boundary degeneracy of topological order
TRANSCRIPT
Boundary Degeneracy of Topological Order
Juven Wang1, 2 and Xiao-Gang Wen2, 1, 3
1Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA2Perimeter Institute for Theoretical Physics, Waterloo, ON, N2L 2Y5, Canada
3Institute for Advanced Study, Tsinghua University, Beijing, 100084, P. R. China
We introduce the notion of boundary degeneracy of topologically ordered states on a compactorientable spatial manifold with boundaries, and emphasize that it provides richer information thanthe bulk degeneracy. Beyond the bulk-edge correspondence, we find the ground state degeneracyof fully gapped edge states depends on boundary gapping conditions. We develop a quantitativedescription of different types of boundary gapping conditions by viewing them as different waysof non-fractionalized particle condensation on the boundary. Via Chern-Simons theory, this allowsus to derive the ground state degeneracy formula in terms of boundary gapping conditions, whichreveals more than the fusion algebra of fractionalized quasiparticles. We apply our results to Toriccode and Levin-Wen string-net models. By measuring the boundary degeneracy on a cylinder, wepredict Zk gauge theory and U(1)k×U(1)k non-chiral fractional quantum hall state at even integer kcan be experimentally distinguished. Our work refines definitions of symmetry protected topologicalorder and intrinsic topological order.
PACS numbers:
Introduction— Quantum many-body systems exhibitsurprising new phenomena where topological order andthe resulting fractionalization are among the centralthemes[1, 2]. One way to define topological order isthrough its ground state degeneracy(GSD) on two spa-tial dimensional(2D) higher genus closed Riemann sur-face, which is encoded by the fusion rules of fractionalizedquasiparticles and the genus number[3]. However, on a2D compact manifold with boundaries(Fig. 1), there canbe gapless boundary edge states. For non-chiral topo-logical order states, where the numbers of left and rightmoving modes equal, we show there are rules that edgestates can be fully gapped out. It is the motivation ofthis paper to understand GSD for this type of systemswhere all boundary edge excitations are gapped. In thefollowing, we name this degeneracy as boundary GSD,to distinguish it from bulk GSD of a gapped phase ona closed manifold without boundary. To understand theproperty of boundary GSD is both of theoretical inter-ests and of application purpose where lattice model intopological quantum computation such as Toric code[4]can be put on space with boundaries[5, 6].
In this paper, we demonstrate boundary GSD is notsimply a factorization of the degeneracies of all bound-aries. Not only the fusion rules of fractionalized quasipar-ticles(anyons) and the manifold topology, but also bound-ary gapping conditions are the necessary data. BoundaryGSD reveals richer information than bulk GSD. More-over, gluing edge states of a compact manifold withboundaries to form a closed manifold, enables us toobtain bulk GSD from boundary GSD. We first intro-duce physical notions characterizing this boundary GSDand then rigorously derive its general formula by Chern-Simons theory[1]. For concrete physical pictures, we re-alize specific cases of our result by Z2 Toric code andstring-net model[7]. Interestingly, by measuring bound-
ary GSD on a cylinder, our result predicts distinction be-tween Zk gauge theory(Zk Toric code) and U(1)k×U(1)knon-chiral fractional quantum hall state at even integerk, despite the two phases cannot be distinguished by bulkGSD. By the same idea, we refine definitions of symmet-ric protected topological order(SPT) and intrinsic topo-logical order. Our prediction can be tested experimen-tally.
es
(a) (b)
FIG. 1: (a) Illustration of fusion rules and total neutral-ity, where anyons are transported from one boundary to an-other(red arrows), or when they fuse into physical excita-tions(blue arrows) (b) A higher genus compact surface withboundaries(punctures).
Physical Notions— Topological order on a compactspatial manifold with boundaries have N branches ofgapless edge states[1]. Suppose the manifold has to-tal η boundaries, we label each boundary as ∂α, with1 ≤ α ≤ η. Let us focus on the case the manifold is home-omorphic to a sphere with η punctures(Fig. 1(a)), we willcomment on cases with genus or handles(Fig. 1(b)) later.If electrons condense on the boundary due to edge statesscattering, it introduces mass gap to the edge states. Aset of electrons can condense on the same boundary ifthey do not have relative quantum fluctuation phaseswith each other, thus all condensed electrons are stabi-lized in the classical sense. It requires condensed electrons
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have relative zero Aharonov-Bohm (charge-flux) phase,we call these electrons are null and mutual null[8]. Sinceelectrons have discrete elementary charge unit, we labelthem as a dimension-N(dim-N) lattice Γe, and label con-densed electrons as discrete lattice vectors `∂α(∈ Γe) forboundary ∂α. We define a complete set of condensed elec-trons, labeled as lattice Γ∂α , to include all electrons whichare null and mutual null to each other. Notably thereare different complete sets of condensed electrons, repre-senting different kinds of boundary gapping conditions.Assigning a complete set of condensed (non-fractionalizedbosonic) electrons to a boundary corresponds to assign-ing a type of boundary gapping condition. The numberof types of complete sets equals to the number of typesof boundary conditions. In principle each boundary canassign its own boundary condition independently, thisassignment is not determined from the bulk information.Below we focus on the non-chiral orders, assuming allbranches of edge states can be fully gapped out.
Remarkably there exists a set of compatible anyons donot produce flux effect to condensed electron charge, sotheir Aharonov-Bohm phases are zero. In other words,compatible anyons are mutual null to any elements in thecomplete set of condensed electrons. For boundary ∂α,We label compatible anyons as discrete lattice vectors `∂αqpand find all such anyons to form a complete set labeledas Γ∂αqp . Note that Γ∂α ⊂ Γ∂αqp . Γ∂α and Γ∂αqp have thesame dimension. If compatible anyons can transport be-tween different boundaries of the compact manifold, theymust follow total neutrality - the net transport of compat-ible anyons between boundaries must be balanced by thefusion of physical particles in the system(Fig. 1(a)), so∑α `
∂αqp ∈ Γe. Transporting anyons from boundaries to
boundaries in a fractionalized manner(i.e. not in integralelectron units), changes topological sectors(i.e. groundstates) of the system. Given data: Γe,Γ
∂α ,Γ∂αqp , we thusderive generally GSD counts the number(Num) of ele-ments in a quotient group:
GSD = Num[{(`∂1qp, . . . , `
∂ηqp) | ∀`∂αqp ∈ Γ∂αqp ,
∑α `
∂αqp ∈ Γe}
{(`∂1 , . . . , `∂η ) | ∀`∂α ∈ Γ∂α}]
(1)
Boundary degeneracy of Abelian topological or-der— To demonstrate our above notions, let us takeAbelian topological order as an example, which is be-lieved to be fully classified by K matrix Abelian Chern-Simons theory[9]. For a system lives on a 2D compactmanifoldM with 1D boundaries ∂M, edge states of eachclosed boundary(homeomorphic to S1) are described bymultiplet chiral bosons[1], with bulk and boundary ac-tions:
Sbulk =KIJ
4π
∫Mdt d2x εµνρaIµ∂νa
Jρ (2)
S∂ =1
4π
∫∂M
dt dx KIJ∂tΦI∂xΦJ + VIJ∂xΦI∂xΦJ
+
∫∂M
dt dx∑a
ga cos(`a,I · ΦI) (3)
KIJ and VIJ are symmetric integer N × N matrices,aIµ is the 1-form emergent gauge field’s I-th componentin the multiplet. cos(`a,I · ΦI) is derived from local
Hermitian gapping term ψ + ψ† = ei`a,I ·ΦI + e−i`a,I ·ΦI ,where `a has N -component with integer coefficients.
Canonical quantization of K matrix Abelian Chern-Simons theory edge states— In order to understand theenergy spectrum or GSD of the edge theory, we study the‘quantum’ theory, by canonical quantizing the boson fieldΦI . Since ΦI is the compact phase of a fermion field, itsbosonization has zero mode φ0I and winding momentumPφJ , in addition to non-zero modes[10]:
ΦI(x) = φ0I +K−1IJ PφJ
2π
Lx+ i
∑n 6=0
1
nαI,ne
−inx 2πL (4)
The periodic boundary size is L. The conjugate momen-tum field of ΦI(x) is ΠI(x) = δL
δ(∂tΦI) = 14πKIJ∂xΦJ .
With conjugation relation for zero modes: [φ0I , PφJ ] =iδIJ , and generalized Kac-Moody algebra for non-zeromodes: [αI,n, αJ,m] = nK−1
IJ δn,−m. We thus have canon-ical quantized fields: [ΦI(x1),ΠJ(x2)] = 1
2 iδIJδ(x1−x2).
Gapping Rules, Hamiltonian, and Hilbert Space[11]—Let us determine the properties of `a as condensed elec-trons and the set of allowed boundary gapping latticeΓ∂ = {`a}, a labels the a-th vector in Γ∂ . With-out any symmetry constraint, then any gapping termis allowed if it is[11]: (1) Null and mutual null[8]:∀`a, `b ∈ Γ∂ , `a,IK
−1IJ `b,J = 0, This implies self statis-
tics and mutual statistics are bosonic, and the excitationis local. Localized fields are not eliminated by self ormutual quantum fluctuations, so the condensation sur-vives in the classical sense. (2) ‘Physical’ excitation:`a ∈ Γe = {
∑J cJKIJ | cJ ∈ Z}, is an excitation of
electron degree of freedom since it lives on the ‘physical’boundary. (3) Completeness of Γ∂ , defined by: ∀`c ∈ Γe,if `cK
−1`c = 0 and `cK−1`a = 0 with ∀`a ∈ Γ∂ , then
`c ∈ Γ∂ must be true. It turns out rules (1)(2)(3) are notguaranteed to fully gap out edge states, we add an extrarule (4): The system is non-chiral. The signature of K(≡ Num of positive eigenvalues − Num of negative eigen-values) must be zero, i.e. N is even. Additionally, thedimension of Γ∂ must be N/2 and
√|detK| ∈ N. Phys-
ically, rule (4) excludes violating examples such as oddrank(≡rk) K matrix with central charge c− c̄ 6= 0, whichuniversally has gapless chiral states. For instance, thedim-1 boundary gapping lattice: {n(A,B,C) | n ∈ Z} ofK3×3 = diag{1, 1,−1}, with A2 + B2 − C2 = 0, satisfies(1)(2)(3), but cannot fully gap out edge states.
3
Since any linear combinations of `a ∈ Γe still sat-isfy (1)(2)(3)(4), we can regard Γ∂ as an infinite dis-crete lattice group. As we solve exactly the number oftypes of boundary gapping conditions N ∂
g , at rk(K) =
2, N ∂g = 2. However, for rk(K) ≥ 4, N ∂
g = ∞,which is rather surprising. For example, as K4×4 =diag{1, 1,−1,−1}, one can find dim-2 boundary gappinglattices, {n(A,B,C, 0),m(0, C,B,A) | n,m ∈ Z}, wheredifferent sets of A2 +B2 −C2 = 0 give different lattices.There are infinite solutions of them[11].
To determine the mass gap of boundary states andshow the gap is finite in the large system size limitL → ∞, we will take large g coupling limit of Hamilto-
nian: −ga∫ L
0dx cos(`a,I ·ΦI)→ 1
2ga(`a,I ·ΦI)2L. Solvethe quadratic Hamiltonian exactly, we find the spectrumE(n) =
√∆2 + (nπL )2δ1 + (nπL )δ2, generically indicating
edge states has finite gap ∆ independent of boundary sizeL. To count boundary GSD of this gapped system, weneed to include the full cosine term for the lowest energystates - zero and winding modes:
cos(`a,I · ΦI)→ cos(`a,I · (φ0I +K−1IJ PφJ
2π
Lx)) (5)
The kinetic term Hkin = (2π)2
4πL VIJK−1Il1K
−1Jl2Pφl1Pφl2 has
O(1/L) so can be neglected in L → ∞, the cosine po-tential Eq. (5) dominates, use [`a,Iφ0I , `a,I′K
−1I′JPφJ ] =
`a,IK−1I′J`a,I′δIJ = 0 by rule (1), we can safely expand
the cosine term, then integrate L:
ga
∫ L
0
dx Eq.(5) = gaL cos(`a,I · φ0I)δ(`a,I ·K−1IJ PφJ ,0)
(6)φ0 is periodic, so Pφ forms a discrete lattice. By rule (2),so cos(`a,I ·φ0I) are hopping terms along condensed elec-tron vector `a,I in sublattice of Γe in Pφ lattice. P qpφ rep-resents compatible anyons `qp which is mutual null to con-densed electrons ` by `K−1P qpφ = `K−1`qp = 0. By rule(1), `qp parallels along some ` vector. However, `qp liveson quasiparticle lattice, i.e. unit integer lattice of Pφ lat-tice. So `qp is parametrized by 1
| gcd(`a)|`a,J , with greatest
common divisor | gcd(`a)| ≡ gcd(|`a,1|, |`a,2|, . . . , |`a,N |).For boundary ∂α, a complete set of condensed elec-
trons forms boundary gapping lattice Γ∂α = {∑a I
∂αa `∂αa,I |
I∂αa ∈ Z}. A complete set of compatible anyons forms theHilbert space of winding modes on Pφ lattice:
Γ∂αqp = {`∂αqp,I} = {∑a
j∂αa`∂αa,I
| gcd(`∂αa )|| j∂αa ∈ Z} (7)
or simply anyon hopping lattice. Anyon fusion rules andtotal neutrality essentially imply physical charge excita-tions can fuse into multiple anyon charges. The rulesconstraint the direct sum[12] of anyon hopping lattice Γ∂αqp
over all η boundaries must be on the Γe lattice,
Lqp⋂e ≡ {
η⊕α=1
∑a
j∂αa`∂αa,I
| gcd(`∂αa )|| ∀j∂αa ∈ Z, ∃ cJ ∈ Z,
3∑α,a
j∂αa`∂αa,I
| gcd(`∂αa )|=∑J
cJKIJ} (8)
GSD counts topological sectors distinct by fractionalizedanyons transport between boundaries. With
⊕α Γ∂α is
a normal subgroup of Lqp⋂e, and given input data: K
and Γ∂α (which determines Γ∂αqp ), we derive GSD is thenumber of elements in a quotient finite Abelian group:
GSD = Num[ Lqp⋂ e⊕
α Γ∂α
](9)
analog to Eq. (1). Interestingly Eq. (9) works forboth closed manifolds or compact manifolds withboundaries[13]. By gluing boundaries of compact man-ifold, enlarging K matrix of glued edges states toK2N×2N , creating N scattering channels to fully gapout edge states. For a genus g Riemann surface withη′ punctures(Fig. 1(b)), we start with a number of gcylinders drilled with extra punctures[11], use Eq. (9),remove glued boundaries part which contributes a factor≤ |detK|g, and redefine particle hopping lattices Lqp
⋂e
and⊕
α′ Γ∂α′ only for unglued boundaries (1 ≤ α′ ≤ η′),
GSD ≤ |detK|g ·Num[ Lqp
⋂e⊕η′
α′=1 Γ∂α′] (10)
For genus g Riemann surface(η′ = 0), Eq. (10) becomesGSD ≤ | detK|g. The inequalities are due to differentchoices of gapping conditions for glued boundaries[13].
Apply our algorithm to generic K2×2 matrix case[11],to fully gap out edge states require detK = −k2 withinteger k. Take a cylinder(a sphere with 2 punctures) asan example, Eq. (9) shows GSD =
√|detK| = k when
boundary conditions on two edges are the same Γ∂1 =Γ∂2 . However, GSD ≤ k when Γ∂1 6= Γ∂2 . Specifically,take KZk =
(0 kk 0
)and Kdiag,k =
(k 00 −k
), we obtain KZk
has GSD = 1, while Kdiag,k has GSD = 1 for odd k butGSD = 2 for even k. See Table I.
KZk Kdiag,k
Boundary Γ∂1 6= Γ∂2 1 1 (k ∈ odd) or 2 (k ∈ even)
GSD Γ∂1 = Γ∂2 k k
Bulk GSD k2 k2
TABLE I: Boundary GSD and bulk GSD for KZk and Kdiag,k
This shows a new surprise, we predict a distinctionbetween two types of order KZk(Zk gauge theory) andKdiag,k(U(1)k×U(1)k non-chiral fractional quantum hallstate) at even integer k by simply measuring their bound-ary GSD on a cylinder.
4
Mutual Chern-Simons, Zk gauge theory , Toriccode and String-net model— We now take KZk ex-ample to demonstrate our understanding of two types ofGSD on a cylinder in physical pictures.
(a) (b)
G.S.D.=k
G.S.D.=1
Toric Code String-netChern-Simons
(G.S.D.=2)
RR
R S
S S
[Cx]
[Cz]
z-string
x-string
FIG. 2: (a)Same boundary conditions on two ends of a cylin-der allow a pair of cycles [cx], [cz] of a qubit, thus GSD =2. Different boundary conditions do not, thus GSD = 1.(b)Same boundary conditions allow z- or x-strings connecttwo boundaries. Different boundary conditions do not.
When k = 2, it realizes Z2 Toric code[14] withHamiltonian[4] H0 = −
∑v Av −
∑pBp. There are two
types of boundaries[5] on a cylinder(Fig. 2(a)): x (R:rough) boundary where z-string charge e condenses andz (S: smooth) boundary where x-string charge m con-denses. We can determine GSD by counting the de-gree of freedom of the code subspace: Num of qubits− Num of independent stabilizers. For Γ∂1 = Γ∂2 , wehave the same number of qubits and stabilizers, withone extra constraint
∏allBp = 1 for two x-boundaries
(∏allAv = 1 for two z-boundaries). This leaves 1 free
qubit, thus GSD = 2. For Γ∂1 6= Γ∂2 , still the same num-ber of qubits and stabilizers, but has no extra constraint.This leaves no free qubits, thus GSD = 1. We can alsocount independent logical operators(Fig. 2(a)) in homol-ogy class, with string-net(Fig. 2(b)) picture in mind -there are two cycles [cx1
], [cz1 ] winding around the com-pact direction of a cylinder. If both are x-boundaries, weonly have z-string connects two edges: cycle [cz2 ]. If bothare z-boundaries, we only have x-string (dual string) con-nects two edges: cycle [cx2 ]. Cycles of either case definethe algebra σx, σy of a qubit, so GSD = 2. For bothx-boundaries (z-boundaries), one ground state has evennumber of strings (dual strings), the other ground statehas odd number of strings (dual strings), connecting twoedges. If boundaries are different, no cycle is allowed inthe non-compact direction, no string and no dual stringcan connect two edges, so GSD = 1. Generally, for alevel k doubled model, GSD = kdim[H1(M;Z2)] = kb1(M),k to the power of the 1st Betti number[13, 15].Definitions of Topological Order— Now let us ask afundamental question: what is topological order? We re-alize the original definition of degenerated ground states
on a higher genus Riemann surface can be transplanted todegenerated ground states on a cylinder with two bound-aries. For a non-chiral fully-gapped-boundary state, wedefine: the state is intrinsic topological order, if it hasdegenerated ground states (at least for certain boundarygapping conditions) on a cylinder. The at least statementis due to GSD ≤
√|detK|, only the same boundary con-
ditions on two sides gives GSD =√|detK|. Similarly,
the state is trivial order or SPT, if it has a unique groundstate on a cylinder for any boundary gapping condition.Conclusion— In summary, by KN×N matrix AbelianChern-Simon theory, we derive boundary fully gappingconditions for Abelian topological order states, its lowenergy Hamiltonian and Hilbert space, a general GSDformula, and the number of types of boundary gap-ping conditions N ∂
g . The fully gapped boundaries re-quires N = even and non-chiral(doubled) Chern-Simonstheory, reflecting gapped boundary Quantum Doubledmodel of Toric code and string-net[6, 16]. Our N ∂
g for-mula proves the two boundary types conjecture of Toriccode[5, 15, 16], Zk gauge theory, and a more generalK2×2
Chern-Simon theory. We show gapping edge states tocount boundary GSD is related to bulk GSD. However,we find there are more types of boundary GSD (than aunique bulk GSD) depending on types of boundary gap-ping conditions. A remarkable example is rank-2 KZk
and Kdiag,k with k = even[11], where we predict exper-imentally GSD = k or 1 for KZk and GSD = k or 2 forKdiag,k on a cylinder, though GSD of both phases on aclosed genus g surface are indistinguishable(= k2g). Thisexample is especially surprising because both phases KZk
and Kdiag,k have the same (Zk)2 fusion algebra. Thismeans fusion algebra alone cannot determine boundaryGSD[11]. In the category language[6], the model of uni-tary fusion category C shows that (in Table II) there canbe many different C,D types realizing the same monoidalcenter Z(C) = Z(D).
Physics Category
Bulk excitation objects in unitary modular category
(anyons) Z(C) (monoidal center of C)Boundary type the set of equivalent classes {C,D, . . . }
of unitary fusion category
TABLE II: Dictionary between physics and category
Finally, our definitions of topological orders not onlydeepen understanding of topological GSD, but also easethe experimental platform with only cylinder topologyinstead of higher genus surfaces. To open future re-search avenues, it will be interesting to realize boundaryGSD and N ∂
g without using K matrix, which is restrictedonly to Abelian topological order, also to study them onhigher dimensional space. Physical notions which we in-troduced should still hold universally. Other than fusionrule and total neutrality, whether braiding rule can enter
5
into GSD formula[17]? Additionally, engineer boundariesto line-like defects may synthesize projective non-Abelianstatistics[18]. All these shall inspire generalizing Eq. (9)to boundary GSD of non-Abelian topological order.
It will be illuminating to have more predictions basedon our theory, as well as experimental realizations ofboundary types. One approach is flux insertion througha cylinder, where the adiabatic flux change inducesanyon transport from one boundary to the other by∆ΦB/(h/e) = ∆Pφ[11]. It will be interesting to seehow the same type of boundary conditions allow this ef-fect(unit flux changes topological sectors, with total sec-tors as GSD =
√|detK|), while different type of bound-
ary conditions restrain this effect ‘dynamically’(GSD <√|detK|). To detect this dynamical effect can guide
experiments to distinguish boundary types.We thank Maissam Barkeshli, William Witczak and
Lucy Zhang for comments on the paper. This workis supported by NSF Grant No. DMR-1005541, NSFC11074140, and NSFC 11274192. Research at Perime-ter Institute is supported by the Government of Canadathrough Industry Canada and by the Province of Ontariothrough the Ministry of Research.
[1] X. -G. Wen, Adv. Phys. 44 405 (1995)[2] F. Wilczek, (ed.), Fractional statistics and anyon super-
conductivity (World Scientic, 1990).[3] X. G. Wen, Phys. Rev. B 40, 7387 (1989).[4] A. Y. Kitaev, Annals Phys. 303, 2 (2003)[5] S. B. Bravyi, A. Y. Kitaev, arXiv: quant-ph/9811052.[6] A. Kitaev, L. Kong, Commun. Math. Phys. 313, 351
(2012)[7] M. A. Levin and X. -G. Wen, Phys. Rev. B 71, 045110
(2005)[8] F. D. M. Haldane, Phys. Rev. Lett. 74, 2090 (1995).[9] X. G. Wen and A. Zee, Phys. Rev. B 46, 2290 (1992).[10] X. G. Wen, Phys. Rev. B 41, 12838 (1990).[11] Explicit results in the appendix of this paper.[12] For Abelian group, finite direct product is finite direct
sum, so⊕
α Γ∂α =∏α Γ∂α , with α a finite number of
punctures. Direct sum is indeed the coproduct in the cat-egory of abelian groups.
[13] J. Wang, X.-G. Wen, in preparation.[14] S. -P. Kou, M. Levin and X. -G. Wen, Phys. Rev. B 78,
155134 (2008)[15] M. F. Freedman, D. A. Meyer, arXiv:quant-ph/9810055,[16] S. Beigi, P.W. Shor, D. Whalen, Commun. Math. Phys.
306, 663 (2011)[17] Peculiarly simply apply Eq. (9) to surfaces without
genus(Fig. 1(a)) and with genus(Fig. 1(b)) cases givesthe same GSD, if both have the same boundary num-ber and boundary conditions. However by gluing tech-niques, we learn from Eq. (10) there can be a factorcontribution(| detK|g) from the genus topology. So thenontrivial winding in genus may modify Eq. (9), wherebraiding rules may involve.
[18] M. Barkeshli, C. -M. Jian and X. -L. Qi, arXiv:1208.4834
[cond-mat.str-el].
APPENDIX
In Appendix, we explain detailed physical meanings ofboundary gapping rules in Sec.A, and demonstrate ouralgorithm and GSD formula Eq. (9) for a generic rank-2K matrix in Sec.B. We also give an example why fusionalgebra alone does not provide enough information todetermine boundary GSD from the bulk-edge correspon-dence viewpoint. In Sec.C, we comment more about thenumber of boundary types N ∂
g . In Sec.D, we use gluingtechniques to derive Eq. (10) for a compact manifold withgenus. Lastly, we explain how flux insertion experimentmay help to test boundary types in Sec.E.
A. BOUNDARY GAPPING RULES
We argue boundary gapping rules are:(i)Bosonic: ∀`a ∈ Γ∂ , `a,IK
−1IJ `a,J ∈ even integer,
which means mutual statistics has multiple 2π phase.(ii)Local: ∀`a, `b ∈ Γ∂ , `a,IK
−1IJ `b,J ∈ integer. Winding
one excitation around another yields multiple 2π phase.(iii)Localizing field at classical value without being
eliminated by self or mutual quantum fluctuation:∀`a, `b ∈ Γ∂ , `a,IK
−1IJ `b,J = 0, so the condensation
survives in the classical sense.(iv) `a must be excitations of electron degree of freedomsince it lives on the ‘physical’ boundary, so `a ∈ Γe elec-tron lattice, where Γe = {
∑J cJKIJ | cJ ∈ Z} . This rule
imposes integer charge qIK−1IJ `a,J in the bulk, and in-
teger charge QI =∫ L
01
2π∂xΦIdx = K−1IJ PφJ = K−1
IJ `a,Jfor each branch on the boundary. Here qI is the chargevector coupling to an external field Aµ of gauge or globalsymmetry, by adding AµqIJ
µI to the Sbulk, which causes
qIAµ∂xΦI in the S∂ .(v)Completeness: we define Γ∂ is a complete set by:∀`c ∈ Γe, if `cK
−1`c = 0 and `cK−1`a = 0 with
∀`a ∈ Γ∂ , then `c ∈ Γ∂ must be true, otherwise Γ∂ is notcomplete. The completeness rule is due to no symmetryprotection forbids any possible local gapping terms.
We can summarize all above rules of Γ∂ as (1) Null andmutual null: ∀`a, `b ∈ Γ∂ , `a,IK
−1IJ `b,J = 0, (2) ‘Phys-
ical’ excitation: `a ∈ Γe, (3) Completeness of Γ∂ .(1)(2)(3) are not guaranteed to fully gap out edge states,we add an extra rule: (4) N ∈ even. Dimension of Γ∂
must be N/2, which requires√|detK| ∈ N.
6
B. EXACT RESULTS OF K2×2 CHERN-SIMONSTHEORY
Here we work through a rank-2 K matrix Chern-Simons theory example, to demonstrate our generic algo-rithm in the main text. We derive its low energy Hamil-tonian, Hilbert space, boundary GSD formula, and thenumber of types of boundary gapping conditionsN ∂
g . For
K2×2 =(k1 k3k3 k2
)≡( k1 k3k3 (k23−p
2)/k1
), in order to fully gap
out edge states, we find that detK = −p2, (a) the edgestates need to be non-chiral (K2×2 has equal number ofpositive and negative eigenvalues, so detK < 0 and (b)|detK| needs to be an integer p square. Two indepen-dent sets of allowed gapping lattices Γ∂ = {n`a,I |n ∈ Z},Γ∂′
= {n′`a,I′|n′ ∈ Z} satisfy gapping rules (1)(2)(3)(4),
so N ∂g = 2 at rk(K) = 2, with
n`a,I = n(`a,1, `a,2) =n p
| gcd(k1, k3 + p)|(k1, k3 + p) (11)
n′`′a,I = n′(`′a,1, `′a,2) =
n′ p
| gcd(k3 + p, k2)|(k3 + p, k2) (12)
| gcd(k, l)| stands for the greatest common divisor in|k|, |l|, its absolute value. If k (or l) is zero, we define| gcd(k, l)| is the other value |l| (or |k|). n, n′ ∈ Z areallowed if no other symmetry constrains its values.
Now take specific topology, a disk(a sphere with 1puncture) and a cylinder(a sphere with 2 punctures) asexamples of manifolds with boundary, for K2×2 theirHilbert space of edge states are[1]:
Hdisk = H1,2KM ⊗HPφ1 ⊗HPφ2 (13)
(if K2×2 is diagonal, then H1,2KM = H1
KM ⊗H2KM ).
H cylinder =⊕jA
⊕jB
(Htopdisk ⊗Hbottomdisk )(jA,jB) (14)
= Htopdisk ⊗Hbottomdisk ⊗Hgl (15)
= Htop,1,2KM ⊗Hbottom,3,4KM ⊗HP topφ1,P bottomφ3
⊗HP topφ2,P bottomφ4
HKM stands for Hilbert space of nonzero Fourier modespart with Kac-Moody algebra. We label zero and wind-ing modes’ Hilbert space by winding mode Pφ, whichcan be regarded as a discrete lattice because of ΦI(x)periodicity. Because the bulk cylinder provides channelsconnecting edges states of two boundaries, so fractional-ized quasiparticles (here Abelian anyons) can be trans-ported from one edge to the other. Hgl contains frac-tional sectors |jA, jB〉, the 1st branch jA runs betweenthe top(P topφ1
) and the bottom(P bottomφ3), the 2nd branch
jB runs between the top(P topφ2) and the bottom(P bottomφ4
).Let us explicitly show that edge states of any bound-
ary has a finite gap from ground states at large systemsize L and large coupling g. Without losing general-ity, take V =
(v1 v2v2 v1
)and a gapping term (`a,1, `a,2) =
p| gcd(k1,k3+p)| (k1, k3+p) ∈ Γ∂ , and diagonalize the Hamil-tonian,
H ' (
∫ L
0
dx VIJ∂xΦI∂xΦJ) +1
2g(`a,I · ΦI)2L (16)
we find eigenvalues:
E1,2(n) =
√∆2 + (
nπ
Lp2)2δ1 ± (
nπ
Lp2)δ2 (17)
where the finite mass gap is independent of L:
∆ =
√2πg
(k1(k1 − k2)v1 + 2(k3 + p)(k3v1 − k1v2)
)| gcd(k1, k3 + p)|
,
δ1 = (k1− k2)2v21 + 4(k3v1− k1v2)(k3v1− k2v2) and δ2 =
v1(k1 + k2)− v2(2k3).To count boundary GSD, for generic K2×2 Abelian
topological order on a disk, take `a ∈ Γ∂ in Eq. (11) with-out losing generality(same argument for Γ∂
′)[2], we have
Pφ1 = I1`a,1+jA`a,1
| gcd(`a)| , Pφ2= I1`a,2+jA
`a,2| gcd(`a)| . The
total anyon charge for each branch needs to conserve, buta single boundary of a disk has no other boundaries tolocate transported anyons. This implies: jA = 0, thereis no different topological sector induced by transportinganyons, thus GSD = 1.
On the other hand, if the topology is replaced to acylinder with the top ∂1 and the bottom ∂2 boundariesin Fig. 3, when the gapping terms from boundary gappinglattice Γ∂ are chosen to be the same, the Hilbert spaceon Pφ lattice is:
Pφ1 = I1`a,1 + jA`a,1
| gcd(`a)|, Pφ2 = I1`a,2 + jA
`a,2| gcd(`a)|
(18)
Pφ3= I2`a,1 + jB
`a,1| gcd(`a)|
, Pφ4= I2`a,2 + jB
`a,2| gcd(`a)|
(19)
Anyon fusion rule and charge conservation for eachbranch constrains (Pφ1
+ Pφ3, Pφ2
+ Pφ4) ∈ Γe electron
lattice. With | gcd(`a)| = p, it implies jA = −jB(mod p).0 ≤ jA(mod p) < p has p different topological sectorsinduced by different jA. When ∆jA/p ∈ Z, it transportselectrons, so it brings back to the same sector. Countthe number of distinct sectors, i.e. ground states, wefind GSD = p.
If gapping terms on two boundaries of a cylinder arechosen to be different: Γ∂ for ∂1, Γ∂
′for ∂2, we revise
the second line of Eq. (19) to
Pφ3= I ′2`
′a,1 + j′B
`′a,1| gcd(`′a)|
, Pφ4 = I ′2`′a,2 + j′B
`′a,2| gcd(`′a)|
(20)Anyon fusion rules and conservation implies:
( k1| gcd(`a)|jA + k3+p
| gcd(`′a)|j′B ,
(k3+p)| gcd(`a)|jA + k2
| gcd(`′a)|j′B) ∈ Γe.
This constraint gives a surprise. For exam-ple, KZp =
( 0 pp 0
), GSD = 1. However, when
7
Kdiag,p =( p 0
0 −p), GSD = 1 for p ∈ odd, but GSD = 2
for p ∈ even. This provides a new approach that one candistinguish two types of orders KZp and Kdiag when p iseven by measuring their boundary GSD.
We illustrate this result in an intuitive way in Fig. 3.When boundary types are the same on two sides of thecylinder, Fig. 3(a) is enough to explain GSD = p , wherefractionalized anyons transport from the bottom to thetop. For KZp case, say Γ∂1 = Γ∂2 = Γ∂ = {n(p, 0)}, qp1
with `a = jA(1, 0) for 0 ≤ jA ≤ p − 1. For Kdiag,p case,say Γ∂1 = Γ∂2 = Γ∂ = {n(p, p)}, qp1 with `a = jA(1, 1)for 0 ≤ jA ≤ p− 1. This accounts all p sectors.
When boundary types are different, Fig. 3(a) is not al-lowed for fractionalized anyons transport. Fig. 3(b) iscrucial to account the second ground state of Kdiag,p atp ∈ even. Let us take Γ∂1 = Γ∂ = {n(p, p)} and Γ∂2 =Γ∂′
= {n(p,−p)}, where es represents ` = (p, 0), whileqp1 with `a = (p/2, p/2) and qp2 with `a = (p/2,−p/2)at p ∈ even are allowed fractionalized anyons (with in-teger unit of anyon charge). This process switches theground state to a different sector, so GSD = 2. However,fractionalized anyons transport in Fig. 3(b) is not allowedfor KZp with different boundary types on two sides of thecylinder, which results in GSD = 1.
esqp1
qp1
qp1
qp2
(a) (b)
FIG. 3: (a)Anyon(qp1) is transported from the bottom tothe top of the cylinder. (b)Physical excitations es split into apair of anyons(qp1 to the bottom, qp2 to the top). Note that(b) is crucial to explain GSD = 2 for Kdiag,p at p ∈ even withdifferent boundary types on two sides of the cylinder.
The result remarks that only fusion algebra(both havedoubled fusion algebra (Zp)2) is not sufficient enoughto determine boundary GSD. Let us show this by KZ2
and Kdiag,2 examples from the bulk-edge correspondenceviewpoint. Both orders have the doubled fusion al-gebra, (Z2)2, with bulk excitation such as two anyontypes `a = (0, 1), `b = (1, 0). For KZ2 , two typesof boundary conditions Γ∂ = {n2(0, 1) | n ∈ Z} andΓ∂ = {n2(1, 0) | n ∈ Z} correspond to two of `a, `b, thusthis is one to one correspondence. However, for Kdiag,2,two boundary conditions Γ∂ = {n2(1, 1) | n ∈ Z} andΓ∂′
= {n2(1,−1) | n ∈ Z} correspond to one bulk ex-citation ` = (1(mod2), 1(mod2)) = (1(mod2),−1(mod2))in fusion algebra. It is two (on the boundary) to one (inthe bulk) correspondence. Hence Kdiag,2 shows an ex-ample where anyon fusion algebra from the bulk cannotdistinguish different boundary types.
C. NUMBER OF TYPES OF BOUNDARYGAPPING CONDITIONS
For the number of types of boundary gapping condi-tionsN ∂
g , at rk(K) = 2, we showedN ∂g = 2, which proves
the conjecture[3] nicely. rk(K) ≥ 4, N ∂g is infinite, is sur-
prising. We may wonder whether extra rules are possibleto restrict N ∂
g . Let us assume |detK| = 1, the canon-ical form of this unimodular indefinite symmetric inte-gral KN×N matrix exists[4]. For the odd matrix(wherequadratic form has some odd integer coefficient, so withfermionic statistics), the canonical form is composed byN/2 blocks of
(1 00 −1
)along the diagonal blocks of KN×N .
For the even matrix(where quadratic form has only eveninteger coefficient, so with only bosonic statistics), thecanonical form is composed by blocks of
(0 11 0
)and a set
of all positive(or negative) coefficients E8 lattices, KE8,
KE8=
2 −1 0 0 0 0 0 0
−1 2 −1 0 0 0 0 0
0 −1 2 −1 0 0 0 −1
0 0 −1 2 −1 0 0 0
0 0 0 −1 2 −1 0 0
0 0 0 0 −1 2 −1 0
0 0 0 0 0 −1 2 0
0 0 −1 0 0 0 0 2
(21)
along the diagonal blocks of KN×N . A positive defi-nite KE8
with eight chiral bosons cannot be gapped out.Thus, in order to have non-chiral states, the even matrixcanonical form must be composed by N/2 blocks of
(0 11 0
)Now it is useful to rethink about the number of boundarytypes N ∂
g in this canonical form when |detK| = 1. Wehad claimed under rule (1)(2)(3)(4) when rk(K) ≥ 4,N ∂g = ∞. If we add an extra gapping rule (5): Each
gapping term is formed by only two branches of chiral-antichiral edge states scattering. There are no three ormore multi-channel scatterings. With (5), the bound-ary gapping lattices Γ∂ are formed by a set of ` termsof two branches of edge states with opposite chirality.This modifies our result to N ∂
g = (N/2)!2N/2. Its (N/2)!factor is due to each chiral state can find another statefrom antichiral ones to form a pair. The relative sign oftheir bosonized phase is ±, gives an extra 2N/2 factor.We comment that (5) seems to be artificial without deepreasoning, and the formula only works for |detK| = 1with the canonical form. However, one can postulatesome extra rule like (5) can restrict N ∂
g .
D. SURGERY TO GLUE CYLINDERS TO FORMA GENUS g RIEMANN SURFACE WITH
PUNCTURES
Here we show how to glue the boundaries of puncturedcylinders to form a genus g Riemann surface with punc-
8
tures, and determine its GSD of the topological order.For a genus g Riemann surface with η′ punc-
tures(Fig. 1(b) and Fig. 4(c)), we start from Fig. 4(a), anumber of g cylinders drilled with total puncture num-ber η = η′ + 2g + 2(g − 1), where 2g count two punc-tures on top and bottom for each cylinder(hi,T and hi,B ,1 ≤ i ≤ g) , drill an extra puncture on both the 1st(h1,L) and the last gth cylinder (hg−1,R), and drill twoextra punctures(hj−1,R and hj,L) for j-th cylinder for2 ≤ j ≤ g − 1. There are thus 2(g − 1) extra punc-tures. Glue the boundaries of hj,L and hj,R togetherfor 1 ≤ j ≤ g − 1, and glue hi,T and hi,B together for1 ≤ i ≤ g, results in Fig. 4(c). Use Eq. (9), remove gluedboundaries part(1 ≤ α ≤ 2g+2(g−1)) which contributes
a factor ≤ | detK|g, and redefine particle hopping lat-tices Lqp
⋂e and
⊕α′ Γ
∂α′ only for unglued boundaries(1 ≤ α′ ≤ η′), we derive
GSD ≤ |detK|g ·Num[ Lqp
⋂e⊕η′
α′=1 Γ∂α′] (22)
For a genus g Riemann surface(η′ = 0), this becomesGSD ≤ |detK|g, where rk(K) = N for a closed manifoldcase is relaxed to any natural number N, which works forboth odd and even number of branches. The inequalitieshere are due to different choices of gapping conditions forglued boundaries.
......
......
......
(a)
(b)
(c)
h1,T
h1,B
h2,T
h2,B
hg-1,T
hg-1,B
hg,T
hg,Bh1,L h1,R h2,L h2,R hg-2,L hg-2,Rhg-1,L hg-1,R
...
...
...
FIG. 4: Glue punctured cylinders to form a genus g Riemann surface with η′ punctures. Start from (a), firstly identify left andright B arrows of each square to form number g of punctured cylinders. Then glue hj,L and hj,R (red dotted circles) togetherfor 1 ≤ j ≤ g − 1, and glue hi,T and hi,B (blue arrows) together for 1 ≤ i ≤ g, which yields (b), equivalently as a genus gRiemann surface (c). The extra η′ punctures are indicated here as a shaded blue puncture in the left most handle.
E. FLUX INSERTION ARGUMENT ANDEXPERIMENTAL TEST ON BOUNDARY TYPES
Consider an artificial-designed or external gauge field(such as electromagnetic field) coupled to topologicallyordered states by a charge vector qI . An adiabatic fluxinsertion ∆ΦB inside the cylinder induces Ex, causinga perpendicular current Jy flows to the boundary, fromthe bulk term JµJ = −qI e
2πK−1IJ
c~εµνρ∂νAρ, so qI∆ΦB =
−qI∫dt∫~E · d~l = − 2π
e KIJ~∫Jy,Jdtdx = − 2π
e KIJ~eQJ .
QJ is the charge condensed on the edge of the cylin-der. On the other hand, the edge state dynamics affectswinding modes by QI =
∫J0∂,Idx = −
∫e
2π∂xΦIdx =
−eK−1IJ Pφ,J , so
qI∆ΦB/(h
e) = ∆Pφ,I (23)
For the same types of boundaries: Γ∂1 = Γ∂2 , there are√|detK| sectors by ∆Pφ(mod
√|detK|), the
√|detK|
units of flux bring the state back to the original sector.For Γ∂1 6= Γ∂2 , we had shown GSD <
√|detK| (such
as GSD = 1). This motivates an interesting questionif one inserts flux into the cylinder, what dynamical ef-fect, which repulses anyons transporting from one edgeto the other, will be detected. The detection of this effectmay guide experiments to distinguish different types ofboundary gapping conditions.
[1] X. G. Wen, Phys. Rev. B 41, 12838 (1990).[2] Notice if we take |n| ≥ 2 or |n′| ≥ 2 larger-size hopping
term for n`a,I ∈ Γ∂ ,n′`′a,I ∈ Γ∂′, the lattice can break
into more sublattices of different sectors: the calculation
9
on a disk is altered to GSD = |n|. However, genericallyno symmetry forbids n = n′ = 1 least-size hopping terms,which dominant cosine potential can minimize the energy.
[3] S. B. Bravyi, A. Y. Kitaev, arXiv: quant-ph/9811052.[4] http://mathoverflow.net/questions/97448