Renormalization of Tensor Network States

Tao Xiang

Institute of Physics

Chinese Academy of Sciences

txiang@iphy.ac.cn

II. RG of Tensor Network States

Tensor-Network Ansatz

of the ground state wavefunction

: 1D: Matrix Product State (MPS)

| =

𝑚1,…𝑚𝐿

(𝑚1, …𝑚𝐿)|𝑚1, …𝑚𝐿

m1

m2

m3mL-2

mL-1

mL

…

Parameter number grows

exponentially with L

𝑑𝐿 parameters

1

1 1[ ]... [ ] ...L

L L

m m

Tr A m A m m m

𝑑𝐷2𝐿 parameters

m1 m2 m3 … … mL-1 mL

A[m2 ]

D

d

Parameter number grows

linearly with L

Virtual basis state

MPS is the wave function generated by the DMRG

1

1 1[ ]... [ ] ...

0 0 1 0 0 2[ 1] [0] [1]

0 12 0 0 0

L

L L

m m

Tr A m A m m m

A A A

2

1 1

1 1 2

2 3 3i i i i

i

H S S S S

Affleck, Kennedy, Lieb, Tasaki, PRL 59, 799 (1987)

Example：S=1 AKLT valence bond solid state

A[m]

m

virtual S=1/2 spin

A[m] :

To project two virtual

S=1/2 states, and ,

onto a S=1 state m

A[m]

m

=

1 =1

21

2

m1 m2 m3 … … mL-1 mL

1

1 1[ ]... [ ] ...L

L L

m m

Tr A m A m m m A[m]

m

Matrix product state (MPS)

Gauge Invariance

MPS wavefunction is unchanged if one replaces A[m] by

𝐴 𝑚 → 𝐴′ 𝑚 = 𝑃𝐴 𝑚 𝑃-1

MPS as a Projection of 2D Tensor-Network Model

Ai

Matrix Product Operator

𝑚1′ 𝑚2

′ 𝑚3′ … … 𝑚𝐿−1

′ 𝑚𝐿′

M

𝑚1 𝑚2 𝑚3 … … 𝑚𝐿−1 𝑚𝐿

𝑂 =

𝑚

𝑇𝑟 𝑀 𝑚1, 𝑚1′ ⋯𝑀 𝑚𝐿 , 𝑚𝐿

′ | 𝑚1 ⋯𝑚𝐿 𝑚1′⋯𝑚𝐿′ |

=

Ground state eigen-operator: 𝑂＝｜ ｜

𝑥 𝑥′

2D: Projected Entangled Pair State (PEPS)

𝑇𝑥𝑥′𝑦𝑦′ [𝑚] =

𝑦

y'

𝑚

Physical

basis

Local

tensor

Virtual

basis

D

Key parameter: virtual basis dimension D

Virtual spins at each bond form a maximally entangled state

Tensor product states

H. Niggemann and J. Zittarz, Z. Phys. B 101, 289 (1996)

G. Sierra and M. Martin-Delgado, 1998

Variational approach:

Nishino, Okunishi, Maeshima, Hieida, Akutsu, Gendiar (since 1998)

Projected entangled pair states (PEPS) : Area law obeys

F. Verstraete and J. Cirac, cond-mat/0407066

Tensor-Network State as a Variational Ansatz

PEPS: exact representation of Valence Bond Solid

𝑇𝑎𝑏𝑐𝑑 [𝑚𝑖] =𝑎 𝑏

𝑐

d𝑚𝑖

Physical stateVirtual basis state

S = 2

𝐻 =

𝑖𝑗

𝑃4(𝑖, 𝑗)

To project two S=2 spins on sites

i and j onto a total spin S=4 state

1/2

Bond dimension dependence of the ground state energy

and magnetization

Gapless: Power law dependent on D

Gapped: Exponential law dependent on D

Bond Dimension Dependence of Physical Quantities

1. Determine the local tensors

2. Evaluate the physical observables using the TRG

or other tensor network RG methods

Two Problems Need to Be Solved

Evaluation of Expectation Values

|

|

𝐷

𝑑 = 𝐷2

|

𝐷2

How Large Is the Virtual Bond Dimension Needed?

|

𝐷2

𝑑 = 𝐷2

In the DMRG or all tensor-related

approached, small physical dimension d

is much earlier to study than a larger 𝑑

(accuracy drops quickly with 𝑑)

The virtual bond dimension needed to

obtain a converged result is roughly of

order 𝑑, i.e. 𝐷2, of course, the larger the

better

Computational Cost

Method CPU Time Minimum Memory

TMRG/CTMRG 𝐷12𝐿 𝐷10

TEBD 𝐷12𝐿 𝐷10

TRG 𝐷12ln𝐿 𝐷8

SRG 𝐷12ln𝐿 𝐷8

HOTRG 𝐷14ln𝐿 𝐷8

HOSRG 𝐷16ln𝐿 𝐷12

TNR 𝐷14ln𝐿 𝐷10

Loop-TNR 𝐷12ln𝐿 𝐷8

𝐷: bond dimension of PEPS bond dimension kept ~ 𝐷2 𝐿: lattice size

Tensor Network States

in the Frustrated Lattices

Z. Y. Xie et al, PRX 4, 011025 (2014)

Geometric frustration

1

1

N

i i

i

H J S S

J

J

J

Quantum frustration

e.g. S=1 bilinear-biquadratic Heisenberg model

2

cos sini j i j

ij

H S S S S

Two Kinds of Frustrations

More than two-body correlations/entanglements are important

PEPS on Kagome or Other Frustrated Lattices

There is a serious cancellation in the tensor elements if three

tensors on a simplex (triangle here) are contracted

3-body (or more-body) entanglement is important

Max ( ) ~ 1

Max ( ) < 10-6

Cancellation in the PEPS

1

1

N

i i

i

H J S S

Projected Entangled Simplex States (PESS)

Projection tensor

Simplex tensor

Virtual spins at each simplex form a maximally entangled state

Remove the geometry frustration: The PESS is defined on the

decorated honeycomb lattice

Only 3 virtual bonds, low cost

Z. Y. Xie et al, PRX 4, 011025 (2014)

PESS as an exact representation of Simplex Solid States

D. P. Arovas, Phys. Rev. B 77, 104404 (2008)

Example: S = 2 spin model on the Kagome lattice

A S = 2 spin is a symmetric superposition of two virtual S = 1 spins

Three virtual spins at each triangle form a spin singlet

Projection tensor

Simplex tensor

S=2 Simplex Solid State on the Kagome Lattice

𝐴𝑎𝑏[𝜎] =1 1 2𝑎 𝑏 𝜎

antisymmetric tensor

C-G coefficients

Local tensors

Pn : projection operator

Parent Hamiltonians

or

Projection tensor

Simplex tensor

To enlarge each simplex so that it contains more physical spins

5-PESS: a decorated

square lattice

9-PESS: a honeycomb

lattice

Larger Simplex PESS

P. Corboz et al, PRB 86,

041106 (2012).

PESS on Triangular Lattice

PEPS PESS

Order of local tensors: dD6 Simplex tensor: D3

Projection tensor: dD3

PESS on Square Lattice

J1 only J1- J2 model

Vertex-sharing Edge-sharing

How to Determine

the Tensor-Network Wave Function

Determination of Tensor Network Wavefunction

1. Imaginary time evolution

Simple update (entanglement mean-field approach)

Jiang, Weng, Xiang, PRL 101, 090603 (2008)

the solution can be used as the initial input of local tensors in I or

in the full update calculation

Full update

Murg, Verstraete, Cirac, PRA 75, 033605 (2007)

2. Minimize the ground state energy

Nishino et al, Nuclear Physics B 575 [FS] 504 (2000)

F. Verstraete and J. Cirac, cond-mat/0407066

Variational Minimization of Ground State Energy

Determine local tensors by minimizing the ground state energy

𝐸 = 𝐻

|

Accurate

Cost is high

D is generally less than 13 without using symmetries

[ ] [ ] i i i i i i j j jx y z x y z i x y z j i j

i blackj white

Tr A m B m m m

Bond vectors: measure approximately the “entanglement”

on the corresponding bonds

Simple Update: Entanglement Mean-Field Approach

Simple Update: Entanglement Mean-Field Approach

state groundlim

He

The local tensors are determined by projection

Converge fast

D as large as 100 can be calculated (more if symmetry is

considered)

Exact on the Bethe lattice

Li, von Delft, Xiang, PRB 86, 195137 (2012)

Canonical Form on the Bethe Lattice

Li, von Delft, Xiang,

PRB 86, 195137 (2012)

ij x y z

ij

ij i j

H H H H H

H JS S

Heisenberg model

Simple Update: Imaginary Time Evolution

state groundlim

state groundlim

MH

M

H

e

e

1. One iteration

2. Repeat the above iteration until converged

20

12

01

~

z

y

x

H

H

H

e

e

e

Trotter-Suzuki

decomposition),,(

)(

,

2

zyxHH

oeeee

blacki

ii

HHHH xyz

Imaginary Time Evolution

,

ˆ

[ ] [ ] i jx

i i i i i i j j j

HH

i j i j x y z x y y i x y y j i j

i blackj i x

e Tr m m e m m A m B m m m

Step I

Step II

Step III

SVD: singular value decomposition

Step I

Step II

Step III Truncate basis space

One Step of Evolution

To use bond vector as effective

fields to take into account the

environment contribution

The projection is done locally. This

keeps the locality of wavefunction,

making the calculation efficient

Truncation error not accumulated

,

ˆ

[ ] [ ] i jx

i i i i i i j j j

HH

i j i j x y z x y y i x y y j i j

i blackj i x

e Tr m m e m m A m B m m m

One Step of Evolution

Step I

Step II

Step III

SVD: singular value decomposition

1

1

N

i i

i

H J S S

Application: Heisenberg Model on the Husimi Lattice

Locally similar to the Kagome lattice, but

less frustrated

Helpful to understand the Kagome

Heisenberg model

Kagome lattice Husimi lattice

Heisenberg Model on the Husimi Lattice

Husimi lattice

1

1

N

i i

i

H J S S

PESS is defined on a unfrustrated

lattice and can be easily studied

Simple update is rigorous

Is the Ground State a Spin Liquid?

Kagome lattice

If yes, then the g.s of the Kagome Heisenberg

should also be a spin liquid since the Kagome

is more frustrated than the Husimi lattice

1

1

N

i i

i

H J S S

Husimi lattice

E0 converges algebraically with D the excitation is gapless

Ground State Energy

1200 Neel ordered at any finite 𝐷, vanishes in the limit 𝐷 → ∞

𝑀 ~ 𝐷𝛼 with 𝛼 = −0.588(2))

Magnetization: Ground State is a Gapless Spin Liquid

Kagome Lattice

Z. Y. Xie et al, PRX 4, 011025 (2014)

1

1

N

i i

i

H J S S

Issue under debate:

Is the ground state a spin liquid?

S=1/2 Kagome Heisenberg Model

Possible Ground States

Valence bond crystal

R. R. P. Singh and D. A. Huse, PRB 77, 144415 (2008) series expansion

G. Evenbly and G. Vidal, PRL 104, 187203 (2010) MERA

Y. Iqbal, F. Becca, and D. Poilblanc, PRB 83, 100404 (2011) VMC

Z2 Gapped spin liquid

S. Yan, D. A. Huse, and S. R. White, Science 332, 1173 (2011) DMRG

Depenbrock,McCulloch,Schollwock, PRL 109, 067201 (2012) DMRG

………….

U(1) Gapless spin liquid

Y. Ran, M. Hermele, P. A. Lee, and X.-G. Wen, PRL 98, 117205 (2007) Gutzwiller

Y. Iqbal, F. Becca, S. Sorella, and D. Poilblanc, PRB 87, 060405 (2013) VMC+Lanczos

………….

Difficulty: Lack of Good Numerical Methods

Mean field or variational approach:

need accurate guess of trial wavefunction

Quantum Monte Carlo:

suffers from the minus sign problem on frustrated systems

Density Matrix Renormalization Group (DMRG):

limited to small lattice systems (area law), the number of states

need to be retained grows exponentially with the circumference

Upper bound: iDMRG (cylinder Ly=12), D=5000, E=-0.4332

Yan et al: -0.4379(3)

Depenbrock et al: -0.4386(5), D=16000, Ly=17

DMRG results

Valence bond solid

Gapped spin liquid

Gapless spin liquid

Ground state energy of the Kagome Heisenberg model

DMRG

Valence bond solid

Gapped spin liquid

Gapless spin liquid

3-PESS

Ground state energy of the Kagome Heisenberg model

Ground state energy of the Kagome Heisenberg model

The ground state energy converges algebraically with D,

indicating that the system is gapless.

Comparison between Kagome and Husimi Lattices

Bond Dimension D

No long-range magnetic order in the infinite D limit

The ground state is more likely a gapless spin liquid

120 Degree Neel Magnetization

Deconfined Quantum Point

2

cos sini j i j

ij

H S S S S

Honeycomb lattice: each site has only 3 neighbors,

quantum fluctuation is strong

Quantum Monte Carlo has the minus sign problem

when sin > 0

S = 1 Bilinear-biquadratic Heisenberg model on Honeycomb Lattice

2

cos sini j i j

ij

H S S S S

FQ

IVFerro-magnetic

Antiferro-magnetic

Ferro-quadrupolar

Antiferro-quadrupolar

( )i

iQ

iQ

AFQ

Classical phase diagram

Quantum fluctuation may

lead to a deconfined

quantum critical point

Ground State Phase Diagram in the Classical Limit

2nd order

There are four phases, but the AFQ phase is killed by quantum fluctuation

d 0.19

Ground State Energy and Orders

d

FQ

Magnetic Orders

Continuous phase transition point:

deconfined critical point?

red bon

black o

d

b nd

12

i j

i j

S SP

S S

d

Plaquette Valence Bond Crystal

Proper tensor-network wave function in treating a

frustrated quantum lattice system

The simple update an approximate and efficient

algorithm for determining the local tensors

Kagome Heisenberg model is likely to be a gapless

spin liquid, but more study needed

Summary