
2mod 1
M : h(M) =L2
2mod 1, T 2
M= (−1)L1+L2 .
(6.70)
Thus we find
Be = π(B + J2w2), Bm = π(L1w
21 + (L2 + J2 + 1)w2
). (6.71)
Notice that the Yang-Mills couples to U(1)e,[1] and U(1)m,[1] background fields Be and
Bm as
S = − 1
4e2
∫
M4
(f−Be)∧?(f−Be)+2π
8π2
∫
M4
(f−Be)∧ (f−Be)+π
2π
∫
M4
(f−Be)Bm.
(6.72)
Substituting (6.71) into (6.72), we obtain the U(1) gauge theory coupled to B,w1, w2
as
S =− 1
4e2
∫
M4
(f − πB − J2πw2) ∧ ?(f − πB − J2πw2)
+2π
8π2
∫
M4
(f − πB − J2πw2)(f − πB − J2πw2)
+π
2π
∫
M4
(f − πB − J2πw2)(L1w
21 + (L2 + J2 + 1)w2
).
(6.73)
The anomaly can be derived by examining the terms in (6.73) that are not well-
defined on an unorientable manifold M4. Such terms are 2π8π2
∫M4
(πB + J2πw2)2 −π2π
∫M4
(πB + J2πw2)(L1w21 + (L2 + J2 + 1)w2) due to the fractional coefficients. To
make sense of these terms, we promote these terms to a 5d integral.
Sanom = π
∫
M5
BSq1B+Sq2Sq1B+(L2+1)Sq1(w2B)+L1w21Sq1B+J2(L2+J2+1)w2w3.
(6.74)
185
As commented in section 6.2, the term Sq1(w2B) is a WZW-like counter term.
We summarize the U(1) spin liquids with θ = 2π and their genuine ’t Hooft
anomalies (i.e. excluding the WZW-like counter terms) as follows:
(J2, L2, L1) = (0, 0, 0) (EbMb)2π π
∫
M5
BSq1B + Sq2Sq1B,
(J2, L2, L1) = (0, 0, 1) (EbMbT )2π π
∫
M5
BSq1B + Sq2Sq1B + w21Sq1B,
(J2, L2, L1) = (0, 1, 0) (EbMfT )2π π
∫
M5
BSq1B + Sq2Sq1B,
(J2, L2, L1) = (0, 1, 1) (EbMf )2π π
∫
M5
BSq1B + Sq2Sq1B + w21Sq1B,
(J2, L2, L1) = (1, 0, 0) (EfMb)2π π
∫
M5
BSq1B + Sq2Sq1B,
(J2, L2, L1) = (1, 0, 1) (EfMbT )2π π
∫
M5
BSq1B + Sq2Sq1B + w21Sq1B,
(J2, L2, L1) = (1, 1, 0) (EfMfT )2π π
∫
M5
BSq1B + Sq2Sq1B + w2w3,
(J2, L2, L1) = (1, 1, 1) (EfMf )2π π
∫
M5
Sq1B + Sq2Sq1B + w21Sq1B + w2w3.
(6.75)
We use the subscript 2π to emphasize that both E and M lines are charged under
Z2,[1]. By rotating the charge-monopole lattice by π/2 (i.e. performing the S-duality),
we are also able to map the U(1) spin liquids in (6.75) to those discussed in [167]. One
simply exchange E ↔ M and M ↔ E. The correspondence has been enumerated in
(6.68). We emphasize that the Z2,[1] one form symmetry background field couples to
both E and M lines in the dual theory.
Notice the WZW-like counter term does not have to be matched along the RG
flow. By matching the genuine ’t Hooft anomalies in (6.75) and the anomalies of
SU(2) Yang-Mills at θ = π, we can enumerate the U(1) spin liquids for each sibling
186
of SU(2) Yang-Mills as follows:
(K1, K2) = (0, 0), (0, 1) : (EbMb)2π, (EbMfT )2π, (EfMb)2π.
(K1, K2) = (1, 0), (1, 1) : (EbMbT )2π, (EbMf )2π, (EfMbT )2π.
(6.76)
The remaining two U(1) spin liquids can not emerge under the RG flow of any sibling
of SU(2) Yang-Mills due to the additional w2w3 anomaly. Merely from matching the
’t Hooft anomalies, we are not able to determine which among the three U(1) spin
liquids in each row of (6.76) is realized for a given (K1, K2). However, by imposing
more physical requirements as we will discuss in section 6.6, we are able to determine
which U(1) spin liquid phase is realized.
6.6 Application II: Gauge Enhanced Quantum
Critical Point Nf ≥ NCFT
In this section, we discuss an application of the deconfinement scenario in section 6.5.
Assuming the SU(2) Yang-Mills at θ = π can flow to a deconfined U(1) gauge theory
which describes the low energy physics of the U(1) quantum spin liquid, it opens up
the possibility of exotic quantum phase transitions between different U(1) spin liquids
and/or trivial paramagnet, where the gauge group is enhanced to SU(2) at and only
at the critical point. We denote such transition as a gauge enhanced quantum critical
point (GEQCP).
187
6.6.1 SU(2) QCD4 and Higher Order Interactions: U(1) Spin
Liquid Phases From Higgsing
We consider the SU(2) QCD4 with Nf fermions, described by the following action
S =
∫
M4
Nf∑
i=1
Ψi(iγµDµ −m)Ψi + Lhigh
− 1
4g2
∫
M4
Tr(F ∧ ?F ). (6.77)
where Ψi is the four component Dirac fermion with a flavor index i = 1, ..., Nf and
a SU(2) color index a = 1, 2 which is suppressed. For the sake of the following dis-
cussion, we have also included a phenomenological four and eight-fermion interaction
term Lhigh,
Lhigh = u3∑
a=1
Nf∑
i=1
ΨiτaΨi
2
+ λ
3∑
a=1
Nf∑
i=1
ΨiτaΨi
2
2
, (6.78)
where τa (a = 1, 2, 3) denotes the generator of the SU(2) gauge group. We will always
take λ > 0 and allow u to be either sign. Throughout, we assume there is a flavor
symmetry Sp(Nf ) or U(Nf ) such that the masses of all the flavors of fermions are
degenerate.
We work in the parameter regime of Nf ≥ NCFT such that the QCD4 with m = 0
flows to a conformal field theory which can describe a second order phase transition
between the two semi-classical phases (which we will discuss in detail below). In
particular, when Nf > 11, the QCD4 with m = 0 is in the infrared free phase and the
coupling constant g flows to zero under RG. At this RG fixed point, the only relevant
perturbation is the fermion mass m, and the terms in Lhigh are irrelevant. Thus for
m = 0, adding the higher order terms Lhigh in (6.77) does not affect the dynamics
in the IR. In particular, u, g and λ all flow to zero, as shown in the middle panel of
figure 6.3.
188
Figure 6.3: Schematic RG flow diagram around the QCD4 fixed point for odd Nf andNf > 11. Possible IR fates are listed for completeness, although some (such as theU(1) SL on the θ = 0 side) may be extremely unlikely.
We proceed to discuss the mass deformation by allowing m to be either positive or
negative.12 We focus on the case when Nf is an odd integer. Then depending on the
sign of m, the QCD flows to the SU(2) Yang-Mills theory with either θ = 0 (for m > 0)
or θ = π (for m < 0). The SU(2) Yang-Mills theory does not describe the ultimate
IR fate of the system. It continues to flow towards different possible IR fixed points
as we have discussed in previous sections. One possibility is that the system enters
the confinement phase, where the coupling g flows large away from the m = 0 QCD4
fixed point. In the confinement phase on the θ = π (m < 0) side, Z2,[1] is unbroken
and time reversal symmetry is spontaneously broken [143, 180]. Another possibility
is that the system remains deconfined with a reduced gauge group, which can lead
to either a U(1) or a Z2 spin liquid phase. The possible U(1) spin liquid phases that
saturate the ’t Hooft anomalies are provided in section 6.5. In the rest of this section,
we provide a potentially possible mechanism for the deconfinement scenario to take
place, and we further determine, if so, which type of U(1) spin liquid (among the
candidates in (6.76)) is indeed realized for a given sibling of SU(2) Yang-Mills.
12In general, the mass parameter in 4d QCD can be complex, which is obvious when we rewritethe Dirac fermions into Weyl fermions with both chirality. In this work, we focus on the real massfor simplicity.
189
Viewing the SU(2) Yang-Mills as the large mass deformation limit of a SU(2)
QCD4 allows us to propose a natural mechanism to realize the deconfinement scenario
in section 6.5. When |m| is nonzero, it is possible that the interaction strength u and
λ in (6.77) and (6.78) could flow strong. Assuming u < 0, the higher order term (6.78)
drives the condensation of SU(2) gauge triplet ΨiτΨi and consequently Higgses the
SU(2) gauge group to its subgroup. If only one component of the gauge triplet acquires
expectation value, e.g. 〈Ψiτ3Ψi〉 6= 0, the SU(2) gauge group will be Higgsed down to
its U(1) subgroup. The remaining low-energy theory will be a U(1) Maxwell theory
that describes the U(1) spin liquid. It will be important in section 6.6.2 that after
Higgsing, each flavor of Ψi gives rise to two types of fermions Ψ1i,Ψ2i which carry
opposite U(1) gauge charge. Ψ1i carries U(1) charge 1, while Ψ2i carries U(1) charge
−1. If more than one components of the gauge triplet acquire expectation values
(depending on the details of higher order interactions), e.g. 〈Ψiτ1Ψi〉, 〈Ψiτ
2Ψi〉 6= 0,
then the remaining gauge group will be Z2, realizing the TQFT description of the
topologically ordered Z2 spin liquid phase.(See [108] for such Z2 spin liquid phases.)
In the following, we will take the U(1) spin liquid as the example to illustrate the
deconfined phase. The schematic RG flow diagram is shown in figure 6.3.
We comment on the possibilities of the signs of m and u in (6.77) and (6.78), and
their consequences.
1. u > 0 for both m > 0 and m < 0: In this scenario, the gauge group SU(2) is
not Higgsed. When m is positive, the theory flows to a trivial gapped phase, in
accord with the standard lore.[148] When m is negative, the theory flows to a
strongly coupled confining phase where time reversal is spontaneously broken.
2. u < 0 for both m > 0 and m < 0: In this scenario, the gauge group SU(2)
is Higgsed for both signs of m, with the only exception at m = 0. The SU(2)
Yang-Mills with both θ = 0 and π flow to certain U(1) spin liquids. We will
determine on the U(1) spin liquid in section 6.6.2. We emphasize that although
190
it is extremely unlikely that SU(2) Yang-Mills with θ = 0 flows to a deconfined
U(1) gauge theory and is beyond the standard lore [148], this scenario is still
not completely ruled out rigorously.
3. u > 0 for m > 0, and u < 0 for m < 0: The signs of m and u are correlated.
In this scenario, the gauge group SU(2) is Higgsed only for θ = π. While for
θ = 0, the SU(2) Yang-Mills flows to a trivial gapped phase, consistent with the
lore [148]. However, the underlying mechanism for the sign correlation between
u and m still needs to be understood.
6.6.2 Symmetries Realizations and Symmetry Enriched U(1)
Spin Liquids in the Infrared
The specific type of the U(1) spin liquid that is realized under the gauge triplet con-
densation depends on how the time-reversal symmetry is implemented in the QCD
theory (6.77). We consider the following two possibilities of time reversal implemen-
tation, where the gauge and global symmetries are
CI :SU(2)× Sp(Nf )× ZT4
Zc2 × Zf2, (6.79)
CII :SU(2)× U(Nf )
Zf2× ZT2 . (6.80)
For (6.79), the SU(2) ≡ Sp(1) gauge transformation and the time reversal sym-
metry act on the fermionic matter field as SU(2) : Ψi → eiθ·τΨi and
CI : T : Ψi → Kγ5γ0Ψ†i . (6.81)
In particular T 2 = −1 on Ψi. Here the Zc2 center of SU(2) is the same as the fermion
parity Zf2 ; we mod out Zc2 = Zf2 twice because SU(2), Sp(Nf ) and ZT4 all share the
same normal subgroup Zc2 = Zf2 . Sp(Nf ) is the flavor symmetry. If we just focus on
191
the SU(2) and time reversal (i.e. ignore the flavor symmetry Sp(Nf )), this symmetry
coincides with the CI symmetry class in the ten fold classification of the fermionic
SPT. This motivates an alternative way to understand the SU(2) QCD4 (6.77): The
SU(2) QCD4 with symmetry class (6.79) can be understood as from gauging the
SU(2) global symmetries of Nf copies of free fermions in symmetry class CI.
For (6.80), the SU(2) gauge transformation acts in the same way as in the CI
class. However, time reversal acts on the fermionic matter field differently:
CII : T : Ψi → Kiγ5γ0Ψ†i . (6.82)
Compared with (6.81), there is an additional U(1) ⊂ U(Nf ) flavor transformation.
In particular, T 2 = 1. The quotient in (6.80) is to identify the common normal
subgroup of SU(2) and U(Nf ). The SU(2) QCD4 with symmetry class (6.80) can be
understood as from gauging the SU(2) global symmetries of Nf copies of free fermions
in symmetry class CII.
Under the condensation of 〈Ψiτ3Ψi〉 6= 0, the remaining U(1) gauge group acts as
U(1) : Ψi → eiθτ3Ψi. The U(1) generator commutes with the time reversal transfor-
mation, which forms the AIII symmetry class. The class AIII fermionic SPT state is
Z8 × Z2 classified, where only the phases associated with Z8 can be represented by
the free fermion theories.13 Turning on the fermion mass m effectively put the Ψi
field in the class AIII fermionic SPT states labeled by the topological index ν = 0
(m > 0) or ν = 2Nf (m < 0). Connecting with the U(1) gauge theories in section 6.5,
θ = νπ. If Ψi is in the ν = 0 phase, the U(1) monopole is simply a boson. If Ψi is in
the ν = 2Nf phase, the U(1) monopole will carry will carry 2Nf fermion zero modes.
However, these zero modes carry U(1) gauge charge. To form U(1) gauge invariant
monopole operator, we need to consider only those monopole that are neutral under
13Before gauging, the AIII SPT theory is simply Nf free fermions coupled to U(1) backgroundfields. Hence only the Z8 part is relevant for our purpose.
192
U(1). We note that under Higgsing, both CI and CII classes reduce to AIII classes,
Higgsing : CI→ AIII, CII→ AIII. (6.83)
(6.83) can also be interpreted as different ways of embedding AIII symmetry class into
CI and CII classes. See [164] for extensive discussions of the embedding in (6.83) and
other examples among the ten Cartan symmetry classes. The difference between the
two reduced AIII classes are that the U(1) neutral monopoles have different symmetry
quantum numbers, which we determine below.
We proceed to determine the time reversal properties (Kramers degeneracy) of
the time reversal symmetric monopole operators of charge (qe, qm) = (0, 1). For
illustrative purposes, we first determine the time reversal properties of the monopole
in AIII class ν = 2 (i.e. Nf = 1 copy of AIII system and the topological theta
parameter in the U(1) Mexwell theory is θ = 2π) with the global symmetry
U(1)× ZT4Zf2
. (6.84)
The time reversal properties of the fermion zero modes descend from the time reversal
transformations in (6.81) and (6.82). In (6.81), time reversal maps a fermion to its
conjugate, and only the spinor indices are rotated. Hence the fermion zero mode ca
(for Nf = 1), where a = 1, 2 is the SU(2) index, maps under time reversal as
CI : T : ca → c†a, c†a → ca. (6.85)
In (6.82), time reversal maps a fermion to its conjugate, accompanied by a Z4 ⊂ U(1)
transformation generated by i. Hence the fermion zero mode ca maps under time
193
reversal as
CII : T : ca → ic†a, c†a → −ica. (6.86)
Using the operator-state correspondence, the monopoleM without any fermion zero
mode occupied is mapped to a state |0〉 with ca|0〉 = 0 for a = 1, 2. Under T , the
empty state |0〉 is mapped to a fully occupied state c†1c†2|0〉, i.e. T |0〉 = c†1c
†2|0〉. We
further notice that the two fermion zero modes has opposite gauge charge. c1 carries
U(1) charge 1, while c2 carries U(1) charge −1. Thus the U(1) neutral monopole
operators are associated with the states
|0〉, c†1c†2|0〉 (6.87)
rather than the half filled states c†1|0〉, c†2|0〉. Combined with (6.85) and (6.86), we can
compute T 2 of the empty and full states in (6.87) as
CI : T 2|0〉 = c1c2c†1c†2|0〉 = −|0〉, T 2c†1c
†2|0〉 = c†1c
†2c1c2c
†1c†2|0〉 = −c†1c†2|0〉,
CII : T 2|0〉 = −c1c2c†1c†2|0〉 = |0〉, T 2c†1c
†2|0〉 = −c†1c†2c1c2c
†1c†2|0〉 = c†1c
†2|0〉.
(6.88)
In short, for Nf = 1 (or ν = 2), the (qe, qm) = (0, 1) monopole is Kramers doublet
(T 2 = −1) in the AIII class descended from CI, while Kramers singlet (T 2 = 1)
in the AIII class descended from CII. Moreover, in both cases, the (qe, qm) = (0, 1)
monopole is a boson, which is obvious from (6.70).
Using similar analysis for the monopole quantum numbers in (6.88) for the AIII
class ν = 2, it is straightforward to obtain the monopole quantum numbers for AIII
194
class ν = 2Nf , which is
CI : T 2 = (−1)Nf ,
CII : T 2 = 1,
(6.89)
for monopoles associated with the U(1) neutral states |0〉, c†1ic†2j|0〉, ..., (c†1i1 ...c†1iNf
c†2j1 ...c†2jNf
)|0〉
where the number of 1 and 2 of the SU(2) indices should balance. Since we focus on
the case where Nf is odd, the time reversal Kramers degeneracy for the two cases in
(6.89) are different.
We emphasize that the quantum numbers in (6.89) are for the probe monopoles in
the AIII symmetry classes. Further gauging the U(1) global symmetries of the AIII
fermionic SPTs lead to different U(1) spin liquids. Thus (6.89) also characterizes the
quantum numbers of the dynamical monopoles in the U(1) spin liquids.
We are ready to identify the U(1) spin liquid phases in the IR. We first determine
the candidate U(1) spin liquid for SU(2) Yang-Mills with θ = 0. Since in SU(2) QCD4,
the SU(2) gauge field is coupled to fermions, the SU(2) Yang-Mills theories should
have fermionic Wilson lines, i.e., K2 = 1. (See an similar discussion in section 6.4.1.
) On the other hand, the U(1) charges should also be fermionic because they descend
from Higgsing the fermionic SU(2) charges, i.e. E should be fermionic. Combining
with the U(1) monopole quantum numbers in (6.89), we find that, when m < 0,
the QCD in the CI class flows to (EfMbT )2π, while the QCD in CII class flows to
(EfMb)2π.
The make contact with the siblings of SU(2) Yang-Mills at θ = π, we further
need relate the U(1) spin liquids determined above to the labels of the siblings, i.e.
(K1, K2). As we find above, K2 = 1. Furthermore, K1 can be determined by matching
the anomaly of the U(1) spin liquids in (6.75) with the anomaly of the SU(2) Yang-
Mills (6.23). Thus we we determine the U(1) spin liquids as well as the siblings of
195
the Lorentz symmetry enriched SU(2) Yang-Mills as
ν = 2Nf : CI : (K1, K2) = (1, 1), → AIII : (EfMbT )2π. (6.90)
ν = 2Nf : CII : (K1, K2) = (0, 1), → AIII : (EfMb)2π. (6.91)
The U(1) spin liquids on the m > 0 side is simply
ν = 0 : CI,CII : (K1, K1) = (0, 1), (1, 1) → AIII : EfMb. (6.92)
Thus we have singled out a particular symmetry enriched U(1) spin liquid as the
low energy of SU(2) Yang-Mills from the anomaly matched candidates in (6.76), by
embedding the SU(2) Yang-Mills into a SU(2) QCD4 with the assumed SU(2) triplet
Higgsing pattern. (6.90) and (6.91) are precisely the time reversal CFTs initially
proposed [164].
We finally comment that although Ψi in (6.81) satisfies T 2 = −1, this does not
mean Ψi is Kramers doublet, because Ψi is not mapped to itself under time reversal.
See [103] for an analogue discussion in 2 + 1d. A priori, it seems to be difficult to
determine the (K1, K2) from the symmetry assignment (6.81). Here, we provide a way
to determine it through identifying the U(1) spin liquid (EfMbT )2π and via anomaly
matching. Analogue comments also apply to (6.82).
6.6.3 Gauge Enhanced Quantum Critical Points
From the U(1) spin liquids determined in section 6.6.2, we are able to predict a series
of gauge enhanced quantum critical points (GEQCP) using SU(2) QCD4. We will
focus on the second and third scenarios in section 6.6.1 which involve U(1) spin liquid
phases, and finally comment on the first scenario where no U(1) spin liquid phases
are involved.
196
We first discuss the second scenario in section 6.6.1 where the fermion bilinear
condensation takes place for both m > 0 and m < 0, realizing EfMb and (EfMbT )2π
respectively for the sibling (K1, K2) = (1, 1), while EfMb and (EfMb)2π respectively
for the sibling (K1, K2) = (0, 1). For simplicity, we will mainly discuss the sibling
(K1, K2) = (1, 1) below. The transition between EfMb and (EfMbT )2π spin liquids
can be realized by tuning the mass m in (6.77), assuming the SU(2) Yang-Mills theory
can flow to the deconfined U(1) Maxwell theory on both sides. At m = 0, both the
gauge coupling g and the interaction u are irrelevant (if Nf > 11), such that the
transition is controlled by the IR free QCD fixed point. This provides a novel GEQCP
scenario for the Kramer-changing quantum criticality between EfMb and (EfMbT )2π
spin liquids as a QCD theory, where the gauge group is enhanced from U(1) to
SU(2) at the critical point, which is different from the QED description proposed in
Ref.[167]. Nevertheless, similar to Ref.[167], additional symmetries must be imposed
to guarantee a single direct transition, otherwise the critical point can be interrupted
by other time reversal invariant terms such as the alternating chemical potential term
ψ†iγ5ψi or can be split to multiple transitions if different fermion flavors have different
masses. One simple way is to demand an inversion symmetry I : ψi → γ0ψi,x→ −x
together with the Sp(Nf ) flavor symmetry.
We proceed to the third scenario in in section 6.6.1 where the fermion bilin-
ear condensation takes place only for m < 0, realizing (EfMbT )2π for the sibling
(K1, K2) = (1, 1), while (EfMb)2π for the sibling (K1, K2) = (0, 1). On the m > 0
side, the theory flows to a trivial vacua. For simplicity, we only discuss the sibling
(K1, K2) = (1, 1). The QCD theory also afford a GEQCP scenario for the phase tran-
sition between the (EfMbT )2π U(1) gauge theory and the trivially confined vacuum.
The conventional transition from a EfMb U(1) spin liquid to a trivial paramagnet
can happen by monopole condensation (as a confinement transition). Note that E
is a fermion and can not be condensed, unless condensing in pairs which would lead
197
to a Z2 topological order. However for the (EfMbT )2π spin liquid, if we condense
the monopole, the time reversal symmetry will be spontaneously broken because the
monopole is a Kramers doublet. It seems difficult to drive a direct transition from the
(EfMbT )2π spin liquid to a trivial paramagnet. Nevertheless, our analysis provides
a compelling possibility by first enlarging the gauge group from U(1) to SU(2) and
then allowing the SU(2) to confine trivially by removing tuning to the θ = 0 side.
As shown in the flow diagram Fig.6.3, it is possible to connect the (EfMbT )2π spin
liquid and the trivial paramagnet in the parameter space by going through the plane
of m = 0, which is controlled by the QCD fixed point, where an enlarged SU(2)
gauge group together with gapless fermionic partons will emerge. This constitutes
yet another example of the GEQCP.
We finally comment on the first scenario, where time reversal is spontaneously
broken for m < 0, and a trivial gapped phase is realized for m > 0. If this scenario
takes place, the SU(2) QCD4 with odd Nf fundamental fermions can access as a
second order deconfined phase transition, where deconfinement is realized at and only
at the critical point. This scenario is discussed in [180]. See [181, 182, 183, 184] for
other deconfined quantum critical points (DQCP) between various confining phases.
198
Appendix A
Appendices for Chapter 2
A.1 Conventions for MPS and Canonical MPS
A.1.1 Conventions for MPS and Transfer Matrix
Since each unit cell contains q spins-12’s, it is natural to start with the translational
invariant MPS in Eq. (2.38), i.e.,
|GS〉 =∑
{gri }
Tr
( L−1∏
r=0
T gr1 ...g
rq
)|{gri }〉. (A.1)
For convenience, we introduce the notation of the physical operators acting on the
MPS tensors. Denoting Xri and Zr
i as the Pauli X and Z operators acting on i-th
orbital (i = 1, . . . , q) in the r-th unit cell, their action on the MPS matrices are
defined as:
Xri ◦ T g
r′1 ...g
r′i ...g
r′q =
T gr′1 ...(1−gr
′i )...gr
′q if r′ = r
T gr′1 ...g
r′i ...g
r′q if r′ 6= r,
(A.2)
and
Zri ◦ T g
r′1 ...g
r′i ...g
r′q = (−1)δrr′g
r′i T g
r′1 ...g
r′i ...g
r′q . (A.3)
199
For other, more complex operators, the notation ◦ can be naturally generalized.
To make the equations more compact, let hi ∈ {1, ..., D} be the virtual indices of
the MPS matrices, where D is the bond dimension. Notice that the bold font hi is
different from the Z2 valued virtual indices h’s in the main text. For instance, the
MPS matrix elements of Eq. (2.31) become Tgr1g
r2gr3
h1h2,h3h4≡ (T g
r1gr2gr3)h1,h2 , so we identify
h1 and h2 as the composite of Z2 valued h indices, i.e., h1h2 and h3h4 respectively.
Given the MPS matrix elements (T gr1 ...g
rq )h1,h2 , where h1,h2 ∈ {1, ..., D} are the left
and right virtual indices, we can construct the MPS transfer matrix Th1h3,h2h4 by
contracting over the physical indices,
Th1h3,h2h4 =∑
gr1 ...grq
(T gr1 ...g
rq )h1,h2(T g
r1 ...g
rq )∗h3,h4
. (A.4)
Here, h1h3 is regarded as a composite left virtual index of the transfer matrix, of
dimension D2. The same applies to h2h4. The transfer matrix T is a D2×D2 matrix.
A.1.2 Review of Canonical MPS
We now review the definition and the properties of canonical MPS, and apply the
canonical MPS to stabilizer codes. The MPS matrix T gr1 ...g
rq is called “canonical” if
its transfer matrix satisfies:
∑
h2
(T)h1h3,h2h2
= δh1h3 ,
∑
h1,h3
Λh1h3
(T)h1h3,h2h4
= Λh2h4 ,
(A.5)
where T is the transfer matrix of T gr1 ...g
rq , and Λ is a full-rank diagonal matrix whose
diagonal elements are the entanglement spectrum of a single cut. In Ref. [185], it was
shown that a generic MPS matrix T gr1 ...g
rq on an open chain can be mapped to the
200
canonical form T gr1 ...g
rq via a similarity transformation
T gr1 ...g
rq = S · T gr1 ...grq · S−1, (A.6)
where S is an invertible matrix. We use ˘ to denote the canonical form of the MPS
matrix and the MPS transfer matrix throughout the appendix.
In Ref. [186], it was proved that when there is a non-degenerate ground state on
any compact space, the entanglement spectrum of a stabilizer code ground state is
flat. The reduced density matrices are, in fact, projectors. Their original proof was
formulated in 2 spatial dimensions, but it can be directly generalized to arbitrary
dimensions. See Ref. [87] for the application to 3 spatial dimensions. Here we apply
their conclusion to the case of 1 spatial dimension. Hence, the entanglement spectrum
of a 1D stabilizer code with PBC is flat.
The reduced density matrix on a local and contractible region of a gapped state
should not depend on the boundary condition far away from the local region. Thus
the entanglement spectrum does not depend on the boundary condition either. Thus
for the 1D stabilizer code with OBC, the entanglement spectrum is flat. Hence Λ
in Eq. (A.5) is also flat for one of the ground states with OBC. Since Λ is full-rank,
there are no zero diagonal elements in Λ and Λ is proportional to an identity matrix.
Hence the canonical MPS of a stabilizer code satisfies the following conditions
∑
h2
(T)h1h3,h2h2
= δh1h3 ,
∑
h1
(T)h1h1,h2h4
= δh2h4 .
(A.7)
The two conditions in Eq. (A.7) are graphically represented in Fig. A.1.
Hence we can use Eq. (A.7) to solve for the MPS with OBC. By Assumption 2.0.3,
the MPS matrices for the OBC shall also be the MPS matrices for the PBC.
201
Figure A.1: Graphical representation of Eq. (A.7).
A.2 Correlation Functions and Transfer Matrix
Eigenvalues
In this appendix, we derive the eigenvalue structure of the transfer matrix of a general
translational invariant stabilizer code. As we will prove, there is only one nonzero
eigenvalue of the MPS transfer matrix, obtained by Jordan decomposition. Moreover,
a finite power of the MPS transfer matrix can be decomposed as a tensor product of
two vectors. The lemmas and theorems will be used in App. A.3.
Lemma A.2.1. Suppose an operator O anti-commutes with some of the Hamiltonian
terms in Eq. (2.33), i.e., H = −∑L−1r=0
∑tα=1Orα, its expectation value of the ground
state of Eq. (2.33) satisfies
〈GS|O|GS〉 = 0. (A.8)
Proof. Without loss of generality, suppose O anti-commutes with O01 in Eq. (2.33).
Since the ground state |GS〉 satisfies the stabilizer condition Eq. (2.37), we have
〈GS|O|GS〉 =〈GS|OO01|GS〉
=− 〈GS|O01O|GS〉
=− 〈GS|O|GS〉.
(A.9)
202
Hence
〈GS|O|GS〉 = 0. (A.10)
Consider two operators σi, i = 1, 2. We denote p1 (resp. p2) the support of σ1
(resp. σ2) on the unit cells r1 ≤ r ≤ r1 + p1 − 1 (resp. r2 ≤ r ≤ r2 + p2 − 1). We
define the distance d(σ1, σ2) of the two operators as the number of unit cells between
the two operators plus one, i.e.,
d(σ1, σ2) =
r2 − r1 − p1 + 1 , r2 ≥ r1 + p1
r1 − r2 − p2 + 1 , r1 ≥ r2 + p2
0 , r1 + p1 > r2 > r1 − p2.
(A.11)
In particular, when two operators overlap even only on one site, their distance is zero.
When the distance of two operators σ1 and σ2 are larger than P , where P is the range
of another operator O, then O can not overlap simultaneously with σ1 and σ2.
Lemma A.2.2. Suppose σ1 and σ2 are products of Pauli matrices supported on dif-
ferent regions of distance larger than the maximal interaction range, i.e.:
d(σ1, σ2) > max{P1, . . . , Pt}, (A.12)
where Pα is the support of α-th type of the Hamiltonian term Orα. Then, their expec-
tation values satisfy
〈GS|σ1σ2|GS〉 = 〈GS|σ1|GS〉〈GS|σ2|GS〉. (A.13)
203
Proof. σ1 and σ2 either commute or anti-commute with the Hamiltonian terms, be-
cause σ1, σ2 and stabilizer operators are all products of Pauli matrices. We prove this
lemma case by case:
1. σ1 and σ2 both commute with all stabilizer operators.
[H, σi] = 0, i = 1, 2. (A.14)
Hence for any excited eigenstate |E, k〉 of the Hamiltonian H, i.e., H|E, k〉 =
E|E, k〉 (E is the energy and k labels the degeneracy within the energy
eigenspace), σi|E, k〉 is also an excited eigenstate of H. One can see this from
Eq. (A.14): [H, σi]|E, k〉 = 0 for i = 1, 2, which implies σi|E, k〉 is an energy
eigenstate of H with energy E. So
〈GS|σi|E, k〉 = 0, i = 1, 2. (A.15)
Then
〈GS|σ1|GS〉〈GS|σ2|GS〉
= 〈GS|σ1
(1−
∑
E,k
|E, k〉〈E, k|)σ2|GS〉
= 〈GS|σ1σ2|GS〉 −∑
E,k
〈GS|σ1|E, k〉〈E, k|σ2|GS〉
= 〈GS|σ1σ2|GS〉,
(A.16)
where in the first equality, we have used Assumption 2.0.1, and in the last
equality, we have used Eq. (A.15). Hence Eq. (A.13) holds true in this case.
2. σ1 commutes with all stabilizer operators while σ2 anti-commutes with some of
the stabilizer operators. Hence, σ2 and σ1σ2 both satisfy Lemma A.2.1. Their
204
expectation values are both 0:
〈GS|σ2|GS〉 = 0, 〈GS|σ1σ2|GS〉 = 0. (A.17)
Therefore, Eq. (A.13) holds true in this case.
3. σ1 anti-commutes with some of the stabilizer operators while σ2 commutes with
all stabilizer operators. This is the same situation as the last one. Both sides
of Eq. (A.13) vanish.
4. σ1 and σ2 both anti-commute with some of stabilizer operators. Using Lemma
A.2.1, their expectation values both vanish. There does not exist a stabi-
lizer operator which overlaps simultaneously with σ1 and σ2, because σ1 and
σ2 are separated with a distance larger than the maximal interaction range
max{P1, . . . , Pt}. Hence, σ1σ2 still anti-commutes with some of the stabilizer
operators. So both sides of Eq. (A.13) vanish.
This completes the proof.
Theorem A.2.3. Suppose two arbitrary operators O and O are supported on different
regions separated by a distance larger than max{P1, . . . , Pt}. Then we have
〈GS|OO|GS〉 − 〈GS|O|GS〉〈GS|O|GS〉 = 0. (A.18)
Proof. First we can expand the two operators as the summations of the products of
Pauli matrices:
O =∑
i
φiσi
O =∑
j
θjσj,
(A.19)
205
where the terms σi and σj are products of Pauli matrices supported in two separated
regions, and φi and θj are complex coefficients. Recall our assumption that O and O
are supported on different regions separated by a distance larger than the maximal
interaction range max{P1, . . . , Pt}. Then, σi and σj are also supported on regions
with a distance larger than max{P1, . . . , Pt}. Hence, σi and σj satisfy Lemma A.2.2.
Therefore
〈GS|OO|GS〉 =∑
i,j
φiθj〈GS|σiσj|GS〉
=∑
i,j
φiθj〈GS|σi|GS〉〈GS|σj|GS〉
=〈GS|O|GS〉〈GS|O|GS〉.
(A.20)
This completes the proof.
Theorem A.2.4. Let Tgr1 ...g
rq
h1,h2be the MPS matrix element of a translational invariant
stabilizer code where h1 and h2 are the virtual indices, and Th1h3,h2h4 be the MPS
transfer matrix of T defined in Eq. (A.4). Then T has only 1 nonzero eigenvalue.
Proof. For convenience, we introduce the notation:
T[Or]h1h3,h2h4
=∑
gr1 ...grq
(Or ◦ T gr1 ...grq )h1,h2(T gr1 ...g
rq )?h3,h4
.(A.21)
Moreover, the transfer matrix T can always be decomposed into Jordan blocks:
T = U(Pλ0 + Pλ1 + Pλ2 + · · · )U−1, (A.22)
where |λ0| > |λ1| > |λ2| > · · · are the eigenvalues of T, and Pλi is the corresponding
Jordan block. By a proper scaling of T, we let λ0 = 1. Using this normalization, Pλ0 ≡
P1 is non-degenerate due to the gap and non-degeneracy of the ground state.[187, 4]
206
Without loss of generality, let us consider the special basis of the virtual indices such
that U is an identity matrix, i.e.,
T = Pλ0 + Pλ1 + Pλ2 + · · · . (A.23)
Suppose we have two operators Or and Or+l with a sufficiently large (but finite) l
such that they satisfy Theorem A.2.3. The expectation value of Or and Or+l can be
written in terms of transfer matrices:
〈GS|OrOr+l|GS〉 =Tr[Tr(T[Or])Tl−1(T[Or+l])TL−1−r−l
]
Tr (TL)
=Tr[TL−l−1(T[Or])Tl−1(T[Or+l])
]
Tr (TL).
(A.24)
By Assumption 2.0.2 in the beginning of Sec. 2, the MPS matrices is independent of
the system size when L is sufficient large. For simplicity, let us take the limit:
limL→∞
TL−l−1 = limL→∞
TL = P1. (A.25)
Eq. (A.24) then simplifies to
〈GS|OrOr+l|GS〉 =Tr[P1(T[Or])Tl−1(T[Or+l])
]
Tr (P1)
=Tr[P1(T[Or])Tl−1(T[Or+l])
].
(A.26)
Using the Jordan blocks decomposition of T (λ0 = 1), we have
Tl−1 = P1 +∑
|λ|<1
P l−1λ . (A.27)
207
Substituting to Eq. (A.26), we have
〈GS|OrOr+l|GS〉
=Tr
P1(T[Or])(P1 +
∑
|λ|<1
P l−1λ )(T[Or+l])
=Tr[P1(T[Or])P1(T[Or+l])
]+∑
|λ|<1
Tr[P1(T[Or])P l−1
λ (T[Or+l])].
(A.28)
Since P1 is 1 dimensional (unique gapped ground state),
Tr[P1(T[Or])P1(T[Or+l])
]=Tr (P1T[Or]) Tr
(P1T[Or+l]
)
=〈GS|Or|GS〉〈GS|Or+l|GS〉.(A.29)
Hence
〈GS|OrOr+l|GS〉 = 〈GS|Or|GS〉〈GS|Or+l|GS〉+∑
|λ|<1
Tr[P1(T[Or])P l−1
λ (T[Or+l])].
(A.30)
Theorem A.2.3 implies that:
∑
λ 6=1
Tr[P1(T[Or])P l−1
λ (T[Or+l])]
= 0 (A.31)
for any operators Or and Or+l with a sufficiently large but finite l. Then the only
possibility is that for all λ 6= 1, λ = 0. In other words, the only nonzero eigenvalue
of T is 1. This completes the proof.
We numerically checked the Zq−1XZq−1 models with 2 ≤ q ≤ 6 and found that
the transfer matrix indeed has only 1 nonzero eigenvalue.
208
Lemma A.2.5. For a Jordan block P0 of size m ×m with zero diagonal elements,
then
(P0)n = 0, (A.32)
where the integer n ≥ m.
Proof. In terms of matrix elements, P0 is:
P0 =
0 1 0 . . . 0
0 0 1 . . . 0
. . . . . .
0 0 0 . . . 1
0 0 0 . . . 0
(A.33)
Denote ei as the vector of size m whose i-th entry is 1 and 0 otherwise. Then we can
show that:
P0 · e1 = 0,
P0 · ei = ei−1, ∀i = 2, 2, . . . ,m.
(A.34)
Hence, for any vector ei (i = 1, 2, . . . ,m), we can prove that:
(P0)m · ei = (P0)m−1 · ei−1 . . . = (P0)m−i+1 · e1 = 0 (A.35)
Therefore, we conclude that:
(P0)m = 0. (A.36)
For any integer n ≥ m, we also have:
(P0)n = 0. (A.37)
209
Theorem A.2.6. Suppose the transfer matrix T of size D2 × D2 satisfies Theorem
A.2.4. In other words, its nonzero eigenvalues contain a unique 1. Then
(T)D2
= uv (A.38)
for a column vector u of size D2 and a row vector v of size D2 such that
v · u = 1, (A.39)
where · represents the vector multiplication. In terms of matrix elements, Eq. (A.38)
is(
(T)D2)h1h3,h2h4
= uh1h3vh2h4 , (A.40)
and Eq. (A.39) isD∑
h1,h2=1
uh1h2vh1h2 = 1. (A.41)
Proof. Using the fact that T satisfies Theorem A.2.4, its Jordan decomposition is:
T = U(P1 + P0)U−1, (A.42)
where P1 is the projector into the 1 dimensional Jordan block for eigenvalue 1 and
P0 is the projector into the Jordan block for eigenvalue 0. Therefore,
TD2
= U(PD2
1 + PD2
0 )U−1 = UP1U−1, (A.43)
where we have used Lemma A.2.5 and the fact that the size of P0 is smaller than
D2 ×D2:
PD2
0 = 0. (A.44)
210
Since the Jordan block with eigenvalue 1 is 1 dimensional, there is only one nontrivial
matrix element which locates at the diagonal of P1. Without loss of generality, we
assume that the only nonzero element of P1 locates at 1-th row and 1-th column.
Hence, we can write this equation in terms of matrix elements
TD2
h1h3,h2h4=Uh1h3,1
(U−1
)1,h2h4
≡uh1h3vh2h4 ,
(A.45)
where we define
uh1h3 ≡ Uh1h3,1
vh2h4 ≡(U−1
)1,h2h4
.
(A.46)
From these definitions
v · u = (U−1 · U)1,1 = 1. (A.47)
This completes the proof.
Now we explore the properties for the canonical MPS with the tensor T and
Eq. (A.7).
Lemma A.2.7. For a stabilizer code, the transfer matrix of the ground state canonical
MPS satisfies Eq. (A.7). We prove that:
∑
h1
(Tn)h1h1,h2h4
= δh2h4 ,∑
h2
(Tn)h1h3,h2h2
= δh1h3 , (A.48)
for any integer n > 0.
211
Proof. Using the definition of the canonical MPS in Eq. (A.7), we first show that
∑
h1
(Tn)h1h1,h2h4
=∑
h1,h5,h6
Th1h1,h5h6
(Tn−1
)h5h6,h2h4
=∑
h5,h6
δh5h6
(Tn−1
)h5h6,h2h4
=∑
h1
(Tn−1
)h1h1,h2h4
.
(A.49)
Then we repeatedly apply this equation until there is only 1 T matrix.
∑
h1
(Tn)h1h1,h2h4
=∑
h1
(Tn−1
)h1h1,h2h4
=∑
h1
(Tn−2
)h1h1,h2h4
...
=∑
h1
(T)h1h1,h2h4
=δh2h4 .
(A.50)
Similarly, we can prove the other equation. This completes the proof.
Lemma A.2.8. For a stabilizer code, the transfer matrix of its ground state canonical
MPS satisfies Theorem A.2.6. We prove that the elements of u and v are
uh1h2 =δh1h2
Tr(v), vh1h2 =
δh1h2
Tr(u), (A.51)
where
Tr(u) =∑
h
uhh, Tr(v) =∑
h
vhh. (A.52)
In other words,
(TD2
)h1h3,h2h4
=1
Tr(u)Tr(v)δh1h3δh2h4 =
1
Dδh1h3δh2h4 . (A.53)
212
Proof. Using Theorem A.2.6 for a canonical MPS, we have:
(TD2
)h1h3,h2h4
= uh1h3vh2h4 . (A.54)
Applying Lemma A.2.7 with n = D2, we obtain:
δh2h4 =∑
h1
(TD2
)h1h1,h2h4
= Tr(u)vh2h4 . (A.55)
Hence, the second equation of Eq. (A.51) is proved. Similarly, we can prove the first
one. Using Eqs. (A.39) and (A.51), we find that
v · u =D
Tr(u)Tr(v)= 1. (A.56)
This yields
Tr(u)Tr(v) = D. (A.57)
Hence, Eq. (A.53) is proved.
Note that Lemma A.2.8 is not true for a general MPS transfer matrix. Indeed,
using the similarity transformation Eq. (A.6), a general MPS transfer matrix is related
to a canonical one:
Th1h2,h3h4 =∑
h5,6,7,8
Sh1,h5S?h2,h6
Th5h6,h7h8S−1h7,h3
S−1?h8,h4
(A.58)
213
where S is the similarity transformation. Applying Lemma A.2.8, we get:
(TD2
)h1h2,h3h4
=∑
h5,6,7,8
Sh1,h5S?h2,h6
(TD2
)h5h6,h7h8
S−1h7,h3
S−1?h8,h4
=∑
h5,6,7,8
Sh1,h5S?h2,h6
1
Dδh5h6δh7h8S
−1h7,h3
S−1?h8,h4
=1
D
∑
h5,7
Sh1,h5S?h2,h5
S−1h7,h3
S−1?h7,h4
(A.59)
The similarity transformation S is required to be invertible, but does not have to be
unitary. Hence, we conclude that Lemma A.2.8 is not true for a general MPS transfer
matrix.
Lemma A.2.9. For a stabilizer code, the transfer matrix of the ground state canonical
MPS T satisfies:
(Tn)h1h3,h2h4
=1
Dδh1h3δh2h4 , ∀ n > D2 ∈ N. (A.60)
Proof. Using Lemma A.2.8, we have
(Tn)h1h3,h2h4
=(TD2Tn−D2
)h1h3,h2h4
=∑
h5,h6
1
Dδh1h3δh5h6
(Tn−D2
)h5h6,h2h4
=1
Dδh1h3
∑
h5
(Tn−D2
)h5h5,h2h4
.
(A.61)
Using Lemma A.2.7, we obtain
(Tn)h1h3,h2h4
=1
Dδh1h3δh2h4 . (A.62)
This completes the proof.
214
We further remark that the Lemma A.2.9 holds only when n > D2, which is more
restricted than the condition, i.e., n > 0, for the Lemma A.2.7 holds true. However,
when we contract over the two virtual indices h1 and h3 (or h2 and h4) in Eq. (A.62),
we get Eq. (A.48).
A.3 Stabilizer Operator Acts on MPS Locally
In this appendix, we prove that Eq. (2.12) (and its generic case Eq. (2.39)) is a
sufficient and necessary condition satisfied by any MPS description of the 1D stabilizer
codes fulfilling the 3 assumptions of Sec. 2.
Theorem A.3.1. Eq. (2.39) is a necessary and sufficient condition for Eq. (2.37)
when the system size L ≥ D2 + max{P1, . . . , Pt} where P1, P2, . . . , Pt is defined in
Lemma A.2.2.
Proof. By substituting Eq. (2.39) into the left hand side of Eq. (2.37), it is trivial to
show that Eq. (2.39) is a sufficient condition for Eq. (2.37). Hence, our focus in the
rest of the proof is to show that Eq. (2.39) is also a necessary condition for Eq. (2.37).
It suffices to prove this statement for a particular operator O01. The proof generalizes
to other operators.
The strategy of this proof is to first establish this statement for the canonical MPS
T and then for a general MPS T . Typically, we will encounter many long equations
where there are T -matrices with their physical indices uncontracted on both sides.
Using the properties of the canonical MPS, i.e., Eq. (A.7), we are able to shorten the
equations by contracting out those T -matrices. We will use this trick many times
below. Similar to Sec. 2.2, Eq. (2.37) for O01 implies: (Notice that O0
1 is supported
215
from r = 0 to r = P1 − 1)
Tr
(O0
1 ◦(P1−1∏
r=0
T gr1 ...g
rq
)·(
L−1∏
r=P1
T gr1 ...g
rq
))
=Tr
(L−1∏
r=0
T gr1 ...g
rq
).
(A.63)
Multiplying both sides with(∏L−1
r=P1T g
r1 ...g
rq
)?h1h2
and summing over their physical
indices, we obtain:
∑
gP11 ...g
P1q ...gL−1
1 ...gL−1q
Tr
(O0
1 ◦(P1−1∏
r=0
T gr1 ...g
rq
)·(
L−1∏
r=P1
T gr1 ...g
rq
))(L−1∏
r=P1
T gr1 ...g
rq
)?
h1h2
=∑
gP11 ...g
P1q ...gL−1
1 ...gL−1q
Tr
(L−1∏
r=0
T gr1 ...g
rq
)(L−1∏
r=P1
T gr1 ...g
rq
)?
h1h2
.
(A.64)
Summing over the physical indices gives rise to transfer matrices. We rewrite this
equation with explicit virtual indices as follows
∑
h3h4
O01 ◦(P1−1∏
r=0
T gr1 ...g
rq
)
h3h4
(TL−P1
)h4h1,h3h2
=∑
h3h4
(P1−1∏
r=0
T gr1 ...g
rq
)
h3h4
(TL−P1
)h4h1,h3h2
.
(A.65)
Using Lemma A.2.9 and considering L ≥ D2 +max{P1, . . . , Pt} as stated, we simplify
∑
h3h4
O01 ◦(P1−1∏
r=0
T gr1 ...g
rq
)
h3h4
δh4h1δh3h2
=∑
h3h4
(P1−1∏
r=0
T gr1 ...g
rq
)
h3h4
δh4h1δh3h2 .
(A.66)
216
Figure A.2: Graphical representation of (a) Eq. (A.64) and (b) Eq. (A.67).
Equivalently,
O01 ◦(P1−1∏
r=0
T gr1 ...g
rq
)
h2h1
=
(P1−1∏
r=0
T gr1 ...g
rq
)
h2h1
. (A.67)
See Fig. A.2 for the graphical representation of Eqs. (A.64) and (A.67). Notice that
a general MPS tensor T differs from T by a similarity transformation in Eq. (A.6),
then after doing a similarity transformation on both sides of Eq. (A.67), we find that
an analogue equation for non-canonical MPS also holds,
O01 ◦(P1−1∏
r=0
T gr1 ...g
rq
)
h2h1
=
(P1−1∏
r=0
T gr1 ...g
rq
)
h2h1
. (A.68)
This completes the proof.
Applying the theorem A.3.1 to the ZZXZZ model, we find that Eq. (2.12) is a
necessary and sufficient condition for Eq. (2.4) when the system size is large enough,
i.e., L ≥ 16 + 3 = 19.
217
A.4 The Action of L and R Operators on the MPS
Matrices
Theorem A.4.1. Eq. (2.40) is a necessary and sufficient condition of Eq. (2.39).
Proof. It is trivial to show that Eq. (2.40) is a sufficient condition of Eq. (2.39). Our
focus in this proof is to show that it is also a necessary condition. Without loss of
generality, we only need to prove this for a particular pair of L and R operators, Lr1,1and Rr
1,1.
The strategy of this proof is to first establish this statement for the canonical
MPS T and then for a general MPS T . The matrix element Tgr1 ...g
rq
h1,h2of a canonical
MPS satisfies Eq. (A.7). We start with Eq. (2.39), and restore the virtual indices as
follows,
∑
h2
(Lr1,1 ◦ T g
r1 ...g
rq
)h1,h2
Rr
1,1 ◦(r+P1−1∏
r′=r+1
T gr′1 ...g
r′q
)
h2,h3
=∑
h2
T gr1 ...g
rq
(r+P1−1∏
r′=r+1
T gr′1 ...g
r′q
)
h2,h3
.
(A.69)
Multiplying(∏r+P1−1
r′=r+1 Tgr′
1 ...gr′q
)?h4,h3
on both sides of the Eq. (A.69), and summing
over both the physical indices gr′
1 , . . . , gr′q with r+ 1 ≤ r′ ≤ r+P1− 1 and the virtual
218
index h3, we find that
∑
h2,h3,gr′
1 ,...,gr′q |r+1≤r′≤r+P1−1
(Lr1,1 ◦ T g
r1 ...g
rq
)h1,h2
(Rr
1,1 ◦(r+P1−1∏
r′=r+1
T gr′1 ...g
r′q
))
h2,h3
×(r+P1−1∏
r′=r+1
T gr′1 ...g
r′q
)?
h4,h3
=∑
h2,h3,gr′
1 ,...,gr′q |r+1≤r′≤r+P1−1
Tgr1 ...g
rq
h1,h2
(r+P1−1∏
r′=r+1
T gr′1 ...g
r′q
)
h2,h3
(r+P1−1∏
r′=r+1
T gr′1 ...g
r′q
)?
h4,h3
=∑
h2,h3
Tgr1 ...g
rq
h1,h2
(TP1−1
)h2h4,h3h3
=∑
h2
Tgr1 ...g
rq
h1,h2δh4,h2
=Tgr1 ...g
rq
h1,h4,
(A.70)
where in the third equality, we use Lemma A.2.7. Let us define
(U r1,1)h2,h4 ≡
∑
h3,gr′
1 ,...,gr′q |r+1≤r′≤r+P1−1
(Rr
1,1 ◦(r+P1−1∏
r′=r+1
T gr′1 ...g
r′q
))
h2,h3
(r+P1−1∏
r′=r+1
T gr′1 ...g
r′q
)?
h4,h3
.
(A.71)
LHS of Eq. (A.70) becomes
∑
h2
(Lr1,1 ◦ T gr1 ...g
rq )h1,h2(U r
1,1)h2,h4 . (A.72)
Eq. (A.70) and the definition of U r1,1 are graphically represented in (a) and (b) of
Fig. A.3 respectively. Combining Eqs. (A.70), (A.71) and (A.72), we find
∑
h2
(Lr1,1 ◦ T gr1 ...g
rq )h1,h2(U r
1,1)h2,h4 = Tgr1 ...g
rq
h1,h4. (A.73)
219
Figure A.3: Graphical representation of (a) Eq. (A.70) and (b) the virtual operatorU r
1,1.
Applying Lr1,1 on both sides, since (Lr1,1)2 is an identity operator1, we obtain
∑
h2
(T gr1 ...g
rq )h1,h2(U r
1,1)h2,h4 = (Lr1,1 ◦ T gr1 ...g
rq )h1,h4 . (A.74)
This is one of the first set of equations in Eq. (2.40) when the tensors are canonical.
Substituting the RHS of Eq. (A.74) into the LHS of Eq. (A.73), we find
∑
h2
(T gr1 ...g
rq )h1,h2 [(U r
1,1)2]h2,h3 = (T gr1 ...g
rq )h1,h3 . (A.75)
Using the property of the canonical form Eq. (A.7), we obtain that (U r1,1)2 = I is an
identity operator, hence
U r1,1 = (U r
1,1)−1. (A.76)
1This is because the Hamiltonian terms are Hermitian, and should be product of Hermitianoperators, i.e. Pauli operators X,Y and Z.
220
In particular, the U matrices are invertible. Since the product Lr1,1Rr1,1 leaves T g
r1 ...g
rq ·
∏r+P1−1r′=r+1 T
gr′
1 ...gr′q invariant, Rr
1,1 has to transform∏r+P1−1
r′=r+1 Tgr′
1 ...gr′q as
(Rr
1,1 ◦(r+P1−1∏
r′=r+1
T gr′1 ...g
r′q
))
h1,h4
=∑
h2
(U r1,1)−1
h1,h2
(r+P1−1∏
r′=r+1
T gr′1 ...g
r′q
)
h2,h4
, (A.77)
which is one of the second set of equations in Eq. (2.40) when the tensors are canonical.
In Eq. (A.77), we use (U r1,1)−1 explicitly to manifest the fact that (Lr1,1Rr
1,1) leaves
the MPS invariant. Similarly, we can prove for other pairs of L and R operators.
Therefore, we have completed the proof for the canonical MPS T .
For a generic MPS T gr1 ...g
rq , it is related to its canonical form via a similarity
transformation, Eq. (A.6). The equations that T obeys can be inferred from those T
obeys in Eqs. (A.74) and (A.77):
∑
h2
(T gr1 ...g
rq )h1,h2(U r
1,1)h2,h4 = (Lr1,1 ◦ T gr1 ...g
rq )h1,h4
∑
h2
(U r1,1)−1
h1,h2
(r+P1−1∏
r′=r+1
T gr′1 ...g
r′q
)
h2,h4
=
(Rr
1,1 ◦(r+P1−1∏
r′=r+1
T gr′1 ...g
r′q
))
h1,h4
,
(A.78)
where
U r1,1 = S · U r
1,1 · S−1. (A.79)
where S is the similarity transformation defined in Eq. (A.6). Similarly for other
pairs of L and R operators. Therefore, Eq. (2.40) also holds. This completes the
proof.
Applying Theorem A.4.1 to the ZZXZZ model, we find that Eqs. (2.17), (2.18)
and (2.19) are necessary and sufficient conditions for Eq. (2.12).
221
A.5 Commutation Relations of U Operators
Theorem A.5.1. (Eq. (2.40)) U rα,τ operators have the same commutation/anti-
commutation relation as the Lrα,τ operators or Rrα,τ operators.
Proof. For convenience, we first denote:
Lrα,τLrα′,τ ′ = (−1)trατ,α′τ ′Lrα′,τ ′Lrα,τ . (A.80)
where trατ,α′τ ′ is an integer. Consider these two operators acting on the tensors of the
canonical MPS T :
Lrα,τLrα′,τ ′ ◦(r+τ−1∏
r′=r
T gr′1 ...g
r′q
)
=(−1)trατ,α′τ ′Lrα′,τ ′Lrα,τ ◦
(r+τ−1∏
r′=r
T gr′1 ...g
r′q
).
(A.81)
Apply Eq. (2.40) to both sides of the equation twice when the tensor in Eq. (2.40) is
the canonical one T :
(r+τ−1∏
r′=r
T gr′1 ...g
r′q
)U rα′,τ ′U
rα,τ
=(−1)trατ,α′τ ′
(r+τ−1∏
r′=r
T gr′1 ...g
r′q
)U rα,τ U
rα′,τ ′ .
(A.82)
Multiply both sides with(∏r+τ−1
r′=r T gr′1 ...g
r′q
)†and sum over the physical indices:
∑
gr1 ...grq ...g
r+τ−11 ...gr+τ−1
q
(r+τ−1∏
r′=r
T gr′1 ...g
r′q
)†(r+τ−1∏
r′=r
T gr′1 ...g
r′q
)U rα′,τ ′U
rα,τ
=(−1)trατ,α′τ ′
∑
gr1 ...grq ...g
r+τ−11 ...gr+τ−1
q
(r+τ−1∏
r′=r
T gr′1 ...g
r′q
)†(r+τ−1∏
r′=r
T gr′1 ...g
r′q
)U rα,τ U
rα′,τ ′ .
(A.83)
222
Using the canonical conditions Eq. (A.7), we can find that:
U rα′,τ ′U
rα,τ = (−1)
trατ,α′τ ′ U r
α,τ Urα′,τ ′ . (A.84)
Hence, we have completed the proof that U rα,τ operators form the same commutation
relations as the Lrα,τ does. Similarly, we can prove that the U rα,τ operators form the
same commutation relations as the Rrα,τ does.
We further discuss the case where the MPS matrix T gr1 ...g
rq is not canonical. Since
T gr1 ...g
rq is related to its canonical form via a similarity transformation Eq. (A.6), the
virtual U operator is related to U via the same similarity transformation, S, i.e.,
U rα,τ = S · U r
α,τ · S−1. Hence
U rα′,τ ′U
rα,τ = S · U r
α′,τ ′ · S−1 · S · U rα,τ · S−1
= S · U rα′,τ ′U
rα,τ · S−1
= (−1)trατ,α′τ ′S · U r
α,τ Urα′,τ ′ · S−1
= (−1)trατ,α′τ ′U r
α,τUrα′,τ ′ .
(A.85)
So the virtual U operators (associated to the non-canonical MPS) also satisfy the
same commutation relation as the physical L operators.
A.6 Linear Equations for Local Tensors
In this appendix, we prove that Eq. (2.29) (and its generalization Eq. (2.46)) is a
necessary and sufficient condition of Eqs. (2.17), (2.18) and (2.19) (and their gener-
alization Eq. (2.40)).
Theorem A.6.1. Eq. (2.46) is a necessary and sufficient condition of Eq. (2.40).
223
Figure A.4: Graphical representation of (a) Eq. (A.86) and (b) Eq. (A.88), and (c)Eq. (A.91).
Proof. It is trivial to show that Eq. (2.46) is a sufficient condition for Eq. (2.40).
Our focus in this proof is to show that Eq. (2.46) is also a necessary condition for
Eq. (2.40). We start with the leftmost L operator in Eq. (2.40).
By shifting the positions of Eq. (2.40), we can obtain Eq. (2.41), and we will
mainly use Eq. (2.41). We first consider the case when the MPS is canonical, and
then discuss the general case. To prove that Eq. (2.46) is necessary of Eq. (2.41),
we use an recursive method. In particular, let us focus on the first two equations of
Eq. (2.41) when the tensor is the canonical one T : (See Fig. A.4 (a) for the graphical
224
representation)
Lrα,1 ◦(T g
r1 ...g
rq
)= T g
r1 ...g
rq U r
α,1
Lr−1α,2 ◦
(T g
r−11 ...gr−1
q T gr1 ...g
rq
)= T g
r−11 ...gr−1
q T gr1 ...g
rq U r−1
α,2 .
(A.86)
We can apply (Lr−1α,1 )−1 to the second equation:
(Lr−1α,1 )−1Lr−1
α,2 ◦(T g
r−11 ...gr−1
q T gr1 ...g
rq
)
=(Lr−1α,1 )−1 ◦
(T g
r−11 ...gr−1
q T gr1 ...g
rq U r−1
α,2
).
(A.87)
Using the first equation of Eq. (A.86) at the (r − 1)-th site, we continue to simplify:
(See Fig. A.4 (b) for the graphical representation )
(Lr−1α,1 )−1Lr−1
α,2 ◦(T g
r−11 ...gr−1
q T gr1 ...g
rq
)
=T gr−11 ...gr−1
q (U r−1α,1 )−1T g
r1 ...g
rq U r−1
α,2 .
(A.88)
Notice that the physical operator (Lr−1α,1 )−1Lr−1
α,2 only acts on the r-th site. We can
rewrite this equation as:
T gr−11 ...gr−1
q
((Lr−1
α,1 )−1Lr−1α,2 ◦ T g
r1 ...g
rq
)
=T gr−11 ...gr−1
q (U r−1α,1 )−1T g
r1 ...g
rq U r−1
α,2 .
(A.89)
Multiply both sides with(T g
r−11 ...gr−1
q
)†and sum over the physical indices:
∑
gr−11 ...gr−1
q
(T g
r−11 ...gr−1
q
)†T g
r−11 ...gr−1
q
((Lr−1
α,1 )−1Lr−1α,2 ◦ T g
r1 ...g
rq
)
=∑
gr−11 ...gr−1
q
(T g
r−11 ...gr−1
q
)†T g
r−11 ...gr−1
q (U r−1α,1 )−1T g
r1 ...g
rq U r−1
α,2 .
(A.90)
225
Now we can apply Eq. (A.7) at the (r− 1)-th site: (See Fig. A.4 (c) for the graphical
representation)
(Lr−1α,1 )−1Lr−1
α,2 ◦ T gr1 ...g
rq = (U r−1
α,1 )−1T gr1 ...g
rq U r−1
α,2 . (A.91)
Hence, we have proved the following equations for the canonical MPS T :
Lrα,1 ◦ T gr1 ...g
rq = T g
r1 ...g
rq · U r
α,1
((Lr−1
α,1 )−1Lr−1α,2
)◦ T gr1 ···grq = (U r−1
α,1 )−1 · T gr1 ···grq · U r−1α,2 .
(A.92)
By iterating the process, we can prove the rest of the equations in Eq. (2.46) for
the canonical MPS with tensor T . The same statement is true for a general MPS
with a tensor T , since the tensor T and T are related by the similarity transformation
in Eq. (A.6). Therefore, we have completed our proof.
Applying the theorem A.6.1 to the ZZXZZ model, we find that Eq. (2.29) is the
necessary and sufficient condition for Eqs. (2.17), (2.18) and (2.19).
Theorem A.6.2. If Lrα,1, U rα,1 and V r
α,1 satisfy
Lrα,1 ◦ T gr1 ...g
rq = T g
r1 ...g
rq · U r
α,1
Lrα,1 ◦ T gr1 ...g
rq = T g
r1 ...g
rq · V r
α,1,
(A.93)
then U rα,1 = V r
α,1.
Proof. We first prove when the T matrix is canonical. Since Lrα,1 and Lrβ,1 are identical
physical operators, LHS of Eq. (A.93) are the same. Hence
T gr1 ...g
rq · U r
α,1 = T gr1 ...g
rq · V r
α,1, (A.94)
226
where T gr1 ...g
rq is the canonical MPS matrix, and U r
α,1 and V rα,1 are the associated virtual
operator. In components,
∑
h2
(T gr1 ...g
rq )h1,h2(U r
α,1)h2,h3 =∑
h2
(T gr1 ...g
rq )h1,h2(V r
α,1)h2,h3 . (A.95)
Multiplying (T gr1 ...g
rq )∗h1,h4
on both sides, and summing over h1 as well as the physical
indices gr1, . . . , grq , and using the canonical condition Eq. (A.7), we find
(U rα,1)h4,h3 = (V r
α,1)h4,h3 . (A.96)
When the MPS is not canonical, we apply the similarity transformation Eq. (A.6):
U rα,1 = S · U r
α,1 · S−1, V rα,1 = S · V r
α,1 · S−1. (A.97)
So
U rα,1 = S · U r
α,1 · S−1 = S · V rα,1 · S−1 = V r
α,1. (A.98)
This completes the proof.
A.7 Virtual U Operators as Tensor Products of
Pauli Matrices
In this appendix, we show that the virtual U operators can be constructed as tensor
products of Pauli matrices.
As discussed in the paragraph before Eq. (2.44) in Sec. 2.4 and proved in Ref. [110],
the anti-symmetric integer matrix t can be block diagonalized by a unimodular integer
matrix V , such that each nontrivial block is a 2×2 anti-symmetric matrix with integer
off-diagonal matrix elements. Consider a general set of operators {Ui} (i = 1, ..., N)
227
which either commute or anti-commute,
UiUj = (−1)tijUjUi. (A.99)
Let us define a new set of operators using the unimodular integer matrix V as follows
Ui = UVi11 UVi2
2 ...UViNN , (A.100)
where Vij are the entries of the unimodular integer matrix V . It is straightforward to
compute the commutation relations of {Ui},
UiUj = (−1)∑k,l Viktkl(V
T )lj UjUi
= (−1)(V ·t·V T )ij UjUi.
(A.101)
Due to Eq.(2.44), V · t · V T is block diagonalized. Since V · t · V T appears on the
exponent of (−1), only the modulo 2 values of the matrix elements matter. Hence the
nontrivial 2× 2 blocks have off-diagonal elements ±1 where we keep the minus signs
to make the anti-symmetry manifest. Suppose n is the number of nontrivial blocks of
the V ·t ·V T . Then one can find the representations of Ui by using the Pauli matrices,
because each 2× 2 block corresponds to a pair of anti-commuting operators. For an
irreducible representation, we can assign for instance
Ui =
I ⊗ ...⊗ I︸ ︷︷ ︸i−1
2
⊗X ⊗ I ⊗ ...⊗ I︸ ︷︷ ︸2n−i−1
2
, i is odd, 1 ≤ i ≤ 2n
I ⊗ ...⊗ I︸ ︷︷ ︸i−2
2
⊗Z ⊗ I ⊗ ...⊗ I︸ ︷︷ ︸2n−i
2
, i is even, 1 ≤ i ≤ 2n
I ⊗ ...⊗ I ⊗ I ⊗ I ⊗ ...⊗ I︸ ︷︷ ︸n
, 2n+ 1 ≤ i ≤ N,
(A.102)
228
where n = rank(t)2
, and each Ui is a tensor product of n Pauli matrices, forming a
2rank(t)
2 = 2n dimensional representation. Since V is unimodular, we can do an inverse
transformation from {Ui} to {Ui}.
Ui = U(V −1)i11 ...U
(V −1)iNN . (A.103)
Since {Ui} are tensor product of Pauli matrices, {Ui} are also tensor product of Pauli
matrices. This generalizes the construction of Sec. 2.2.
229
Appendix B
Appendices for Chapter 3
‘
B.1 Projective Representations and 1D Symmetry
Protected Topological Phases
B.1.1 Projective Representations and Cocycles
In this section, we describe projective representations and cocycles. Suppose G is a
discrete group and ρ(g) is a matrix representation of the group element g ∈ G. ρ is
the projective representation of G if
ρ(g1)ρ(g2) = ω2(g1, g2)ρ(g1g2), ∀ g1, g2 ∈ G, (B.1)
where ω2(g1, g2) is a U(1) phase. As a result of Eq. (B.1) being associative, i.e.,
(ρ(g1)ρ(g2)
)ρ(g3) = ρ(g1)
(ρ(g2)ρ(g3)
). (B.2)
230
ω2(g1, g2) satisfies:
ω2(g1, g2)ω2(g1g2, g3) = ω2(g2, g3)ω2(g1, g2g3). (B.3)
We further require that ρ(g) and ρ(g)µ1(g) belongs to the same class of the projective
representation, where µ1(g) is a U(1) phase. This yields that if two cocycles, ω2 and
ω2, are related by µ1 as follows:
ω2(g1, g2) = µ1(g1)µ1(g2)µ1(g1g2)−1ω2(g1, g2), (B.4)
then they give rise to the same projective representation. The conditions Eqs. (B.3)
and (B.4) require the U(1) phase ω2 belongs to the group cohomology H2(G,U(1))
and is a cocycle.[111, 188, 15].
Throughout the paper, G is an Abelian group of the form (Z2)q, and the group
element g is parametrized by g = (g1, g2, . . . , gq) with gi ∈ Z2 for i = 1, 2, . . . , q. All
the cocycles in H2(G,U(1)) are parametrized as in Eq. (3.20)[112, 111].
B.1.2 Cocycle States
In this subsection, we summarize the construction of a class of short range entangled
states which we dub as the the cocycle states, following Ref. [15]. These states are
interesting because they are the states describing the symmetry protected topological
(SPT) phase, protected by the on-site unitary symmetry G. We first set up the
notations, and then review their results with Abelian groups for simplicity.
Consider a 1D lattice with L unit cells. In each unit cell, the local Hilbert space
basis can be labeled by the elements of G: |gr〉,∀ gr ∈ G, (r = 0, 1, ..., L−1). Besides
the group elements {gr}, Ref. [15] also introduced an auxiliary group element g? ∈ G
which does not belong to the Hilbert space, but nevertheless enables one to cons. The
231
cocycle state is constructed as follows (see Eq. (54) of Ref. [15])
|ψ〉G,ω2 =∑
{gr}
(∑
g?
L−1∏
r=0
ω2(gr − gr−1, g? − gr))|{gr}〉. (B.5)
We further restrict Eq. (B.5) to the (Z2)q group. As introduced in App. B.1.1,
each unit cell contains q number of Z2 group elements/spins, i.e., gr = (gr1, ..., grq). A
generic ω2 is in Eq. (3.20), i.e.,
ω2(gr − gr−1, g? − gr)
= exp
(− iπ
∑
1≤i<j≤q
Pij(grj − gr−1
j )(g?i − gri )).
(B.6)
Plugging Eq. (B.6) to (B.5), the cocycle state of (Z2)q global symmetry becomes:
|ψ〉(Z2)q ,ω2 =∑
{gri }
(∑
{g?i }
exp
(− iπ
L−1∑
r=0
∑
1≤i<j≤q
Pij(grj − gr−1
j )(g?i − gri )))|{gri }〉.
(B.7)
Notice that in the exponent, the coefficient of g?i with fixed j, i.e., −iπ∑L−1r=0 Pij(g
rj −
gr−1j ), vanishes due to PBC. This further simplifies the cocycle state Eq. (B.7) to
|ψ〉(Z2)q ,ω2 =
∑
{gri }
exp
(iπ
L−1∑
r=0
∑
1≤i<j≤q
Pij(grj − gr−1
j )gri
)|{gri }〉.
(B.8)
B.1.3 Cocycle Hamiltonians
We now construct a cocycle Hamiltonian H(Z2)q ,ω2 whose ground state is Eq. (B.8).
The cocycle Hamiltonian has been constructed in Refs. [189, 190]. We present a
simplified construction.
232
Lemma B.1.1. There exist qL operators Orα defined by
Orα ≡∏
1≤k<α
(Zr+1k Zr
k)PkαXr
α
∏
α<l≤q
(Zrl Z
r−1l )Pαl (B.9)
satisfying
Orα|ψ〉(Z2)q ,ω2 = |ψ〉(Z2)q ,ω2 ,∀r ∈ [0, L− 1], α ∈ {1, . . . , q}. (B.10)
In the main text, we adopt a slightly different but equivalent convention to label
all the operators Orα using translation symmetry. See Eq. (3.22). In the main text, the
convention adopted in Eq. (3.22) is consistent with the discussion of general stabilizer
code Eq. (2.36). In this appendix, Orα in Eq. (B.9) shares the same label with Xrα in
its expression. The convention in Eq. (B.9) will simplify the proof without repeating
the same equations for different labels.
Proof. We first act Xrα on |ψ〉(Z2)q ,ω2 (B.8),
Xrα|ψ〉(Z2)q ,ω2 =
∑
{grk}
exp
(iπ
L−1∑
r=0
∑
1≤k<l≤q
Pkl(grl − gr−1
l )grk
)Xrα|{grk}〉
=∑
{grk}
exp
(iπ
L−1∑
r=0
∑
1≤k<l≤q
Pkl(grl − gr−1
l )grk
)|{grk + δrrδkα}〉.
(B.11)
In the second line, we used the fact that since the group element grα is defined mod
2, grk + δrrδkα is equivalent to flipping the value of the spin grα. We further redefine
233
the spins as grk = grk + δrrδkα, and rewrite the equation as
Xrα|ψ〉(Z2)q ,ω2
=∑
{grk}
exp
(iπ
L−1∑
r=0
∑
1≤k<l≤q
Pkl(grl − gr−1
l − δrrδlα + δ(r−1)rδlα)(grk − δrrδkα)
)|{gkr}〉
=∑
{grk}
exp
(iπ
L−1∑
r=0
∑
1≤k<l≤q
Pkl(grl − gr−1
l )grk − iπ∑
1≤k<α
Pkα(grk − gr+1k )
− iπ∑
α<l≤q
Pαl(grl − gr−1
l )
)|{gkr}〉
=∏
1≤k<α
(Zr+1k Zr
k)Pkα
∏
α<l≤q
(Zrl Z
r−1l )Pαl
∑
{grk}
exp
(iπ
L−1∑
r=0
∑
1≤k<l≤q
Pkl(grl − gr−1
l )grk
)|{grk}〉
=∏
1≤k<α
(Zr+1k Zr
k)Pkα
∏
α<l≤q
(Zrl Z
r−1l )Pαl |ψ〉(Z2)q ,ω2 .
(B.12)
In the second line, the first term on the exponent has exactly the same form as the
original |ψ〉(Z2)q ,ω2 , while the second and the third terms on the exponent are extra
terms. They can be reproduced by the acting with the product of Pauli Z operators,∏
1≤k<α(Zkr+1Z
kr )Pkα
∏α<l≤q(Z
lrZ
lr−1)Pαl . This observation directly leads to the third
line. Hence we find the following combination leaves |ψ〉(Z2)q ,ω2 invariant:
Orα ≡∏
1≤k<α
(Zr+1k Zr
k)PkαXr
α
∏
α<l≤q
(Zrl Z
r−1l )Pαl . (B.13)
This completes the proof.
Lemma B.1.2. The operators Orα in Lemma B.1.1 mutually commute, i.e,
[Orα,Or′
α′ ] = 0, ∀r, r′, α, α′. (B.14)
234
Proof. Without loss of generality, we assume r = 1. Then O1α only acts on the
unit cells at 0, 1 and 2. O1α and Or′α′ trivially commute unless X1
α overlap with a
Pauli Z operator of Or′α′ and/or Xr′
α′ overlap with a Pauli Z operator of O1α. It is
straightforward to check that the Pauli X and Z operators overlap when
1. r′ = 2 and α > α′.
2. r′ = 1 and α 6= α′.
3. r′ = 0 and α < α′.
When r′ = 2 and α > α′,
O1αO2
α′ = (−1)Pα′α(−1)Pα′αO2α′O1
α = O2α′O1
α. (B.15)
When r′ = 1 and α > α′,
O1αO1
α′ = (−1)Pα′α(−1)Pα′αO1α′O1
α = O1α′O1
α. (B.16)
When r′ = 1 and α < α′,
O1αO1
α′ = (−1)Pαα′ (−1)Pαα′O1α′O1
α = O1α′O1
α. (B.17)
When r′ = 0 and α < α′,
O1αO0
α′ = (−1)Pαα′ (−1)Pαα′O0α′O1
α = O0α′O1
α. (B.18)
In summary, we have proven that for any r′, α, α′, [O1α,Or
′
α′ ] = 0. By translational
invariance, [Orα,Or′
α′ ] = 0, ∀r, r′, α, α′. This completes the proof.
Lemma B.1.3. The operators Orα in Lemma B.1.1 are all independent for different
r = 0, ..., L− 1 and α = 1, ..., q.
235
Proof. The observation is that each Orα involves only one Pauli X operator, Xrα. Then
all operators Orα are independent.
Lemma B.1.4. The commuting Hamiltonian
H(Z2)q ,ω2 = −L−1∑
r=0
q∑
α=1
Orα. (B.19)
has only one ground state.
Proof. We prove by counting the degrees of freedom and the number of independent
constraints. Since each unit cell contains q spins and there are L unit cells, the total
dimension of the Hilbert space is 2qL. From Lemma B.1.2, all the operators in the
Hamiltonian commute. Thus the ground state |ψ〉(Z2)q ,ω2 must be stabilized by all the
operators satisfying
Orα|ψ〉(Z2)q ,ω2 = |ψ〉(Z2)q ,ω2 . (B.20)
From Lemma B.1.3, all the operators Orα are independent. Hence each Eq. (B.20)
provides one independent constraint for the ground state Hilbert space. Because Orαis a product of Pauli operators, each equation in Eq. (B.20) eliminates half of the
Hilbert space dimension. Since there are qL independent equations, the number of
ground state is 2qL−qL = 1. Hence there is only one ground state.
Summarizing Lemma B.1.1, B.1.2, B.1.3 and B.1.4, we have constructed the co-
cycle Hamiltonian:
Theorem B.1.5. The cocycle state Eq. (B.8) is stabilized by the cocycle Hamiltonian
H(Z2)q ,ω2 = −L−1∑
r=0
q∑
α=1
Orα, (B.21)
236
where
Orα ≡∏
1≤k<α
(Zr+1k Zr
k)PkαXr
α
∏
α<l≤q
(Zrl Z
r−1l )Pαl . (B.22)
The Hamiltonian satisfies
1. All the operators Orα are products of Pauli operators, and mutually commute.
2. There is a unique ground state |ψ〉(Z2)q ,ω2 with PBC.
B.2 Some Useful Identities
In this appendix, we prove that Eq. (3.35) holds. We first prove a Lemma which
turns out to be useful in proving Eq. (3.35).
Lemma B.2.1. If x is an integer, then the following equation holds.
exp
(iπ
1
2x2
)=
1 + exp (iπx)
2+ i
1− exp (iπx)
2. (B.23)
Proof. When x is an even integer, both sides are 1. When x is an odd integer, both
sides are i. Hence Eq. (B.23) holds.
Lemma B.2.2. Eq. (3.35) holds.
Proof. We start with the LHS of Eq. (3.35). Using∑
i<j gigj = 12
((∑
i gi)2 −∑i g
2i ),
we reduce the LHS to
exp
(iπ∑
i<j
gigj
)= exp
(iπ
1
2
((∑
i
gi)2 −
∑
i
g2i
)). (B.24)
237
If we further restrict the value of gi as gi ∈ {0, 1}, we have g2i = gi, hence
∑i g
2i =
∑i gi. Applying Lemma B.2.1 with x =
∑i gi, we further reduce Eq. (B.24) to
(1 + eiπ
∑ni=1 gi
2+ i
1− eiπ∑ni=1 gi
2
)e−
iπ2
∑ni=1 gi
=√
2 cos
(π
2
(n∑
i=1
gi −1
2
)).
(B.25)
Introducing a hidden variable h to write the RHS in the RBM form, we find the RHS
is precisely
1√2
1∑
h=0
exp
(iπ
2(1− 2h)
n∑
i=1
gi − iπ
4(1− 2h)
). (B.26)
This completes the proof.
Two simple examples of Eq. (3.35) are:
exp
(iπg1g2
)
=1√2
1∑
h=0
exp
(iπ
2(1− 2h)(g1 + g2)− iπ
4(1− 2h)
) (B.27)
for n = 2 and
exp
(iπ(g1g2 + g1g3 + g2g3)
)
=1√2
1∑
h=0
exp
(iπ
2(1− 2h)(g1 + g2 + g3)− iπ
4(1− 2h)
) (B.28)
for n = 3.
238
B.3 More Examples of RBM for Cocycle Model
In this appendix, we exemplify the construction of the RBM state in Sec. 3.4.3 by
the cocycle model with P12 = P13 = · · · = P1q = 1 and Pij = 0 with i ≥ 2 and j > i.
The Hamiltonian of the model is
H(Z2)q ,ω2 =−L−1∑
r=0
(q∏
i=2
ZriX
r+11
q∏
i=2
Zr+1i +
q∑
i=2
Zr1Z
r+11 Xr
i
). (B.29)
The ground state is
|GS〉(Z2)q ,ω2 =∑
{gri }
L−1∏
r=0
exp
(iπ
q∑
i=2
(gri − gr−1i )gr1
)|{gri }〉. (B.30)
The q × q Γ matrix (defined in Eq. (3.53)) is
Γ =
0 0 · · · 0 0
1 0 · · · 0 0
1 0 · · · 0 0
......
. . .... 0
1 0 · · · 0 0
. (B.31)
Applying the procedures introduced in the proof of Lemma. 3.4.2, we first use row
operations to set the all the rows of Eq. (B.31) to zero except the first row. Recall
G1 and G2 defined in Eq. (3.47). The row operation is
GT = G2(1, q − 1)G2(1, q − 2) · · ·G2(1, 2)G1(1, q). (B.32)
239
The visible spins transform as
gr1
gr2...
grq−1
grq
→
gr1
gr2...
grq−1
grq
= G−1 ·
gr1
gr2...
grq−1
grq
=
∑qi=2 g
ri
gr2...
grq−1
gr1
. (B.33)
The Γ matrix is transformed to
Γ→ Γ = GT · Γ ·G =
0 0 · · · 0 1
0 0 · · · 0 0
0 0 · · · 0 0
......
. . .... 0
0 0 · · · 0 0
. (B.34)
Hence the rank of the Γ matrix is
rank(Γ) = rank(Γ) = 1. (B.35)
Using the identity Eq. (3.35), we only need to introduce one hidden spin of type h
and type h respectively to express the exponent in Eq. (B.30) in terms of RBM,
L−1∑
r=0
q∑
i=2
(gri − gr−1i )gr1 =
L−1∑
r=0
(− Sym(gr1,
q∑
i=2
gr−1i ) + Sym(gr1,
q∑
i=2
gri )
).
(B.36)
240
The ground state Eq. (B.30) can be written as an RBM state
|GS〉(Z2)q ,ω2 =∑
{gri },{hr1},{hr1}
L−1∏
r=0
exp
(− iπ
2(1− 2hr1)(gr1 +
q∑
i=2
gr−1i ) + i
π
4(1− 2hr1)
+ iπ
2(1− 2hr1)
q∑
i=1
gri − iπ
4(1− 2hr1)
)|{gri }〉.
(B.37)
This RBM can be casted into an MPS with bond dimension 2, and the matrix elements
of the RBM-MPS are:
Tgr1 ,...,g
rq
hr1,hr+11
= exp
(− iπ
2(1− 2hr1)gr1 − i
π
2(1− 2hr+1
1 )
q∑
i=2
gri + iπ
4(1− 2hr1)
)
×1∑
hr1=0
exp
(iπ
2(1− 2hr1)
q∑
i=1
gri − iπ
4(1− 2hr1)
).
(B.38)
We also present the RBM for two examples in Fig. B.1 and B.2 corresponding to
q = 3 and q = 4.
Figure B.1: RBM network for cocycle model with q = 3, P12 = P13 = 1, P23 = 0.
Figure B.2: RBM network for cocycle model with q = 4, P12 = P13 = P14 = 1, P23 =
P24 = P34 = 0.
241
Appendix C
Appendices for Chapter 4
C.1 Proof for the Concatenation Lemma for the
3D Toric Code Model
In this section, we prove the Concatenation lemma for the 3D toric code model
by induction. First of all, we propose and prove two lemmas:
(A) Let Tt1t2t3... be a contraction of a network of local T tensors, whose (i.e. open)
indices {t1t2t3...} are un-contracted virtual indices. If Tt1t2t3... satisfies the Con-
catenation lemma of the 3D toric code model, then the contraction of Tt1t2t3...
over a subset of its open virtual indices, say contracting over t1 and t2, i.e.,∑
t1t2Tt1t2t3...δt1t2 still satisfies the Concatenation lemma of the 3D toric
code model.
(B) If Tt1t2t3... and Tt1 t2 t3... are two networks of contracted local T tensors both
of which satisfying the Concatenation lemma of the 3D toric code model,
then the contraction over one pair of indices, say∑
t1 t1Tt1t2t3...Tt1 t2 t3...δt1 t1 , still
satisfies the Concatenation lemma of the 3D toric code model.
Proof:
242
(A): Since T satisfies the Concatenation lemma, its elements T{t} are:
T{t} =
0 if∑
i ti = 1 mod 2
N if∑
i ti = 0 mod 2
(C.1)
where N is a constant independent of the open virtual indices in the Concatenation
lemma. Suppose that we contract two indices of T, tm, tn ∈ {t}, and we denote the
contraction as T′ and the remaining open virtual indices after the contraction {t′}.
Then we have:
T′{t′} =∑
tm,tn
T{t}δtm,tn
=∑
tm,tn
T...tm...tn...δtm,tn
=∑
tm
T...tm...tm...
=T...0...0... + T...1...1...
(C.2)
Hence, the contraction still satisfies the Concatenation lemma:
T′{t′} =
0 if∑
i t′i = 1 mod 2
2N if∑
i t′i = 0 mod 2
(C.3)
243
(B): Since T and T satisfy the Concatenation lemma, their elements T{t} and
T{t} are:
T{t} =
0 if∑
i ti = 1 mod 2
N if∑
i ti = 0 mod 2
T{t} =
0 if∑
i ti = 1 mod 2
N if∑
i ti = 0 mod 2
(C.4)
where N and N are the constants independent of the indices {t} and {t} respectively.
Suppose that we contract two indices tm ∈ {t} and tn ∈ {t}, and we denote the
contraction as T′ and the remaining open virtual indices after the contraction {t′}.
Then we have:
T′{t′} =∑
tm,tn
T{t}T{t}δtm,tn
=∑
tm,tn
T...tm...T...tn...δtm,tn
=T...0...T...0... + T...1...T...1...
(C.5)
The last line is nonzero if and only if∑
i 6=m ti and∑
j 6=n tn have the same parity. If
this parity is even (resp. odd), only T...0...T...0... (resp. T...1...T...1...) is nonzero and
equal to NN . Since∑
i t′i =
∑i 6=m ti +
∑j 6=n tn, we conclude that:
T′{t′} =
0 if∑
i t′i = 1 mod 2
NN if∑
i t′i = 0 mod 2
(C.6)
T′{t′} still satisfies the Concatenation lemma. 2
Having proved Lemma (A) and (B), we can further prove that:
244
(C) If T and T are two networks of contracted local T tensors which both satisfy
the Concatenation lemma of the 3D toric code model, then their contraction
over any pairs of indices still satisfies the Concatenation lemma of the 3D
toric code model.
Proof:
We can decompose the contraction process into two steps: (1) contract T and T
over one pair of indices; (2) contract the rest of the indices. Lemma (B) guarantees
that the outcome tensor of the contraction (1) still satisfies the Concatenation
lemma. Lemma (A) guarantees that the outcome tensor of the contraction (2) also
satisfies the Concatenation lemma. Hence, Lemma (C) is proved. 2
Now we can complete the induction proof for the Concatenation lemma of the
3D toric code model: First of all, we point out the a single local T tensor satisfies
the Concatenation lemma. Next, we assume that two networks of contracted local
T tensors satisfy the Concatenation lemma, and prove that their contraction also
satisfies the Concatenation lemma. This induction step is, in fact, Lemma (C).
Therefore, we have completed the induction proof for the Concatenation lemma
of the 3D toric code model.
C.2 Numerics for Haah Code
In this appendix, we present various numerical evidences for the entanglement entropy
of the Haah code.
In this appendix, we present the results of numerical calculations for the entan-
glement entropies using various cuts.
We start with the TNS wave function of the Haah code defined on T 3 of size
Lx × Ly × Lz. We choose a bipartition of the TNS, |TNS〉 =∑{t} |{t}〉A ⊗ |{t}〉A
where {t} is the set of open virtual indices. For a given choice of region A, we then
245
compute the reduced density matrix (RDM) ρA = TrA|TNS〉〈TNS|, and diagonalize
the RDM. For all the cases we computed, the non-zero eigenvalues of the RDM is
fully degenerate. This degenerate eigenvalue of the normalized RDM is denoted as λ.
The cases No.1 and No.2 in Table. C.1 corresponds to type 1 exact SVD regions
with l = 2 and l = 3 respectively. We see that this is consistent with the general
formula Eq. (4.66). When l = 2, 6l−5 = 6×2−5 = 7; when l = 3, 6l−5 = 6×3−5 =
13. The case No.3 corresponds to the square region A with size 2× 2× 2. The TNS
wave function under such cut is not an SVD (under the TNS basis), however, as we
have shown in Sec. 4.4.4, we can make a change of basis such that in the new basis the
wave function is an SVD. The counting of new basis gives the entanglement entropy of
the cubic cut Eq. (4.87). The brute force numerical calculation in case No.3 of Table.
C.1 is consistent with the formula: S(A)/ log 2 = 6l2−6l+2 = 6×22−6×2+2 = 14.
Note that all these numerical results have been checked using the direct evaluation
of the full GS wave function up to the system sizes 4× 3× 3.
246
System SizeCoordinate of Region A
λ S/ log 2 SVD? No.x y z Left/Right
3× 3× 3
1 1 0 0
1128
7 Yes 1
1 0 1 00 1 1 01 1 1 01 0 0 10 1 0 10 0 1 11 1 1 1
4× 4× 4
1 1 0 0
18192
13 Yes 2
1 0 1 00 1 1 01 1 1 01 2 0 01 2 1 00 2 1 01 0 0 10 1 0 10 0 1 11 1 1 10 2 0 11 2 1 11 1 0 10 1 1 1
3× 3× 3
0 0 0 0
116384
14 No 3
0 0 0 11 0 0 01 0 0 10 1 0 00 1 0 11 1 0 01 1 0 10 0 1 00 0 1 11 0 1 01 0 1 10 1 1 00 1 1 11 1 1 01 1 1 1
Table C.1: Entanglement entropies for various bipartitions of the |TNS〉 of the Haahcode. The second to fourth column list the coordinates of vertices in region A. Thecolumn of ”Left/Right” labels the spin on the left or right position on the vertex(x, y, z), where 0 and 1 corresponds to the left and right position respectively. Weused the coordinate frame as shown in Eq. 4.49 and Fig. 4.6.247
Appendix D
Appendices for Chapter 5
D.1 Review of Entanglement Entropy and Spec-
trum
In this appendix, we review the definition of the entanglement entropy, and review
the notation that we use in this work.
To define the entanglement entropy, we first partition the space into two parts, A,
and its complement, B, via an entanglement surface Σ.1 For a given pure quantum
state |ψ〉, the wave function can be decomposed as
|ψ〉 =∑
ab
Wab|Aa〉|Acb〉, (D.1)
where a labels normalized basis states of the Hilbert space HA localized in region
A and b labels normalized basis states of the Hilbert space HAc localized in region
Ac. We perform a singular value decomposition (SVD) of the matrix W as Wab =
UacDcdV†db and define new bases |A′c〉 = Uac|Aa〉 and |Ac′
d 〉 = V †db|Acb〉. Dcd is a diagonal
matrix with positive entries, but not all the diagonal elements need be nonzero. The
1Because we are interested in (3+1)D systems, the entanglement surface Σ is a two dimensionalsurface.
248
number of nonzero elements is the rank of W , and the nonzero “singular values” are
denoted as e−ξλ/2. ξλ are termed the entanglement energies, and the whole set of
entanglement energies is the entanglement spectrum {ξλ}λ=1,··· ,Rank(W ). Zero singular
values correspond to infinite entanglement energies. Thus,
|ψ〉 =
Rank(W )∑
λ=1
e−ξλ/2|A′λ〉|Ac′
λ 〉. (D.2)
To compute the entanglement entropy, we trace over the states in region Ac to obtain
a reduced density matrix of region A,
ρA = TrHAc |ψ〉〈ψ| =Rank(W )∑
λ=1
e−ξλ|A′λ〉〈A′λ|. (D.3)
The entanglement entropy is defined as the von Neumann entropy of the reduced
density matrix ρA (see Refs. [4] and [191] for a review),
S(A) = −TrHAρA log ρA = −
Rank(W )∑
λ=1
e−ξλ log e−ξλ . (D.4)
Heuristically, the entanglement entropy measures how much the degrees of freedom
in the two regions A and B are correlated.
In this paper, we denote the entanglement entropy of subregion A (whose bound-
ary is Σ) as either S(A) or S[Σ], using either parentheses or square brackets to
highlight the sub region or the entanglement surface, respectively.
D.2 Local Contributions to the Entanglement En-
tropy
In this appendix, we review the general properties of the entanglement entropy. Fol-
lowing the discussions in Ref. [125], we provide some detailed and quantitative analy-
249
ses on how the non-universal and shape dependent terms can enter into the constant
part of the EE.
The simplest property of the EE is S(A) = S(Ac), which says the entropy com-
puted for region A is equal to the entropy computed for its complement Ac. This is
also true for the full entanglement spectrum, and follows directly from Eq. (D.2).
We assume that in a gapped system with finite correlation length, the EE can be
decomposed into a local part and a topological part,
S(A) = Slocal(A) + Stopo(A). (D.5)
The local part Slocal(A) only depends on the local degrees of freedom near the entan-
glement surface, and therefore can be written in the form of an integral over local
variables. Since the only local functions on Σ are the metric hµν , the extrinsic curva-
ture (second fundamental form) Kµν , and the covariant derivatives of Kµν (covariant
derivatives of hµν are zero by definition), Refs. [125, 192, 193] argued that Slocal should
be expressible in terms of local geometric quantities of the entanglement surface Σ,
i.e.,
Slocal(A) =
∫
Σ
d2x√hF (Kµν ,∇ρKµν , ..., hµν), (D.6)
where F is a local function of Kµν and hµν and their covariant derivatives. 2
In contrast, the topological part of the EE, Stopo(A), is precisely the contribution
that cannot be written as an integral of local variables near the entanglement surface.
(In particular, the Euler characteristic term does not contribute to Stopo(A).) Stopo(A)
should be invariant under smooth deformations of the entanglement surface, and
2Suppose the submanifold is given by the embedding φ : Σ → M , concretely, φ : yi → xµ =(z∗, yi) where z∗ is a fixed number specifying the position of hypersurface in the perpendiculardirection of the embedded space. Let the metric in M be gµν , the induced metric therefore ishij ≡ (φ∗g)ij = ∂xµ
∂yi∂xν
∂yj gµν . Let nµ be the normal unit vector of the surface Σ, then the extrinsic
curvature Kµν of Σ is Kµν = ∇µnν − nµnρ∇ρnν . See Appendix D of Ref. [194] for more details.
250
should also be invariant under smooth deformations of the Hamiltonian of the system
(provided the gap does not close). Therefore, reminiscent of two-dimensional systems,
Stopo(A) is expected to be the constant part of the EE. However, in three spatial
dimensions, there are subtleties as we will explore below.
Before moving on, it is important for us to first specify for which systems the EE
separates into a local and a topological part. Systems such as the toric code and its
generalizations (e.g. Dijkgraaf Witten models), as well as the Walker-Wang models
[133] and their generalizations (e.g., the generalized Walker-Wang models which we
study in Sec. 5.2) satisfy this decomposition. There are some systems for which this
decomposition is obviously not valid. For instance, the systems constructed by layer
stacking of two-dimensional systems do not satisfy Eq. (D.5). The constant part of
entropy depends on the thickness Lz of the layered direction, i.e., −γ2DLz, where γ2D
is the topological entropy of a two-dimensional layer. Another class of systems beyond
our discussion are fracton models[195], whose entanglement entropy does not satisfy
Eq. (D.5). Apart from the area law term and the constant term, the entanglement
entropies of these model generically contain a term linearly proportional to the size of
the subregion [196, 47]. Since the decomposition Eq. (D.5) does not lead to a linear
subleading term, its presence in the layered models and the fracton models suggest
the decomposition Eq. (D.5) does not hold.
Since the definition of the EE dictates that S(A) = S(Ac), this should also be true
of the local part of the EE. To compute S(A), one can expand F (Kµν ,∇ρKµν , ..., hµν)
as
F (Kµν ,∇ρKµν , ..., hµν)
= F0 + F1Kµµ + F2[KµνK
µν − (Kµµ)2]
+ F ′2(Kµµ)2 + F3∇µ∇νK
µν + ...,
(D.7)
251
where ∇µ is the covariant derivative induced from hµν , and the indices are raised and
lowered via hµν and its inverse hµν . All indices are contracted so that the formula
Eq. (D.7) is independent of the choice of the coordinates. Demanding that S(A) =
S(Ac) constrains the form of the function F . To see this, we may simply transform
x1 → −x1 and x2 → x2, under which Kµν → −Kµν and hµν → hµν .3 Then
S(A) = S(Ac) implies
F (Kµν ,∇ρKµν , ..., hµν) = F (−Kµν ,∇ρKµν , ..., hµν). (D.8)
After integration, keeping only those terms even under reflection, we find that the
local part of the EE has the form
Slocal(A) = F0|Σ| − F24πχ+ 4F ′2
∫
Σ
d2x√hH2 + ..., (D.9)
where |Σ| is the area of the entanglement surface. The part proportional to F2
gives the Euler characteristic χ(Σ) of the surface Σ, defined by∫
Σd2x√h[KµνK
µν −
(Kµµ)2] = −4πχ(Σ). This term is invariant under any smooth deformation of the
entanglement surface because the Euler characteristic is a topological invariant of Σ.
The part proportional to F ′2 gives the integral of the square of the mean curvature
H = (k1 + k2)/2 (since 2H = Kµµ), where k1, k2 are the two principal curvatures
of Σ, i.e., the eigenvalues of Kµν . This term, though independent of the size of Σ,
depends on its shape. This shows that the local part of the EE has constant terms,
which contrasts with the familiar case in (2+1)D. Therefore, computing the EE and
3x1 → −x1 and x2 → x2 changes the orientation of the entanglement surface Σ. Since theprinciple curvature is an odd function of the orientation of the surface and the eigenvalues of theextrinsic curvature are two principle curvatures, we conclude that the extrinsic curvature is oddunder x1 → −x1 and x2 → x2.
252
extracting the constant part is not a promising way to extract topological information
about the underlying theory.4
The above analysis shows that for a generic gapped system (which is not at an
RG fixed point), the structure of the entanglement entropy is
S(A) = F0|Σ|+ Stopo(A)− 4πF2χ(Σ)
+4F ′2
∫
Σ
d2x√hH2 +O(1/|Σ|). (D.10)
In the main text, we denote the constant part of the EE as Sc(A) = Stopo(A) −
4πF2χ(Σ) + 4F ′2∫
Σd2x√hH2.
The above analysis gives all the possible terms that can exist, but does not require
that they are non-vanishing for a given theory. In Ref. [197], the authors computed
the entanglement entropy for massive bosons and massive fermions in (3+1)D across
S2. Their results show a constant term in the entanglement entropy. For a massive
scalar with mass m and curvature coupling term 12ξRφ2, Sc(A) = (ξ − 1
6) log(mδ),
where δ is the cut off. For a massive Dirac fermion with mass m, Sc(A) = 118
log(mδ).
Obviously, these entropies are not topological (they depend on the cutoff and on mass
parameters), which shows that non-universal contributions to the local term in fact
do exist.
D.3 Derivation of the Reduction Formula
In this appendix we present the complete derivation of the entropy reduction formula
Eq. (5.10). We will use the SSA inequality in two steps. First, in Subsection D.3.1
we derive and solve a recurrence relation for the dependence of STQFTc on the genus of
4In (2 + 1)D, by applying the same analysis, one can show that there is no constant term in theEE which can be written as an integral of local curvature when the space dimension d is even. Thisis because the term of dimension 1/Ld−1 acquires a minus sign when the coordinates of entangl-ment surface are reversed. In particular, in (2+1)D, the constant term in entanglement entropy istopological.
253
CBA
(a)
C’
B’A’
(b)
Figure D.1: Entanglement surfaces used in the application of strong sub-additivityto derive the recurrence relation Eq. (D.17). In (a), A is a general 3-manifold (as anexample, we draw A with 1 genus 3 surface and 2 genus 0 surfaces), B is 3-ball andC is a solid torus. In (b), A′ is a general 3-manifold (as an example, we draw A′ with1 genus 3 surface and 2 genus 0 surfaces), B′ is a solid torus, and C′ is a 3-ball, whichis located exactly at the hole of B′.
the entanglement cut. Second, in Subsection D.3.2 we derive an additional recurrence
relation for the dependence of STQFTc on the number of disconnected components
of the entanglement surface. We solve this recurrence relation to obtain our main
result Eq. (5.10). Our derivation expands upon the discussion in Ref. [125] in that we
obtain explicit formulas for the entropy of arbitrary multiply-connected entanglement
surfaces.
D.3.1 Recurrence for Genus
In order to find the dependence of the TEE on the data {ng}, we need to consider the
configuration of entanglement surfaces as shown in Fig. C.D.1(a): We start with a
general connected 3-manifold with boundary specified by [(0, n0), . . . , (g∗, ng∗)]. The
3-manifold is cut into three regions A, B and C. B is a 3-ball, C is a solid torus and A
occupies the remainder of the manifold. A is connected to B and disconnected from
C. Suppose A connects with B via a disk (shown as a shaded region) which belongs to
a genus (g∗−1)5 boundary of A and also belongs to the genus 0 boundary of B. Then
5Since C ∪ B has a genus 1 surface boundary.
254
the boundary of region A is specified by [(0, n0), . . . , (g∗− 1, ng∗−1 + 1), (g∗, ng∗ − 1)],
where we adopt the labeling scheme defined in Sec. 5.1.1.
We list the constant part of the EE of all regions by their topologies as follows:
STQFTc (A) = STQFT
c [(0, n0), . . . , (g∗ − 1, ng∗−1 + 1), (g∗, ng∗ − 1)],
STQFTc (B) = STQFT
c [(0, 1)],
STQFTc (C) = STQFT
c [(0, 0), (1, 1)],
STQFTc (AB) = STQFT
c [(0, n0), . . . , (g∗ − 1, ng∗−1 + 1), (g∗, ng∗ − 1)],
STQFTc (BC) = STQFT
c [(0, 0), (1, 1)],
STQFTc (ABC) = STQFT
c [(0, n0), . . . , (g∗ − 1, ng∗−1), (g∗, ng∗)].
(D.11)
Then the SSA inequality for regions A, B, and C in Eq. (5.4) reads
STQFTc [(0, n0), . . . , (g∗ − 1, ng∗−1 + 1), (g∗, ng∗ − 1)]
≥ STQFTc [(0, n0), . . . , (g∗, ng∗)] + STQFT
c [(0, 1)]− STQFTc [(0, 0), (1, 1)].
(D.12)
We could have taken A and B to be connected via a disk which belongs to a genus i
(i ≤ g∗− 1) boundary of A and also belongs to the genus 0 boundary of B. Following
an identical procedure, we conclude:
STQFTc [(0, n0), . . . , (i, ni + 1), (i+ 1, ni+1 − 1), . . . , (g∗, ng∗)] + STQFT
c [(0, 0), (1, 1)]
≥ STQFTc [(0, n0), . . . , (i, ni), (i+ 1, ni+1), . . . , (g∗, ng∗)] + STQFT
c [(0, 1)].
(D.13)
For simplicity, we will only need to adopt the choice where i = g∗ − 1.
We proceed to consider another configuration illustrated in Fig. C.D.1(b): We
start with a general 3-manifold with boundary specified by [(0, n0), . . . , (g∗, ng∗)].
The 3-manifold is cut into two regions, A′ and B′. B′ is a solid torus, and A′ is the
rest of the manifold. We assume A′ connects with B′ via a disk (shown as a shaded
255
region) in the genus (g∗ − 1) boundary of A′ and the genus 1 boundary of B′. Hence
the boundary of A′ is labeled by [(0, n0), . . . , (g∗ − 1, ng∗−1 + 1), (g∗, ng∗ − 1)]. In
addition, we denote the 3-ball located in the “hole” of B′ as C′.
We list the constant part of the EE of all regions as follows:
STQFTc (A′) =STQFT
c [(0, n0), . . . , (g∗ − 1, ng∗−1 + 1), (g∗, ng∗ − 1)],
STQFTc (B′) =STQFT
c [(0, 0), (1, 1)],
STQFTc (C′) =STQFT
c [(0, 1)],
STQFTc (A′B′) =STQFT
c [(0, n0), . . . , (g∗, ng∗)],
STQFTc (B′C′) =STQFT
c [(0, 1)],
STQFTc (A′B′C′) =STQFT
c [(0, n0), . . . , (g∗ − 1, ng∗−1 + 1), (g∗, ng∗ − 1)].
(D.14)
The SSA for A′, B′ and C′ in Fig. C.D.1(b) reads in this case:
STQFTc [(0, n0), . . . , (g∗ − 1, ng∗−1 + 1), (g∗, ng∗ − 1)]
≤ STQFTc [(0, n0), . . . , (g∗, ng∗)] + STQFT
c [(0, 1)]− STQFTc [(0, 0), (1, 1)].
(D.15)
Combining inequalities Eq. (D.12) and Eq. (D.15), we find the following equality
STQFTc [(0, n0), . . . , (g∗ − 1, ng∗−1 + 1), (g∗, ng∗ − 1)]
= STQFTc [(0, n0), . . . , (g∗, ng∗)] + STQFT
c [(0, 1)]− STQFTc [(0, 0), (1, 1)].
(D.16)
This relates the constant part of the EE of a given subsystem to that of a system
whose boundary has lower genus. Applying Eq. (D.16) repeatedly, we find
STQFTc [(0, n0), (1, n1), ..., (g∗, ng∗)] = STQFT
c [(0,
g∗∑
i=0
ni)] +
g∗∑
i=1
ini
(STQFT
c [(0, 0), (1, 1)]− STQFTc [(0, 1)]
).
(D.17)
256
In summary, we can reduce the constant part of the EE of an arbitrary sur-
face STQFTc [(0, n0), (1, n1), ..., (g∗, ng∗)] to a linear combination of STQFT
c [(0, n)] and
STQFTc [(0, 0), (1, 1)].
D.3.2 Recurrence for b0
We can further simplify STQFTc [(0,
∑g∗
i=0 ni)] in Eq. (D.17), by using STQFTc [(0, n)] =
nSTQFTc [(0, 1)]. Here we derive this relation by making use of the SSA in a manner
similar to that of the derivation above.
CBA
(a)
C’
B’A’
(b)
Figure D.2: Entanglement surfaces used in the application of strong sub-additivity
to derive Eq. (D.22). In (a), A is a 3-manifold with multiple genus zero surfaces, B
is a 3-ball, C is a 3-ball with small 3-ball removed. In (b), A′ is an open 3-manifold
with multiple genus zero surfaces, B′ is a 3-ball with a small 3-ball removed and C′
is a 3-ball located exactly in the empty 3-ball inside B′.
We consider the configuration shown in Fig. C.D.2(a), where A is a 3-manifold
with (n − 1) genus zero surfaces, B is a 3-ball and C is a 3-ball with a small 3-ball
257
inside it removed. The constant parts of the EE for these three manifolds are
STQFTc (A) =STQFT
c [(0, n− 1)],
STQFTc (B) =STQFT
c [(0, 1)],
STQFTc (C) =STQFT
c [(0, 2)],
STQFTc (AB) =STQFT
c [(0, n− 1)],
STQFTc (BC) =STQFT
c [(0, 2)],
STQFTc (ABC) =STQFT
c [(0, n)].
(D.18)
The SSA inequality reads
STQFTc [(0, n− 1)] + STQFT
c [(0, 2)] ≥ STQFTc [(0, n)] + STQFT
c [(0, 1)]. (D.19)
We can furthermore consider another configuration shown in Fig. C.D.2(b), where
A′ is a 3-manifold with (n−1) genus-0 surfaces, B′ is a 3-ball with small 3-ball removed,
and C′ is a 3-ball locating exactly in the empty 3-ball inside B′. The constant parts
of the EE for these three manifolds are
STQFTc (A′) = STQFT
c [(0, n− 1)],
STQFTc (B′) = STQFT
c [(0, 2)],
STQFTc (C′) = STQFT
c [(0, 1)],
STQFTc (A′B′) = STQFT
c [(0, n)],
STQFTc (B′C′) = STQFT
c [(0, 1)],
STQFTc (A′B′C′) = STQFT
c [(0, n− 1)].
(D.20)
Then SSA inequality reads
STQFTc [(0, n)] + STQFT
c [(0, 2)] ≤ STQFTc [(0, n+ 1)] + STQFT
c [(0, 1)]. (D.21)
258
Combining Eq. (D.19) and Eq. (D.21), one obtains
STQFTc [(0, n)] + STQFT
c [(0, 2)] = STQFTc [(0, n+ 1)] + STQFT
c [(0, 1)]. (D.22)
Since STQFTc [(0, 0)] = 0, we have
STQFTc [(0, n)] = nSTQFT
c [(0, 1)]. (D.23)
Combining this result with Eq. (D.17), we have6
STQFTc [(0, n0), . . . , (g∗, ng∗)]
=
g∗∑
i=0
niSTQFTc [(0, 1)] +
g∗∑
i=1
ini
(STQFT
c [(0, 0), (1, 1)]− STQFTc [(0, 1)]
)
=
g∗∑
i=0
(1− i)niSTQFTc [(0, 1)] +
g∗∑
i=1
iniSTQFTc [(0, 0), (1, 1)]
= b0STQFTc [(0, 0), (1, 1)] +
χ
2
(STQFT
c [(0, 1)]− STQFTc [(0, 0), (1, 1)]
)
= b0STQFTc [T 2] +
χ
2
(STQFT
c [S2]− STQFTc [T 2]
),
(D.24)
where χ =∑g∗
i=0(2 − 2i)ni is the Euler characteristic of the entanglement sur-
face, which in the previous examples of this appendix is ∂(ABC). This is precisely
Eq. (5.10) in the main text. In the last line, we have changed the notation for clarity:
S2 is a 2-sphere and T 2 is a 2-torus. We emphasize that Eq. (5.10) gives the constant
part of the EE for a TQFT. In particular, Eq. (5.10) shows that the constant part of
the EE across an arbitrary entanglement surface is reduced to that across the sphere
S2 and that across the torus T 2. 7
6As remarked in Sec. D.1, we use S(A) to denote the EE of region A, and S[Σ] to denote theEE of region with boundary Σ, such as S[S2] when entanglement surface is Σ = S2. Both notationsrefer to the same thing.
7Notice that STQFTc (A) is an additive variable, i.e., STQFT
c (A ∪ A′) = STQFTc (A) + STQFT
c (A′)if A ∩ A′ = ∅. This fact also follows from the vanishing of mutual information, i.e., I(A ∪ A′) =S(A) + S(A′)− S(A∪A′) = 0 if A∩A′ = ∅. This is because the area part cancels out in I(A∪A′),
259
D.4 Vanishing of the Mean Curvature Contribu-
tion in KPLW Prescription
In this appendix, we explain why the mean curvature terms cancel in the KPLW
combination Eq. (5.14), therefore justifying Eq. (5.18) in the main text.
In the main text, we argued that the KPLW combination of the area law term
and the Euler characteristic term vanish separately, hence we only need to consider
the topological term and the mean curvature term, i.e.,
SKPLW[T 2] = Stopo[T 2] + 4F ′2
∫∂A+∂B+∂C−∂AB−∂AC−∂BC+∂ABC
d2x√hH2. (D.25)
Eq. (D.25) suggests that the mean curvature term in the KPLW combination is invari-
ant under deformations of the entanglement surface since, as argued in the main text,
both SKPLW[T 2] and Stopo[T 2] in Eq. (D.25) are topological invariants. Therefore, we
only need to show that Eq. (5.18) vanishes for one particular entanglement surface
that is topologically equivalent to that in Fig. 5.1 in the main text, such as Fig. D.3.
Then by topological invariance, Eq. (5.18) vanishes for general configurations.
C
B
A
h1
h2
r1r2r3
Figure D.3: KPLW prescription of regularized entanglement surface T 2.
and I(A∪A′) = 0 yields exactly the additivity of the constant part of the entanglement entropy fora TQFT STQFT
c (A).
260
For the configuration in Fig. D.3, we can compute the mean curvature straight-
forwardly. The mean curvature is H = (k1 + k2)/2, where k1 and k2 are the two
principal curvatures at each point of the entanglement surface. We distinguish three
types of points on the cylinder in Fig. D.3.
Points on the top/bottom of a cylinder : the surface is locally flat, k1 = k2 = 0.
Hence, H = (k1 + k2)/2 = 0.
Points on the side of a cylinder : k1 = ±1/r, k2 = 0, where r is the radius of the
cylinder, and the ± sign depends on whether it is inner or outer side surface. Hence,
H = (k1 + k2)/2 = ±1/2r. In the following, we will pick the + sign.
Points on the hinge of a cylinder : One of the hinges of the regular cylinders in
Fig. D.3 is shown as the thick green loop. On every point of the hinge, the Gauss
curvature is the same. To find it, we apply the Gauss-Bonnet theorem to a cylinder.
Because the Gauss curvature on the side and top/bottom of the cylinder vanishes,
integration over the entire surface of the cylinder is reduced to the integration over
the hinge. Hence the Gauss-Bonnet theorem dictates
2
∫
hinge
1
r3
kdσ = 2πχ[C] = 4π, (D.26)
where C is the full cylinder, r3 is the radius of the cylinder. 1/r3 is the principle
curvature along the hinge and k is the principal curvature along the direction per-
pendicular to the hinge. In order to perform the two-dimensional surface integral, we
need to regularize the one-dimensional hinge by smoothing it into an arc of infinites-
imal radius, as shown in Fig. D.4. Assuming the length of the arc is l0, Eq. (D.26)
implies∫ l0
0kdl = 1, which reduces to k = 1/l0. The principal curvature for an ideal
hinge (which corresponds to l0 → 0) is infinite, and we regularize it with the small
parameter l0 to handle the computation.
261
To compute the integral of the mean curvature squared over various surfaces in
Fig. D.3, we first introduce some notation. Let r1 be the inner radius of region
B/C, r2 be the outer radius of region B/C, r3 be the outer radius of region A, h1
be the height of region B, and h2 be the height of region C. We adopt the same
finite regularization for every hinge, although this is not essential. For region A, the
integration∫∂AH2 splits into three parts: the top/bottom, the side and the hinges.
Since the top/bottom surface are flat, they do not contribute to the mean curvature
integral. The mean curvature of the outer side surface is 1/2r3, and that of the inner
side surface is −1/2r2. The integration of the mean curvature over the outer and
inner side of ∂A is
l0A
C
B
r1
r2
r3
✓
r
1
8
7
6
5
4
3
2
11
10
9
12
Figure D.4: Left: Regularization of a rectangular hinge with small arcs. Right: One
choice of regularization of each hinge in Fig. D.3. The numbers label various hinges.
2πr3(h1 + h2)
(1
2r3
)2
+ 2πr2(h1 + h2)
(−1
2r2
)2
=π(h1 + h2)
2r3
+π(h1 + h2)
2r2
. (D.27)
The mean curvature of the outer hinge is (1/r3+1/l0)/2, while according to our choice
of regularization in Fig. D.4, the mean curvature of the inner hinge is (1/l0− 1/r2)/2
because the principle curvature along the θ direction (the meaning of θ and r are
262
specified in Fig. D.4) is −1/r2 and the principle curvature along the r direction is
1/l0 (because we evaluate the curvature from the inside). The integration of the mean
curvature over the hinges is
2× 2πr3l0
(1
2r3
+1
2l0
)2
+ 2× 2πr2l0
(−1
2r2
+1
2l0
)2
, (D.28)
where the factor of 2 in the front comes from equal contribution of the hinges from
the top and bottom respectively. Collecting the above results, we have
∫
∂A
H2 =π(h1 + h2)
2r3
+π(h1 + h2)
2r2
+π(r3 + l0)2
r3l0+π(r2 − l0)2
r2l0. (D.29)
For convenience, we list the mean curvature of each hinge in the following table.
Hinge Mean curvature
1 1/2r3 + 1/2l0
2 1/2r3 + 1/2l0
3 −1/2r2 + 1/2l0
4 −1/2r2 + 1/2l0
5 1/2r2 + 1/2l0
6 1/2r2 + 1/2l0
7 1/2r2 + 1/2l0
8 1/2r2 + 1/2l0
9 −1/2r1 + 1/2l0
10 −1/2r1 + 1/2l0
11 −1/2r1 + 1/2l0
12 −1/2r1 + 1/2l0
263
where the labels of hinges are shown in Fig. D.4. For region B, the side surface
contribution is
2πr2h1
(1
2r2
)2
+ 2πr1h1
(−1
2r1
)2
=πh1
2r2
+πh1
2r1
(D.30)
The hinge contribution is
2× 2πr2l0
(1
2r2
+1
2l0
)2
+ 2× 2πr1l0
(−1
2r1
+1
2l0
)2
=π(r2 + l0)2
r2l0+π(r1 − l0)2
r1l0(D.31)
Hence the total contribution from region B is
∫
∂B
H2 =πh1
2r2
+πh1
2r1
+π(r2 + l0)2
r2l0+π(r1 − l0)2
r1l0(D.32)
For region C, the side surface contribution is
2πr2h2
(1
2r2
)2
+ 2πr1h2
(−1
2r1
)2
=πh2
2r2
+πh2
2r1
(D.33)
The hinge contribution is
2× 2πr2l0
(1
2r2
+1
2l0
)2
+ 2× 2πr1l0
(−1
2r1
+1
2l0
)2
=π(r2 + l0)2
r2l0+π(r1 − l0)2
r1l0(D.34)
Hence the total contribution from region C is
∫
∂C
H2 =πh2
2r2
+πh2
2r1
+π(r2 + l0)2
r2l0+π(r1 − l0)2
r1l0(D.35)
For region AB, the side surface contribution is
2πr3(h1 + h2)
(1
2r3
)2
+ 2πr1h1
(−1
2r1
)2
+ 2πr2h2
(−1
2r2
)2
=π(h1 + h2)
2r3
+πh1
2r1
+πh2
2r2
(D.36)
264
The hinge contribution is
2× 2πr3l0
(1
2r3
+1
2l0
)2
+ 2× 2πr1l0
(− 1
2r1
+1
2l0
)2
+ 2πr2l0
(− 1
2r2
− 1
2l0
)2
+ 2πr2l0
(− 1
2r2
+1
2l0
)2(D.37)
Notice that the third term corresponds to the opposite of hinge 7 (which is not hinge
6). Hence the total contribution from region AB is
∫
∂AB
H2 =π(h1 + h2)
2r3
+πh1
2r1
+πh2
2r2
+π(r3 + l0)2
r3l0+π(r1 − l0)2
r1l0+π(r2 + l0)2
2r2l0+π(r2 − l0)2
2r2l0(D.38)
For region AC, the side surface contribution is
2πr3(h1 + h2)
(1
2r3
)2
+ 2πr2h1
(−1
2r2
)2
+ 2πr1h2
(−1
2r1
)2
=π(h1 + h2)
2r3
+πh1
2r2
+πh2
2r1
(D.39)
The hinge contribution is
2×2πr3l0
(1
2r3
+1
2l0
)2
+2×2πr1l0
(− 1
2r1
+1
2l0
)2
+2πr2l0
(− 1
2r2
− 1
2l0
)2
+2πr2l0
(− 1
2r2
+1
2l0
)2
(D.40)
Hence the total contribution from region AC is
∫
∂AC
H2 =π(h1 + h2)
2r3
+πh2
2r1
+πh1
2r2
+π(r3 + l0)2
r3l0+π(r1 − l0)2
r1l0+π(r2 + l0)2
2r2l0+π(r2 − l0)2
2r2l0(D.41)
For region BC, the side surface contribution is
2πr2(h1 + h2)
(1
2r2
)2
+ 2πr1(h1 + h2)
(−1
2r1
)2
=π(h1 + h2)
2r2
+π(h1 + h2)
2r1
(D.42)
265
The hinge contribution is
2×2πr2l0
(1
2r2
+1
2l0
)2
+2×2πr1l0
(−1
2r1
+1
2l0
)2
=π(r2 + l0)2
r2l0+π(r1 − l0)2
r1l0(D.43)
Hence the total contribution from region BC is
∫
∂BC
H2 =π(h1 + h2)
2r2
+π(h1 + h2)
2r1
+π(r2 + l0)2
r2l0+π(r1 − l0)2
r1l0(D.44)
Finally, for region ABC, the side surface contribution is
2πr3(h1 + h2)
(1
2r3
)2
+ 2πr1(h1 + h2)
(−1
2r1
)2
=π(h1 + h2)
2r3
+π(h1 + h2)
2r1
(D.45)
The hinge contribution is
2×2πr3l0
(1
2r3
+1
2l0
)2
+2×2πr1l0
(−1
2r1
+1
2l0
)2
=π(r3 + l0)2
r3l0+π(r1 − l0)2
r1l0(D.46)
Hence the total contribution from region ABC is
∫
∂ABC
H2 =π(h1 + h2)
2r3
+π(h1 + h2)
2r1
+π(r3 + l0)2
r3l0+π(r1 − l0)2
r1l0(D.47)
266
In summary, we obtain the contribution of mean curvature squared of seven regions
as follows.
∫
∂A
H2 =π(h1 + h2)
2r3
+π(h1 + h2)
2r2
+π(r3 + l0)2
r3l0+π(r2 − l0)2
r2l0.,
∫
∂B
H2 =πh1
2r2
+πh1
2r1
+π(r2 + l0)2
r2l0+π(r1 − l0)2
r1l0,
∫
∂C
H2 =πh2
2r2
+πh2
2r1
+π(r2 + l0)2
r2l0+π(r1 − l0)2
r1l0,
∫
∂AB
H2 =π(h1 + h2)
2r3
+πh1
2r1
+πh2
2r2
+π(r3 + l0)2
r3l0+π(r1 − l0)2
r1l0+π(r2 + l0)2
2r2l0+π(r2 − l0)2
2r2l0,
∫
∂AC
H2 =π(h1 + h2)
2r3
+πh2
2r1
+πh1
2r2
+π(r3 + l0)2
r3l0+π(r1 − l0)2
r1l0+π(r2 + l0)2
2r2l0+π(r2 − l0)2
2r2l0,
∫
∂BC
H2 =π(h1 + h2)
2r2
+π(h1 + h2)
2r1
+π(r2 + l0)2
r2l0+π(r1 − l0)2
r1l0,
∫
∂ABC
H2 =π(h1 + h2)
2r3
+π(h1 + h2)
2r1
+π(r3 + l0)2
r3l0+π(r1 − l0)2
r1l0.
(D.48)
It is straightforward to check that the combination Eq. (5.18) vanishes. Hence the
relation Eq. (5.17) in the main text holds.
D.5 Review of Lattice TQFT
In this section, we briefly review the lattice formulation of TQFTs. We begin with a
triangulation of spacetime. The letters i, j, k etc. label the vertices of a spacetime
lattice. Combinations of vertices denote the simplicies of the lattice. For instance,
(ij) is the 1-simplex (bond) whose ends are vertices i and j. (ijk) is a 2-simplex
(triangle) whose vertices are i, j and k. Gauge fields live on these simplicies. In
our paper, 1-form gauge fields A live on 1-simplicies; 2-form gauge fields B live on
2-simplicies; etc. In the language of discrete theories, A(ij), B(ijk) are the 1-cochain
and 2-cochain associated with the indicated 1-simplex and 2-simplex, respectively.
267
Exterior derivatives are defined by:
dA(ijk) =A(jk)− A(ik) + A(ij),
dB(ijkl) =B(jkl)−B(ikl) +B(ijl)−B(ijk).
(D.49)
Note that the vertices are ordered such that i < j < k < l.
We further illustrate the values that the cochains A(ij) and B(ijk) can take using
canonical quantization. Let us first consider the GWW model described by Eq. (5.19)
on a continuous spacetime with U(1) gauge group. It is known that there are n surface
operators exp(is∮
ΣB), s = 0, 1, · · · , n− 1[134, 135], and exp(in
∮ΣB) = 1 is a trivial
operator for an arbitrary closed surface Σ. Hence∮
ΣB = 2πq
n, where q ∈ Zn and Σ
is any closed surface. The fact that exp(in∮
ΣB) is a trivial operator can be verified
via canonical quantization. To perform canonical quantization, we first use the gauge
transformation Eq. (5.20) to fix the gauge At = 0, Btx = 0, Bty = 0, Btz = 0. The
commutation relations from canonical quantization are
[Ax(t, x, y, z), Byz(t, x′, y′, z′)] = −i2π
nδ(x− x′)δ(y − y′)δ(z − z′). (D.50)
and similarly for other components. Using Eq. (D.50), we find that exp(in∮
ΣB)
commutes with all other gauge invariant operators. Specifically, we compute the
commutation relation between the surface operator exp(in∮
ΣB) and the line operator
exp(il∮γA+ ilp
∫Σ2B). Here Σ is a closed surface in a spatial slice, and Σ2 is an open
surface with boundary γ. Both Σ2 and γ are living in the spatial slice. We find
ein∮ΣBe
il∮γ A+ilp
∫Σ2
B= ei
2πnnlNΣ,γe
il∮γ A+ilp
∫Σ2
Bein
∮ΣB
= eil∮γ A+ilp
∫Σ2
Bein
∮ΣB,
(D.51)
where NΣ,γ is the intersection number of the surface Σ and the loop γ. Since the
phase factor coming from the commutation relation is always 1, exp(in∮
ΣB) com-
268
mutes with all line operators. Since it also commutes with exp(il∮
Σ′B) for any l and
Σ′, we conclude that exp(in∮
ΣB) commutes with all the gauge invariant operators.
Therefore, it must be a constant operator, ein∮ΣB = eiθ where θ is a constant number.
We further show that ein∮Σ B = 1. To show this, we act ein
∮Σ B on a state |0〉 where
B = 0 everywhere (more concretely, if the spacetime is discrete, B = 0 on every
2-simplex). Since ein∮ΣB measures the value of B-field of the state, and B-field is
zero everywhere,
eiθ|0〉 = ein∮Σ B|0〉 = |0〉 (D.52)
Hence the constant number eiθ = 1 everywhere. This proves that ein∮ΣB = 1.
Similarly, exp(in∮γA+ inp
∫Σ2B) commutes with all other operators as well.
ein
∮γ A+inp
∫Σ2
Beil
∮Σ B
= e−i2πnnlNΣ,γeil
∮ΣBe
in∮γ A+inp
∫Σ2
B
= eil∮ΣBe
in∮γ A+inp
∫Σ2
B.
(D.53)
and
ein
∮γ A+inp
∫Σ2
Beil∮γ′ A+ilp
∫Σ′2
B
= e−i 2π
nnlp(Nγ,Σ′2
−Nγ′,Σ2)eil∮γ′ A+ilp
∫Σ′2
Bein
∮γ A+inp
∫Σ2
B
= eil∮γ′ A+ilp
∫Σ′2
Bein
∮γ A+inp
∫Σ2
B.
(D.54)
Therefore ein
∮γ A+inp
∫Σ2
Bcommutes with all gauge invariant operators as well, which
implies ein
∮γ A+inp
∫Σ2
B= eiη where eiη is a constant. Using the same analysis for the
operator ein∮ΣB, we find e
in∮γ A+inp
∫Σ2
B= 1.
On a triangulated lattice, since Σ is any two dimensional surface, exp(in∮
ΣB) = 1
implies that exp(in∮
(ijkl)B) = 1 for any 3-simplex (ijkl). Using the Stokes formula,
269
∮(ijkl)
B =∫
(ijkl)dB = (dB)(ijkl) = B(ijk)−B(ijl) +B(ikl)−B(jkl) where we used
the fact that integrating dB over the volume of 3-simplex (ijkl) is just evaluating the
dB on (ijkl) itself. Hence exp(in∮
(ijkl)B) = 1 implies that B(ijk)−B(ijl)+B(ikl)−
B(jkl) ∈ 2πnZn for any 3-simplex (ijkl). Since the choice of (ijkl) is arbitrary, we
conclude that on each 2-simplex (ijk), B(ijk) takes values in 2πnZn. Similarly, on
each 1-simplex (ij), A(ij) takes values in 2πnZn for any i, j.
Next, we comment on the delta functions obtained from integrating out the A
fields as in Eq. (5.24). For simplicity, we work with a level n = 2 BF/GWW theory.
On each 4-simplex with vertices labeled by (i, j, k, l, s), the action is
2
2π(AdB)(ijkls) =
2
2πA(ij)dB(jkls). (D.55)
Integrating over A means summing over all configurations of A(ij) = 0, π. Hence the
path integral is
1
2
∑
A(ij)=0,π
exp
[i
2
2πA(ij)dB(jkls)
]
=1
2
{1 + exp
[idB(jkls)
]}≡ δ[dB(jkls)
]. (D.56)
This explains the meaning of the delta function in the discrete theory, and we refer to
the B field as flat if the above delta function constraint is satisfied, i.e. if dB(jkls) = 0
mod 2π.
Although we write TQFT actions as integrals in the continuum in the main text,
they can actually be translated into lattice actions using the conventions we have
introduced in this appendix. The wave functions defined via the path integral in
Eqs. (5.23) and (5.27) are then wave functions on the lattice.
270
D.6 Surfaces in the dual lattice
In this appendix, we argue that the simplices on which B = π in the dual lattice form
continuous surfaces. Continuous means that connected simplices in the dual lattice
join via edges, rather than via vertices. Specifically,
1. In three-dimensional space, if a real space 2-cochain B(ijk) satisfies the flatness
condition dB(ijkl) = B(jkl)− B(ikl) + B(ijl)− B(ijk) = 0 mod 2π then its
dual B = π on a closed loop in the dual lattice.
2. In (3 + 1)-dimensional spacetime, if a real space 2-cochain B(ijk) satisfies the
flatness condition dB(ijkl) = B(jkl) − B(ikl) + B(ijl) − B(ijk) = 0 mod 2π
then its dual B = π on a continuous and closed surface in the dual lattice.
The first statement is proven in the main text. In the following, we will present a
more algebraic proof of the first statement, which is easier to generalize to (3 + 1)-
dimensions, allowing for a proof of the second statement.
i
s
qrp
l
k
j
a
e
dcb
Figure D.5: Dual lattice of a tetrahedron (ijkl). (ijkp), (ijlq), (iklr), (jkls) are four
adjacent tetrahedra to (ijkl), which are dual to (b), (c), (d), (e), (a) respectively. The
red dots are the intersection between 2-simplices in the real lattice and the 1-simplices
in the dual lattice. For example, the red dot on (ab) is the intersection point of (ab)
and (ijk).
271
We first redraw the simplex in Fig. 5.3 with some additional details, as shown in
Fig. D.5. To construct the duals of simplices in three-dimensional space, we begin by
considering the tetrahedron (ijkl), in addition to its neighbors (ijkp), (ijlq), (iklr),
and (jkls). 3-simplices in the real lattice are dual to points in the dual lattice: for
example (ijkl) is dual to the point (a), and similarly (ijkp) is dual to (b), (ijlq) is
dual to (c), (iklr) is dual to (d), and (jkls) is dual to (e). 2-simplices in the real
lattice are dual to 1-simplices (bonds). For example, (ijk) is the intersection of (ijkl)
and (ijkp), i.e., (ijk) = (ijkl)∩ (ijkp). Therefore, the dual of (ijk) is the bond (ab),
joining the dual of (ijkl) and (ijkp). Similarly, we are able to identify the duals of
all other simplices. We list the result in the following table:
Real Dual
(ijkl) (a)
(ijkp) (b)
(ijlq) (c)
(iklr) (d)
(jkls) (e)
Real Dual
(ijk) (ab)
(ijl) (ac)
(ikl) (ad)
(jkl) (ae)
The flatness condition implies that there are even number of 2-simplices among the
four faces of the tetrahedron (ijkl) on which B = π. It follows that there are an even
number B = π bonds among the four dual lattice bonds (ab), (ac), (ad), (ae). Thus
these form closed loops in the dual lattice. This proves the first statement.
We proceed to prove the second statement. In (3 + 1) dimensions, spacetime is
triangulated into 4-simplices. Let us consider a 4-simplex labeled by the five vertices
(ijklm) where m is in the extra dimension compared with 3D case shown in Fig. D.5.
To find the dual of 2-simplices, we will begin – as above – by considering the 4-
simplices adjacent to (ijklm) which share one 3-simplex with (ijklm). Introducing
272
the additional vertices p, q, r, s, and t8, these 4-simplices are: (ijkmp), (ijlmq),
(iklmr), (jklms), and (ijklt). Dual simplices in (3 + 1) dimensional spacetime are
determined as follows: 4-simplices in the real lattice are dual to points in the dual
lattice; (ijklm) is dual to a point (a), (ijkmp) is dual to (b), (ijlmq) is dual to (c),
(iklmr) is dual to (d), (jklms) is dual to (e), and (ijklt) is dual to (f)9. 3-simplices
in the real lattice are dual to bonds in the dual lattice. For instance, since (ijkm) is
the intersection of (ijklm) and (ijkmp), i.e., (ijkm) = (ijklm)∩ (ijkmp), the dual of
(ijkl) is the bond (ab), joining the dual of (ijklm) and (ijkmp). Similarly, (ijlm) is
dual to (ac), (iklm) is dual to (ad), (jklm) is dual to (ae), and (ijkl) is dual to (af).
We further proceed to consider the dual of 2-simplices, applying the same method.
For instance, since the 2-simplex (ijk) is the common simplex of (ijkm) and (ijkl),
i.e., (ijk) = (ijkl) ∩ (ijkm), the dual of (ijk) is the surface (abf) joining the dual of
(ijkl) and (ijkm). Similarly, we can identify the duals of the remaining 2-simplices.
We list all the results in the following table:
Real Dual
(ijklm) (a)
(ijkmp) (b)
(ijlmq) (c)
(iklmr) (d)
(jklms) (e)
(ijklt) (f)
Real Dual
(ijkm) (ab)
(ijlm) (ac)
(iklm) (ad)
(jklm) (ae)
(ijkl) (af)
Real Dual
(ijk) (abf)
(ijl) (acf)
(ijm) (abc)
(ikl) (adf)
(ikm) (abd)
(ilm) (acd)
(jkl) (aef)
(jkm) (abe)
(jlm) (ace)
(klm) (ade)
8Notice that t is in the additional dimension as well.9Notice that (f) is in the additional dimension of the dual lattice.
273
The four surfaces (abf), (acf), (adf), (aef) are dual to the four faces (ijk), (ijl), (ikl), (jkl)
of the tetrahedron (ijkl). All of these dual surfaces share a common link (af). The
flatness condition dB(ijkl) = B(jkl)−B(ikl) +B(ijl)−B(ijk) = 0 mod 2π implies
that an even number of faces of the tetrahedron (ijkl) are occupied. Thus, there
are an even number of surfaces among (abf), (acf), (adf), (aef) occupied in the dual
lattice. Since all these occupied surfaces in the dual lattice share a common edge
(af), it follows from our definition of continuity (at the beginning of this appendix)
that surfaces in the dual lattice are continuous. Furthermore, the continuous surfaces
formed by the occupied simplices in the dual lattice are closed, because for any
bond in the dual lattice, for example (af), there exist even (among four) number
of occupied dual-lattice 2-simplices adjacent to it. While for an open dual-lattice
surface, there exist at least one dual-lattice bond such that there are only odd
number of the adjacent dual-lattice 2-simplices occupied, which violate the flatness
condition for the B-cochain. Hence the dual-lattice surface is closed. This proves the
second statement.
For completeness, we comment on how two loops can intersect in the dual space
lattice, and how two surfaces can intersect in the dual spacetime lattice. We first
prove by construction that two loops in the dual spatial lattice can intersect at a
vertex: suppose one dual lattice loop includes the occupied bonds (ab), (ac), and the
other dual lattice loop includes the occupied bonds (ad), (ae). Hence these two loops
intersect at the vertex (a). We now argue that if two surfaces in the dual spacetime
lattice contain the same point, then they must share a bond. Let us assume two
surfaces intersect (at least) at (a). Since all the 2-simplices in the dual lattice including
the vertex (a) are (abc), (abd), (acd), (abe), (ace), (ade), (abf), (acf), (adf) and (aef),
by enumerating all possibilities, we find the two surfaces must share at least one
bond. Without loss of generality, suppose one surface includes the 2-simplices (abc)
and (abd) (notice that (abc) and (abd) join via the bond (ab) and therefore form a
274
continuous surface in the dual lattice). The surface thus includes the three bonds
(ab), (ac), and (ad) emanating from (a). Any other surface that contains (a), would
include, just like this surface, three of bonds emanating from (a). Thus, as (a) is
the only shared part of five bonds (ab), (ac), (ad), (ae), and (af), two surfaces that
include (a) have to share at least one of these bonds, as they occupy three bonds
each. In summary, two loops can intersect at vertices in the dual space lattice, and
two surfaces can intersect at bonds (but not vertices) in the dual spacetime lattice.
D.7 Mutual and Self-Linking Numbers
In this section, we provide all details needed to evaluate the integral Eq. (5.29). As
a simple case, we assume a configuration where B = π only at two surfaces S1, S2 in
the dual lattice of M4, with their boundaries given by the loops l1 = ∂S1, l2 = ∂S2
on the dual lattice of ∂M4. We can write this succinctly as
B = π ∗4 Σ(S1) + π ∗4 Σ(S2), (D.57)
where ∗4 is the discretized version of Hodge star in four spacetime dimensions; its
meaning is explained pictorially in Fig. D.6. Let us comment on Eq. (D.57) in detail.
On ∂M4, B is a 2-cochain, which can be 0 or π; while on the dual lattice of ∂M4,
the π-valued 1-cochains Σ(li) (which are the dual of real-space 2-cochains) form loops
li, i = 1, 2. Moreover, on the spacetime M4, B is still a 2-cochain valued in 0 or
π; while on the dual lattice of M4, the π-valued 2-cochains Σ(Si) (which are the
dual of the real spacetime 2-cochains) form surfaces Si, i = 1, 2 whose boundaries are
li, i = 1, 2. Notice that the closed dual-lattice surfaces which do not intersect with the
spatial slice do not contribute to the wavefunction. Further ∗4Σ(Si) is a 2-cochain on
the original lattice (dual to Si), which is 1 on the dual of Si, and 0 elsewhere. Hence,
the role of the Hodge star is to transform the cochain defined on the dual lattice to the
275
cochain defined on the real lattice. In Fig. D.6 we illustrate the geometric meaning
of these notions with an example in lower dimensions. Returning to the integral in
the wavefunction Eq. (5.29), we thus have
l1
l2
Figure D.6: We illustrate the geometric meaning of the Hodge dual in a two-
dimensional space example. Suppose A is a 1-cochain, which equals π on 1-simplices
in the dual lattice and 0 elsewhere. A = π∗2Σ(l1)+π∗2Σ(l2), where l1 and l2 are loops
in the dual lattice drawn in dashed lines. Σ(l1) and Σ(l2) are 1-cochains living on the
1-simplices in the dual lattice. ∗2 is a lattice version of Hodge star, which transforms
the 1-cochain living on the dual lattice (dashed lines) to a 1-cochain living on the
lattice (green and purple bold lines). Correspondingly, A = π ∗2 Σ(l1) + π ∗2 Σ(l2) is
a 1-cochain living on the green and purple bold lines. We use the dual lattice con-
figuration Si, li to label the B,A-cochains because the dual lattice configurations are
easier to visualize. The interpretation of the 2-cochain B can be straightforwardly
generalized to three spatial dimensions.
276
∫
M4
B ∧B
=π2
∫
M4
(∗4 Σ(S1) + ∗4Σ(S2)
)∧(∗4 Σ(S1) + ∗4Σ(S2)
)
=2π2
∫
M4
∗4Σ(S1) ∧ ∗4Σ(S2) + π2
∫
M4
∗4Σ(S1) ∧ ∗4Σ(S1) + π2
∫
M4
∗4Σ(S2) ∧ ∗4Σ(S2)
=2π2link(l1, l2) + π2link(l1, l1) + π2link(l2, l2),
(D.58)
where link(l1, l2) is the linking number between two loops l1 and l2. This leads to
Eq. (5.30) in the main text.
We will derive the last equality of Eq. (D.58) in Appendix D.7.1, and provide a
detailed discussion of the self-linking numbers of one single loop in Appendix D.7.2.
D.7.1 Intersection and Linking
We prove a statement relating the intersection form in the bulk and the linking number
on the boundary, which in turn explains the last equality in Eq. (D.58).
As explained below Eq. (D.57), ∗4Σ(Si) is a 2-cochain in the real spacetime, which
equals 1 if it is evaluated on any triangulation of Si (in the dual spacetime lattice)
and 0 if evaluated elsewhere. Similarly, ∗3Σ(li) is still a 2-cochain in the real space,
which equals 1 if it is evaluated on the li (in the dual space lattice) and 0 if evaluated
elsewhere. Furthermore, if li is on the boundary of Si (notice that both li and Si are
277
in the dual lattice), we have a relation between these two 2-simplices,10
∗4Σ(Si) = ∗3Σ(∂Si) = ∗3Σ(li). (D.59)
We also notice that B is flat, i.e., d ∗4 Σ(Si) = d ∗3 Σ(li) = 0, i = 1, 2 which come
from the Gauss law for B-cochain Eq. (5.25). This means the duals of the B = π 2-
simplices form two-dimensional surfaces in the spacetime, and form one-dimensional
loops (which are the boundary of two-dimensional dual lattice surfaces) in the space,
as shown in Fig. 5.2. We want to prove,
∫
M4
∗4Σ(S1) ∧ ∗4Σ(S2) =
∫
l1∩∂−1l2
1 ≡ link(l1, l2), (D.60)
where ∂−1l2 denotes a surface in the dual lattice of ∂M4 whose boundary is l2. In
the last equality, we used the definition of the linking number between two loops.
The relation (D.60) can be shown as follows. Keeping in mind that ∗3Σ(l) is a
delta function that is nonzero on l only, we find
∫
l1∩∂−1l2
1 =
∫
M3
∗3Σ(l1) ∧ d−1 ∗3 Σ(l2). (D.61)
10We can understand this formula by constructing examples using the method in appendix D.6.Let (abf), (acf) ∈ S be two dual-lattice 2-simplices in the dual-lattice open surface S in 4D, whichjoin via (af). The boundary is along (ab) and (ac) direction, joined via (a). (ab), (ac) ∈ l forma loop in 3D, which is the boundary of S. We need to compare the real space configuration of Sand l by taking their duals. From the correspondence of real simplices and dual simplices listedin appendix D.6, in 3D, (ab), (ac) are dual to (ijk), (ijl) respectively, and in 4D, (abf), (acf) aredual to (ijk), (ijl) respectively. We find that their real lattice configurations are the same, hence∗4Σ(S) = ∗3Σ(l).
278
Noticing that M3 = ∂M4,
∫
M3
∗3Σ(l1) ∧ d−1 ∗3 Σ(l2) =
∫
∂M4
∗3Σ(l1) ∧ d−1 ∗3 Σ(l2)
=
∫
M4
d(∗4 Σ(S1) ∧ d−1 ∗4 Σ(S2)
)
=
∫
M4
∗4Σ(S1) ∧ ∗4Σ(S2). (D.62)
In the second equality, we used ∗4Σ(Si) = ∗3Σ(li), i = 1, 2. To get the last equality,
we used the flatness condition d ∗4 Σ(Si) = d ∗3 Σ(li) = 0, i = 1, 2. Hence
∫
M4
∗4Σ(S1) ∧ ∗4Σ(S2) =
∫
l1∩∂−1l2
1. (D.63)
Combining Eqs. (D.60), (D.62) and (D.63), we find
∫
M4
B ∧B = 2π2link(l1, l2) + π2link(l1, l1) + π2link(l2, l2).
D.7.2 Self-linking Number
In this subsection, we define the self-linking number of a loop l, i.e., the link(l, l). To
define the self-linking number, we need to regularize the loop into two nearby loops.
This can be achieved by point splitting regularization11. We separate each point of
the spatial lattice into two points, for example
(x, y, z)→
(x, y, z)
(x+ ax, y + ay, z + az)
, (D.64)
where (ax, ay, az) is a constant vector in space chosen to be the same for all loops.
The original loop l splits into two loops l and la.
11The point splitting method is widely used in studying lattice systems, such as in Ref. [198, 199].
279
(ax, ay, az)
Figure D.7: Regularization of a spatial lattice. The blue arrow represents the con-
stant vector (ax, ay, az). The dashed lattice is obtained from the solid lattice by the
translation (x, y, z)→ (x+ ax, y + ay, z + az).
la
l
Figure D.8: An example of lattice regularization of a trefoil knot. l is a knot (drawn
in the dual lattice), while la is the knot obtained by lattice regularization. The
underlying lattice is omitted for clarity.
See Fig. D.7 for an illustration of lattice regularization and Fig. D.8 for an il-
lustration of the regularization of a loop. The mutual-linking number between two
loops is well defined, and it is natural to identify the self-linking number of l to be
280
the mutual-linking number between l and la, i.e.,
link(l, l) ≡ link(l, la). (D.65)
We notice that the definition Eq. (D.65) depends on the regularization Eq. (D.64).
But as long as we use the same regularization for all the loops l [i.e., (ax, ay, az) is
a position-independent constant vector], Eq. (D.65) is consistent [i.e., translating l
(without change its shape) does not change the self-linking number link(l, l) of l].
The definition of the self-linking number of a loop (knot) depends on the point
splitting regularization [i.e., changing the constant vector (ax, ay, az) changes the reg-
ularization, and hence changes the self linking number], and so does the wavefunction.
However, the entanglement entropy is independent of the self-linking number, hence
it is independent of the point splitting regularization.
D.8 NA(CE)NAc(CE) is Independent of CE
In this appendix, we give a more detailed derivation of Eq. (5.41). We first show that
NA(CE)NAc(CE) is independent of CE. We further explain the fact that the number of
configurations on the entanglement surface Σ is 2|Σ|−1.
We start by establishing a one-to-one correspondence between a configuration
CE and a configuration with no dual lattice loops across the entanglement surface.
We find that it is more illuminating to demonstrate this using a two-dimensional
square lattice (but similar arguments work for triangular lattice as well), as shown
in Fig. D.9, which is a spatial slice of the (2 + 1)D spacetime. For simplicity, we
consider the n = 2 case only, where each bond12 is either occupied (B = π mod 2π)
or unoccupied (B = 0 mod 2π). In panel (a), we present a general configuration
12In this section, we will use bonds instead of 1-simplices because simplices are not defined on thesquare lattice.
281
with one occupied loop13 in the dual lattice (the dotted line). The corresponding
configuration in the real lattice is given by the red bonds. The entanglement cut Σ
consists of the green bonds, where two are occupied (bonds which are both green and
red). In panel (b), we present a related configuration with no bonds occupied on Σ.
We denote the boundary configuration on the entanglement surface Σ with no bonds
occupied as C0. The configuration in (b) is obtained from the configuration in (a) by
cutting the loop at Σ in the dual lattice and completing the loops along Σ within
the two regions A and Ac separately. Therefore, we have shown that every bulk
configuration with non-trivial boundary CE can be reduced to a bulk configuration
with trivial boundary configuration C0. However, we note that there can be multiple
ways of cutting and completing the loops (which is more obvious in three spatial
dimensions), and the reduction may not be unique. Hence we have shown that
NAc(CE)NA(CE) ≤ NAc(C0)NA(C0). (D.66)
13The loop configuration is given by the flatness condition dB = 0 mod 2π. On a 2D spatiallattice, B is a 1-form and the flatness condition is (dB)(i, i+x, i+ y, i+x+ y) = B(i, i+x) +B(i+x, i+ x+ y)−B(i+ y, i+ x+ y)−B(i, i+ y) = 0 mod 2π. On a 3D spatial lattice, B is a 2-formand the flatness condition is (dB)(i, i+ x, i+ y, i+ z, i+ x+ y, i+ x+ z, i+ y + z, i+ x+ y + z) =B(i, i+x, i+x+y, i+y)−B(i+z, i+z+x, i+z+x+y, i+z+y)+B(i, i+z, i+x+z, i+x)−B(i+y, i+y+z, i+y+x+z, i+y+x)+B(i, i+y, i+y+z, i+z)−B(i+x, i+x+y, i+x+y+z, i+x+z) = 0mod 2π.
282
A
A
⌃
c
(a)
A
A
⌃
c
(b)
Figure D.9: A configuration associated with nontrivial CE (on panel (a)) can be
reduced to a configuration associated with trivial CE (on panel (b)).
A
A
⌃
c
(a)
A
A
⌃
c
(b)
Figure D.10: A configuration associated with trivial CE (on panel (a)) can be reduced
to a configuration associated with a nontrivial CE (on panel (b)).
To complete the one-to-one correspondence, we have to consider the opposite
deformation: every bulk configuration with trivial boundary configuration C0 can be
changed to a bulk configuration with a specified non-trivial boundary configuration
CE. We use Fig. D.10 to illustrate this process. In panel (a), we present a configuration
with no bonds occupied on Σ, corresponding to the trivial boundary configuration
C0. In panel (b), we draw a specific configuration in which two bonds are occupied.
283
The two occupied bonds on Σ are connected via a “thin” loop along the two sides of
Σ. Therefore, a bulk configuration with nontrivial boundary configuration CE can be
obtained from a bulk configuration with trivial boundary configuration C0 by adding
a “thin” loop along the two sides of the entanglement cut. However, we note that
starting from a configuration with C0, there can be multiple ways to add the thin
loops to obtain a corresponding configuration with a nontrivial CE. Hence, we have
shown that
NAc(C0)NA(C0) ≤ NAc(CE)NA(CE). (D.67)
Combining the inequalities (D.66) and (D.67), we obtain
NAc(CE)NA(CE) = NAc(C0)NA(C0). (D.68)
Equation (D.68) shows that NAc(CE)NA(CE) is independent of the configuration CE,
as expected.
284
(a)
(g)(f)(e)
(d)(c)(b)
(h)
Figure D.11: Configurations on a 2 × 2 lattice with periodic boundary conditions.
There are two entanglement cuts, denoted by two green lines. The occupied bonds
in the real lattice are shown in red, and occupied bonds in the dual lattice are shown
as dotted lines. (a), (b), (c), (d) are configurations with no bonds occupied on the
entanglement cut. (e), (f), (g), (h) are configurations with two bonds occupied on
the entanglement cut.
In addition to the general arguments, it is beneficial to consider an example. In
Fig. D.11, we present all the configurations on a 2 × 2 lattice associated with C0
(no bonds occupied on the entanglement surface) and with CE (two bonds in the
middle occupied on the entanglement surface). The configuration such as does
not exist because the configuration in the dual lattice is not a loop. In each case,
there are 4 configurations, which agrees with our general analysis NAc(CE)NA(CE) =
NAc(C0)NA(C0).
We further show that the total number of configurations on CE is 2|Σ|−1 for the
n = 2 theory, where |Σ| is the number of simplices (bonds) on Σ. (The discussion
285
in this paragraph works for both triangular and square lattices, and we will use the
notations simplices and cochains here.) Notice that since each B-cochain can take
2 values, i.e., 0 mod 2π or π mod 2π, the naive counting of configurations of CE
is 2|Σ|. However, since the simplices where B = π mod 2π form loops in the dual
lattice, there must be an even number of simplices occupied on Σ. This reduces
the total number of CE configurations by half. Therefore, there are 2|Σ|−1 possible
configurations on the entanglement surface. Applying the normalization condition
Eq. (5.40), we complete the demonstration of Eq. (5.41).
D.9 A Case Study of the Conjecture Between GSD
and TEE
In this appendix, we examine the conjecture Eq. (5.58) for the BF theory with level
n in (d+ 1)D by explicitly computing both the GSD on d-dimensional torus T d and
the constant part of the EE across T d−1 (which we believe is the topological part for
the BF theory).
The action of the BF theory with level n on the spacetime T d × S1 is
SBF =
∫
T d×S1
n
2πB ∧ dA, (D.69)
where A is a 1-form gauge field and B is a (d − 1)-form gauge field. The gauge
transformations are A → A + dg, B → B + dλ where λ is a u(1) valued (d − 1)-
form gauge field, and g is a compact scalar (i.e., g ' g + 2π). The gauge invariant
operators, which wrap around the non-contractible cycles of the spatial torus T d, are
V kTi1···id−1
= exp(ik
∮
Ti1···id−1
B), k ∈ {0, 1, · · · , n− 1},
W lTi
= exp(il
∮
Ti
A), l ∈ {0, 1, · · · , n− 1}, (D.70)
286
and their combinations. In the first equation Ti1···id−1is a (d−1)-dimensional torus ex-
tending along the i1 · · · id−1 directions and in the second equation Ti is a 1-dimensional
circle extending along the i-th direction. (The fact that V nTi1···id−1
and W nTi
are triv-
ial operators will be explained in the following.) We will use canonical quantization
to determine the commutation relation between these operators, from which we can
determine the ground state degeneracy GSD[T d].
To perform the canonical quantization, we first fix the gauge as A0 =
0, B0i1···id−2= 0 for any i1 · · · id−2 using the gauge transformations A→ A+dg, B →
B+ dλ. Moreover, the Gauss constraints are ε0i1···id−1id∂id−1Aid = 0 for any i1 · · · id−2,
and ε0i1···id−1id∂i1Bi2···id = 0 where summation over repeated indices is implied. We
have used the definition of totally anti-symmetric tensor
εi1···id−1 =
+1, if i1 · · · id−1 is an even permutation of 0 · · · d− 2
−1, if i1 · · · id−1 is an odd permutation of 0 · · · d− 2
0 otherwise.
(D.71)
The Lagrangian, after gauge fixing, is
LBF =n
2π
(−1)d−1
(d− 1)!εi1···idBi1···id−1
∂0Aid , (D.72)
where Bi1···id−1and Aid obey the Gauss constraints. The canonical quantization con-
ditions on the gauge fields are
[(−1)d−1
(d− 1)!εi1···idBi1···id−1
(t, ~x), Ajd(t, ~y)
]=
2πi
nδidjdδ(~x− ~y). (D.73)
From this canonical relation, one can determine the commutation relation of the line
and higher volume operators by applying the Baker-Campbell-Hausdorff formula. We
287
find
V kTi1···id−1
W lTid
= e(−1)di2πkl/nW lTidV kTi1···id−1
. (D.74)
From Eq.(D.74), we can see that exp(in∮Ti1···id−1
B) commutes with any line
operator exp(ik∮TiA), and also trivially commutes with any surface operator
exp(ik∮Tj1···jd−1
B). Therefore, exp(in∮Ti1···id−1
B) commutes with any gauge invari-
ant operator and should be a constant. By using the same argument as in App. D.5,
exp(in∮Ti1···id−1
B) = 1. Similarly, we find that exp(in∮TiA) = 1 as well. The
explains that the charges k and l of the non-local operators V kTi1···id−1
and W lTid
only
take n different values.
We can define the ground states |u1 · · ·ud〉 to be the eigenstates of W li , and choose
V kTi1···id−1
as the raising and lowering operators acting on the ground states. Since
W ni = 1, the eigenvalues of Wi should be n-th root of unity, i.e., e−(−1)di2πui/n, where
ui ∈ {0, 1, · · · , n− 1}. Specifically,
W li |u1 · · ·ud〉 = e−(−1)di2πlui/n|u1 · · ·ud〉,
V kT12···(i−1)(i+1)···d
|u1 · · ·ud〉 = |u1 · · ·ui−1(ui + 1)ui+1 · · ·ud〉,(D.75)
where ui ∈ {0, 1, · · · , n − 1} for all i. Therefore, there are nd ground states on the
d-dimensional spatial torus, GSD[T d] = nd.
To obtain the EE, we generalize the calculations of Sec. 5.2. Since most of the
calculations are similar, we will only present the crucial steps.
We start by formulating the theory on the higher dimensional triangulated space-
time lattice Md+1. The ground state wavefunction is still the equal weight superpo-
sition of loop configurations in the dual of the spatial lattice,
|ψ〉 = C∑
C∈L
|C〉, (D.76)
288
where the sum is taken over the set L of all possible loop configurations C at the
dual lattice of spatial slice Sd = ∂Md+1. We choose the entanglement surface to be
a (d − 1)-dimensional torus, separating the space into two regions A and Ac. The
wavefunction is
|ψ〉 = C∑
CE
NA(CE)∑
a=1
NAc (CE)∑
b=1
|ACEa 〉|AcCEb 〉, (D.77)
from which one can obtain the reduced density matrix by tracing over the degrees of
freedom in region Ac,
ρA = |C|2∑
CE
NAc(CE)
NA(CE)∑
a,a′=1
|ACEa 〉〈ACEa′ |. (D.78)
The normalization constant C is determined by TrHAρA = |C|2NA(CE)NAc(CE)n|Σ|−1 =
1, where |Σ| is the number of (d− 1)-simplices on the entanglement surface. The EE
is
S(A) = −TrHAρA log ρA =
d
dN
(− TrHA
ρNA(TrHA
ρA)N
)∣∣∣∣N=1
= − d
dN
(|C|2N
∑
CE
NAc(CE)NNA(CE)N)∣∣∣∣
N=1
= − d
dN
(∑
CE
n−(|Σ|−1)N
)∣∣∣∣N=1
= − d
dN
(n−(|Σ|−1)(N−1)
)∣∣∣∣N=1
= |Σ| log n− log n.
(D.79)
In the second line, we used the normalization TrHAρA = 1, TrHA
ρNA = |C|2N∑CE NAc(CE)NNA(CE)N .
In the third line, we used |C|2NA(CE)NAc(CE) = n−(|Σ|−1). In the fourth line, since
the summand does not depend on CE, we just multiply the summand by the number
of CE n|Σ|−1. In the last line, we take the differential with respect to N and take
289
N = 1. Therefore, the constant part of the EE across T d−1 is − log n, which we
conjecture to be the TEE across T d−1. Combining the results GSD[T d] = nd and
Stopo[T d−1] = − log n, we expect that the conjecture exp(−dStopo[T d−1]) = GSD[T d]
of Eq. (5.58) holds for the (d+ 1)-dimensional BF theory.
290
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