Ground State Degeneracy in the Levin-Wen Model for Topological Phases
aa r X i v : . [ c ond - m a t . s t r- e l ] J a n Ground State Degeneracy in the Levin-Wen Model for Topological Phases
Yuting Hu, ∗ Spencer D. Stirling,
1, 2, † and Yong-Shi Wu
3, 1, ‡ Department of Physics and Astronomy, University of Utah, Salt Lake City, UT 84112, USA Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA Department of Physics and Center for Field Theory and Particle Physics, Fudan University, Shanghai 200433, China (Dated: August 21, 2018)We study properties of topological phases by calculating the ground state degeneracy (GSD) ofthe 2d Levin-Wen (LW) model. Here it is explicitly shown that the GSD depends only on the spatialtopology of the system. Then we show that the ground state on a sphere is always non-degenerate.Moreover, we study an example associated with a quantum group, and show that the GSD on a torusagrees with that of the doubled Chern-Simons theory, consistent with the conjectured equivalencebetween the LW model associated with a quantum group and the doubled Chern-Simons theory.
PACS numbers: 05.30.Pr 71.10.Hf 02.10.Kn 02.20.Uw
I. INTRODUCTION
In recent years two-dimensional topological phaseshave received growing attention from the science com-munity. They represent a novel class of quantum mat-ter at zero temperature whose bulk properties are ro-bust against weak interactions and disorders. Topologicalphases may be divided into two families: doubled (withtime-reversal symmetry, or TRS, preserved), and chiral ( with TRS broken). Either type may be exploited to dofault-tolerant (or topological) quantum computing . Chiral topological phases were first discovered in inte-ger and fractional quantum Hall (IQH and FQH) liquids.Mathematically, their effective low-energy description isgiven by Chern-Simons theory or (more generally) topo-logical quantum field theory (TQFT) . One character-istic property of FQH states is ground state degeneracy(GSD), which depends only on the spatial topology of thesystem and is closely related to fractionization of quasiparticle quantum numbers, including fractional(braiding) statistics . In some cases the GSD has beencomputed in refs. .Chern-Simons theories are formulated in the contin-uum and have no lattice counterpart. Doubled topolog-ical phases, on the other hand, do admit a discrete de-scription. The first known example was Kitaev’s toriccode model .More recently, Levin and Wen (LW) constructeda discrete model to describe a large class of doubledphases. Their original motivation was to generateground states that exhibit the phenomenon of string-netcondensation as a physical mechanism for topologicalphases. The LW model is defined on a trivalent lattice (orgraph) with an exactly soluble Hamiltonian. The groundstates in this model can be viewed as the fixed-pointstates of some renormalization group flow . Thesefixed-point states look the same at all length scales andhave no local degrees of freedom.The LW model is believed to be a Hamiltonian ver-sion of the Turaev-Viro topological quantum field the-ory (TQFT) in three dimensional spacetime and,in particular cases, discretized version of doubled Chern- Simons theory . Like Kitaev’s toric code model , weexpect that the subspace of degenerate ground states inthe LW model can be used as a fault-tolerant code forquantum computation.In this paper we report the results of a recent study onthe GSD of the LW model formulated on a (discretized)closed oriented surface M . Usually the GSD is exam-ined as a topological invariant of the 3-manifold S × M . In a Hamiltonian approach accessible to physi-cists, we will explicitly demonstrate that the GSD in theLW model depends only on the topology of M on whichthe system lives and, therefore, is a topological invari-ant of the surface M . We also show that the groundstate of any LW Hamiltonian on a sphere is always non-degenerate. Moreover, we examine the LW model as-sociated with quantum group SU k (2), which is conjec-tured to be equivalent to the doubled Chern-Simons the-ory with gauge group SU (2) at level k , and compute theGSD on a torus. Indeed we find an agreement with thatin the corresponding doubled Chern-Simons theory .This supports the above-mentioned conjectured equiva-lence between the doubled Chern-Simons theory and theLW model, at least in this particular case.The paper is organized as follows. In Section II wepresent the basics of the LW model, easy to read fornewcomers. In Section III topological properties of theground states are studied, and the topological invarianceof their degeneracy is shown explicitly. In section IV wedemonstrate how to calculate the GSD in a general way.In section V we provide examples for the calculation par-ticularly on a torus. Section VI is devoted to summaryand discussions. The detailed computation of the GSDis presented in the appendices. II. THE LEVIN-WEN MODEL
Start with a fixed (connected and directed) trivalentgraph Γ which discretizes a closed oriented surface M (such as a torus). To each edge in the graph we assign astring type j , which runs over a finite set j = 0 , , ..., N .Each string type j has a “conjugate” j ∗ that describesthe effect of reversing the edge direction. For example j may be an irreducible representation of a finite group or(more generally) a quantum group .Let us associate to each string type j a quantum dimen-sion d j , which is a positive number for the Hamiltonianwe define later to be hermitian. To each triple of strings { i, j, k } we associate a branching rule δ ijk that equals 1if the triple is “allowed” to meet at a vertex, 0 if not(in representation language the tensor product i ⊗ j ⊗ k either contains the trivial representation or not). Thisdata must satisfy (here D = P j d j ) X k d k δ ijk ∗ = d i d j X ij d i d j δ ijk ∗ = d k D (1) j = 0 is the unique “trivial” string type, satisfying 0 ∗ = 0and δ jj ∗ = 1 , δ ji ∗ = 0 if i = j .The Hilbert space is spanned by all configurationsof all possible string types j on edges. The Hamilto-nian is a sum of some mutually-commuting projectors H := − P v ˆ Q v − P p ˆ B p (one for each vertex v and eachplaquette p ). Here each projector ˆ Q v = δ ijk with i, j, k onthe edges incoming to the vertex v . ˆ Q v = 1 enforces thebranching rule on v . Throughout the paper we work onthe subspace of states in which ˆ Q v = 1 for all vertices.Each projector ˆ B p is a sum D − P s d s ˆ B sp of operatorsthat have matrix elements (on a hexagonal plaquette forexample) * j j j j j j j j j j j j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ B sp (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j j j j j j j j j j j j + = v j v j v j v j v j v j v j ′ v j ′ v j ′ v j ′ v j ′ v j ′ (2) G j j ∗ j s ∗ j ′ j ′∗ G j j ∗ j s ∗ j ′ j ′∗ G j j ∗ j s ∗ j ′ j ′∗ G j j ∗ j s ∗ j ′ j ′∗ G j j ∗ j s ∗ j ′ j ′∗ G j j ∗ j s ∗ j ′ j ′∗ Here v j = p d j is real. The symmetrized 6 j symbols G are complex numbers that satisfysymmetry: G ijmkln = G mijnk ∗ l ∗ = G klm ∗ ijn ∗ = ( G j ∗ i ∗ m ∗ l ∗ k ∗ n ) ∗ pentagon id: X n d n G mlqkp ∗ n G jipmns ∗ G js ∗ nlkr ∗ = G jipq ∗ kr ∗ G riq ∗ mls ∗ orthogonality: X n d n G mlqkp ∗ n G l ∗ m ∗ i ∗ pk ∗ n = δ iq d i δ mlq δ k ∗ ip (3)For example, these conditions are known to besatisfied if we take the string types j to be all irre-ducible representations of a finite group, d j to be thedimension of corresponding representation space, and G to be the symmetrized Racah 6 j symbols for the group.In this case the LW model can be mapped to Ki-taev’s quantum double model . More general sets of data { G, d, δ } can be derived from quantum groups (or Hopfalgebras) . We will discuss such a case later using thequantum group SU k (2) ( k being the level). Γ (1) ⇒ Γ (2) FIG. 1: Given any two trivalent graphs Γ (1) and Γ (2) dis-cretizing the same surface, we can always mutate Γ (1) to Γ (2) by a composition of elementary f moves. In general Γ (1) andΓ (2) are not required to be regular lattices. These diagramshappen to be the same as , but in a slightly different context. III. GROUND STATES
Any ground state | Φ i (there may be many) must be asimultaneous +1 eigenvector for all projectors ˆ Q v and ˆ B p .In this section we demonstrate the topological propertiesof the ground states on a closed surface with non-trivialtopology.Let us begin with any two arbitrary trivalent graphsΓ (1) and Γ (2) discretizing the same surface (e.g., a torus).If we compare the LW models based on these two graphs,respectively, then immediately we see that the Hilbertspaces are quite different from each other (they have dif-ferent sizes in general).However, we may mutate between any two given triva-lent graphs Γ (1) and Γ (2) by a composition of the follow-ing elementary moves (see also Fig 1 ): f . ⇒ , for any edge; f . ⇒ , for any vertex. f . ⇒ , for any triangle structure.Suppose we are given a sequence of elementary f movesthat connects two graphs Γ (1) → Γ (2) . We now constructa linear transformation H (1) → H (2) between the twoHilbert spaces. This is defined by associating linear mapsto each elementary f move:ˆ T : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j j j j j + → X j ′ v j v j ′ G j j j j j j ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j j j j j + ˆ T : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j j j + → X j j j v j v j v j √ D G j j j j ∗ j j ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j j j j j j + ˆ T : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j j j j j j + → v j v j v j √ D G j ∗ j ∗ j ∗ j ∗ j j ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j j j + (4)The mutation transformations between H (1) and H (2) are constructed by a composition of these elementarymaps. As a special example, the operator ˆ B p = D − P s d s ˆ B sp is such a transformation. In fact, onthe particular triangle plaquette p as in (4), we haveˆ B p = ▽ = ˆ T ˆ T , by using the pentagon identity in (3).Mutation transformations are unitary on the groundstates. To see this, we only need to check that the el-ementary maps ˆ T , ˆ T , and ˆ T are unitary. First notethat the following relations hold: ˆ T † = ˆ T , ˆ T † = ˆ T , andˆ T † = ˆ T . We emphasize that these are maps between theHilbert spaces on two different graphs. For example, wecheck ˆ T † = ˆ T by comparing matrix elements * j j j j j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ T † (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j j j j j + ≡ * j j j j j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j j j j j +! ∗ = v j v j ′ (cid:16) G j j j j j j ′ (cid:17) ∗ = v j ′ v j G j j j ′ j j j ∗ = * j j j j j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j j j j j + (5)where in the third equality we used the symmetry condi-tion in (3).Similarly, for ˆ T † = ˆ T (or ˆ T † = ˆ T ), we have * j j j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ T † (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j j j j j j + ≡ * j j j j j j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j j j +! ∗ = v j v j v j √ D (cid:16) G j j j j ∗ j j ∗ (cid:17) ∗ = v j v j v j √ D G j ∗ j ∗ j ∗ j ∗ j j ∗ = * j j j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j j j j j j + (6)Now we verify unitary. First, ˆ T † ˆ T = id and ˆ T † ˆ T =ˆ T ˆ T = id by the orthogonality condition in (3) (notethat, since we have not used any information about theground states in this argument, ˆ T and ˆ T are unitaryon the entire Hilbert space). For unitary of ˆ T we checkˆ T † ˆ T = ˆ T ˆ T = 1. The last equality only holds on theground states since we have already seen that ˆ T ˆ T =ˆ B p = ▽ and ˆ B p = ▽ = 1 only on the ground states.As another consequence of the above relations, theHamiltonian is hermitian since all ˆ B p ’s consist of ele-mentary ˆ T , ˆ T , and ˆ T maps. Particularly, on a triangleplaquette, we have ˆ B † p = ▽ = ( ˆ T ˆ T ) † = ˆ T † ˆ T † = ˆ T ˆ T =ˆ B p = ▽ .The mutation transformations serve as the symmetrytransformations in the ground states. If | Φ i is a groundstate then ˆ T | Φ i is also a ground state, where ˆ T is acomposition of ˆ T i ’s associated with elementary f movesfrom Γ (1) to Γ (2) . This is equivalent to the conditionˆ T ( Q p ˆ B p ) = ( Q p ′ ˆ B p ′ ) ˆ T , which can be verified by the conditions in (3). (Here p and p ′ run over the plaquetteson Γ (1) and Γ (2) , respectively. Also note that the ˆ B p ’s aremutually-commuting projectors, i.e., ˆ B p ˆ B p = ˆ B p , andthus Q p ˆ B p is the projector that projects onto the groundstates.)These symmetry transformations look a little differentfrom the usual ones since they may transform betweenthe Hilbert spaces H (1) and H (2) on two different graphsΓ (1) and Γ (2) . In general, Γ (1) and Γ (2) do not have thesame number of vertices and edges. And thus H (1) and H (2) have different sizes. However, if we restrict to theground-state subspaces H (1)0 and H (2)0 , mutation trans-formations are invertible. In fact, they are unitary as wehave just shown.The tensor equations on the 6 j symbols in (3) giverise to a simple result: each mutation that preserves thespatial topology of the two graphs induces a unitary sym-metry transformation. During the mutations, local struc-tures of the graphs are destroyed, while the spatial topol-ogy of the graphs is not changed. Correspondingly, thelocal information of the ground states may be lost, whilethe topological feature of the ground states is preserved.In fact, any topological feature can be specified by a topo-logical observable ˆ O that is invariant under all mutationtransformations ˆ T from H (1) to H (2) : ˆ O ′ ˆ T = ˆ T ˆ O (whereˆ O is defined on the graph Γ (1) and ˆ O ′ on Γ (2) ).The symmetry transformations provides a way to char-acterize the topological phase by a topological observ-able. In the next section we will investigate the GSD assuch an observable.Let us end this section by remarking on uniqueness ofthe mutation transformations. There may be many waysto mutate Γ (1) to Γ (2) using f , f and f moves. Eachway determines a corresponding transformation betweenthe Hilbert spaces of ground states, H (1)0 and H (2)0 . Itturns out that all these transformations are actually thesame if the initial and final graphs Γ (1) to Γ (2) are fixed,i.e., independent of which way we choose to mutate thegraph Γ (1) to Γ (2) . This means that the ground stateHilbert spaces on different graphs can be identified (upto a mutation transformation) and all graphs are equallygood.One consequence of the uniqueness of the mutationtranformation is that the degrees of freedom in theground states do not depend on the specific structureof the graph. In this sense, the LW model is the Hamil-tonian version of some discrete TQFT (actually, Turaev-Viro type TQFT, see ). The fact that the degrees offreedom of the ground states depend only on the topol-ogy of the closed surface M is a typical characteristic oftopological phases . IV. GROUND STATE DEGENERACY
In this section we investigate the simplest nontrivialtopological observable, the GSD. Since Q p ˆ B p is the pro-jector that projects onto the ground states, taking a tracecomputes GSD = tr( Q p ˆ B p ).We can show that GSD is a topological invariant.Namely, in the previous section we mentioned that,by using (3), Q p ˆ B p is invariant under any muta-tion ˆ T between the Hilbert spaces H (1) and H (2) :ˆ T † ( Q p ′ ˆ B p ′ ) ˆ T = Q p ˆ B p . Taking a trace of both sidesleads to tr ′ ( Q p ′ ˆ B p ′ ) = tr( Q p ˆ B p ), where the traces areevaluated on H (2) and H (1) respectively.The independence of the GSD on the local structure ofthe graphs provides a practical algorithm for computingthe GSD, since we may always use the simplest graph(see Fig 2 and examples in the next section).Expanding the GSD explicitly in terms of 6 j symbolsusing (2) we obtainGSD = X j j j j j j ... * j j j j j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( Y p ˆ B p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j j j j j + = D − P X s s s s ... d s d s d s d s ... X j ′ j ′ j ′ j ′ j ′ ... d j ′ d j ′ d j ′ d j ′ d j ′ ... X j j j j j ... d j d j d j d j d j ... (cid:16) G j j j s ∗ j ′ j ′ G j ′ j j ′ s ∗ j j ′ G j j ′ j ′ s ∗ j j (cid:17) (cid:16) G j j j ∗ s ∗ j ′∗ j ′ G j ′ j ′∗ j s ∗ j ′ j ∗ G j ∗ j ′ j ′ s ∗ j j (cid:17) ... (7)The formula needs some explanation. P is the total num-ber of plaquettes of the graph. Each plaquette p con-tributes a summation over s p together with a factor of d sp D . In the picture in (7) the top plaquette is being oper-ated on first by ˆ B s p , next the bottom plaquette by ˆ B s p ,third the left plaquette by ˆ B s p , and finally the right pla-quette by ˆ B s p . Although ordering of the ˆ B sp operators isnot important (since all ˆ B p ’s commute with each other),it is important to make an ordering choice (for all pla-quettes on the graph) once and for all .Each edge e contributes a summation over j e and j ′ e together with a factor of d j e d j ′ e . Each vertex contributesthree 6 j symbols.The indices on the 6 j symbols work as follows: sinceeach vertex borders three plaquettes where ˆ B sp ’s are beingapplied, we pick up a 6 j symbol for each corner. However,ordering is important: because we have an overall order-ing of ˆ B sp ’s, at each vertex we get an induced orderingfor the 6 j symbols. Starting with the 6 j symbol furthestleft we have no primes on the top row. The bottom twoindices pick up primes. All of these variables (primed ornot) are fed into the next 6 j symbol and the same ruleapplies: the bottom two indices pick up a prime with theconvention () ′′ = ().By the calculation of the GSD, we have characterizeda topological property of the phase using local quantitiesliving on a graph discretizing M of nontrivial topology. (a) (b)FIG. 2: All trivalent graphs can be reduced to their simpleststructures by compositions of elementary f moves. (a) on asphere: 2 vertices, 3 edges, and 3 plaquettes. (b) on a torus:2 vertices, 3 edges, and 1 plaquette. V. EXAMPLES (1) On a sphere.
To calculate the GSD, we need to inputthe data { G ijmkln , d j , δ ijm } and evaluate the trace in (7).We start by computing the GSD in the simplest case ofa sphere.Let’s consider the simplest graph as in Fig. 2(a).We show in Appendix A that the ground state is non-degenerate on the sphere without referring to any specificstructure in the model: GSD sphere = 1. In fact, for moregeneral graphs one can write down the ground state as Q p ˆ B p | i up to a normalization factor, where in | i alledges are labeled by string type 0.We notice that the GSD on the open disk (which istopologically the same as the 2d plane) can be studiedusing the same technique. This is because the open diskcan be obtained by puncturing the sphere in Fig 2(a) atthe bottom. Although this destroys the bottom plaque-tte, we notice that the constraint ˆ B p = 1 from the bottomplaquette is automatically satisfied as a consequence ofthe same constraint on all other plaquettes. The factthat GSD sphere (= GSD disk ) = 1 indicates the non-chiraltopological order in the LW model. (2) Quantum double model. When the data are deter-mined by representations of a finite group G , the LWmodel is mapped to Kitaev’s quantum double model .The ground states corresponds one-to-one to the flat G -connections . The GSD isGSD QD = (cid:12)(cid:12)(cid:12)(cid:12) Hom( π ( M ) , G ) G (cid:12)(cid:12)(cid:12)(cid:12) (8)where Hom( π ( M ) , G ) is the space of homomorphismsfrom the fundamental group π ( M ) to G , and G in thequotient acts on this space by conjugation.In particular, the GSD (8) on a torus isGSD torusQD = (cid:12)(cid:12) { ( a, b ) | a, b ∈ G ; aba − b − = e } / ∼ (cid:12)(cid:12) (9)where ∼ in the quotient is the equivalence by conjugation,( a, b ) ∼ ( hah − , hbh − ) for all h ∈ G The number (9) is also the total number of irreduciblerepresentations of the quantum double D ( G ) of thegroup G . On the other hand, the quasiparticles in themodel are classified by the quantum double D ( G ). Thusthe GSD on a torus is equal to the number of particlespecies in this example. (3) SU k (2) structure on a torus. More generally, on atorus any trivalent graph can be reduced to the simplestone with two vertices and three edges, as in Fig 2(b). Onthis graph the GSD consists of six local 6 j symbols.GSD = D − X sj j j j ′ j ′ j ′ d s d j d j d j d j ′ d j ′ d j ′ (cid:16) G j j j ∗ sj ′∗ j ′ G j ′∗ j j ′ sj j ′ G j j ′∗ j ′ sj j ∗ (cid:17) (cid:16) G j ∗ j j ∗ sj ′∗ j ′ G j ′ j ′∗ j ∗ sj ′∗ j ∗ G j ∗ j ′∗ j ′ sj j ∗ (cid:17) (10)Now let us take the example using the quantum group SU k (2). It is known that SU k (2) has k + 1 irreduciblerepresentations, and thus the GSD we calculate is finite.We take the string types to be these representations, la-beled as 0 , , ..., k , and the data { G ijmkln , d j , δ ijm } to bedetermined by these representations (for more details,see ).In Appendix B we show that in this case (for the LWmodel on a torus with string types given by irreps of SU k (2)) we have GSD = ( k + 1) . We argue this bothanalytically and numerically.On the other hand, it is widely believed that when thestring types in the LW model are irreps from a quantumgroup at level k , then the associated TQFT is given bydoubled Chern-Simons theory associated with the corre-sponding Lie group at level ± k . This equivalencetells us that in this case the LW model can be viewed asa Hamiltonian realization of the doubled Chern-Simonstheory on a lattice, and it provides an explicit picture ofhow the LW model describes doubled topological phases.Along these lines, our result is consistent with theresult GSD CS = k + 1 for Chern-Simons SU (2) theoryat level k on a torus. This can be seen since the Hilbertspace associated to doubled Chern-Simons should be thetensor product of two copies of Chern-Simons theory atlevel ± k . VI. SUMMARY AND DISCUSSIONS
In this paper, we studied the LW model that describes2d topological phases which do not break time-reversalsymmetry. By examining the 2d (trivalent) graphs withsame topology which are related to each other by agiven finite set of operations (Pachner moves), we de-veloped techniques to deal with topological properties ofthe ground states. Using them, we have been able toshow explicitly that the GSD is determined only by thetopology of the surface the system lives on, which is a typ-ical feature of topological phases. We also demonstratedhow to obtain the GSD from local data in a general way.We explicitly showed that the ground state of any LWHamiltonian on a sphere is non-degenerate. Moreover,the LW model associated with quantum group SU k (2)was studied, and our result for the GSD on a torus is consistent with the conjecture that the LW model asso-ciated with quantum group is the realization of a doubledChern-Simons theory on a lattice or discrete graph.Finally, let us indicate possible extension of the resultsto more general cases. First, more generally in the LWmodel, an extra discrete degree of freedom, labelled by anindex α , may be put on the vertices. Then the branch-ing rule δ αijk , when its value is 1, may carry an extraindex α . (In representation language this implies thatgiven irreducible representations i , j and k , there may bemultiple inequivalent ways to obtain the trivial represen-tation from the tensor product of i ⊗ j ⊗ k . The index α just labels these different ways.) The 6 j symbols ac-cordingly carry more indices. (For more details see thefirst Appendix in the original paper of the LW model.)The expression (7) for GSD is expected to be generaliz-able to these cases. Secondly, the spatial manifold (e.g.a torus) on which the graph is defined may carry non-trivial charge, e.g. labelled by i ¯ i in the SU k (2) case. Thiscorresponds to having a so-called fluxon excitation (oftype i ¯ i ) above the original LW ground states. The loweststates of this subsector in the LW model coincide withthe ground states for the Hamiltonian obtained by re-placing the plaquette projector ˆ B p = D − P j d j ˆ B jp withˆ B p = D − P j s ij ˆ B jp , where s ij is the modular S -matrix.(See Appendix B.) The GSD in this case is computabletoo, but we leave this for a future paper . Acknowledgments
YH thanks Department of Physics, Fudan Universityfor warm hospitality he received during a visit in summer2010. YSW was supported in part by US NSF throughgrant No. PHY-0756958, No. PHY-1068558 and byFQXi.
Appendix A: GSD = 1 on a sphere
In appendix, we derive GSD = 1 on a sphere for a gen-eral Levin-Wen model, without referring to any specificstructure of the data { d, δ, G } . All we will use in thederivation are the general properties in eq. (1) and eq.(3).The simplest trivalent graph on a sphere has threeplaquettes and three edges, as illustrated in Fig. 2(a).Following the standard procedure as in (7), the GSD isexpanded asGSD sphere = X j j j * j j j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ B p ˆ B p ˆ B p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j j j + = X j j j * j j j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D X t d t ˆ B tp D X s d s ˆ B sp D X r d r ˆ B rp (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j j j + = X j j j j ′ j ′ j ′ D X r d r v j v j v j ′ v j ′ G j ∗ j j ∗ r ∗ j ′ ∗ j ′ G j j j ∗ r ∗ j ′ ∗ j ′ D X s d s v j ′ v j v j v j ′ G j ′ j ′ ∗ j ∗ s ∗ j ′ ∗ j ∗ G j ′ ∗ j j ′ s ∗ j j ′ D X t d t v j ′ v j ′ v j v j G j ∗ j ′ ∗ j ′ t ∗ j j ∗ G j j ′ ∗ j ′ t ∗ j j ∗ (A1)where ˆ B p is acting on the top bubble plaquette, ˆ B p on the bottom bubble plaquette, and ˆ B p on the restplaquette outside the two bubbles.All 6 j symbols can be eliminated by using the orthog-onality condition in eq. (3) three times, X r d r G j ∗ j j ∗ r ∗ j ′ ∗ j ′ G j j j ∗ r ∗ j ′ ∗ j ′ = 1 d j δ j ′ j j ′ ∗ δ j j j ∗ X s d s G j ′ j ′ ∗ j ∗ s ∗ j ′ ∗ j ∗ G j ′ ∗ j j ′ s ∗ j j ′ = 1 d j ′ δ j ′ j j ′ ∗ δ j j ′ j ′ ∗ X t d t G j ∗ j ′ ∗ j ′ t ∗ j j ∗ G j j ′ ∗ j ′ t ∗ j j = 1 d j δ j j j ∗ δ j j ′ j ′ ∗ (A2)and the GSD is a summation in terms of { d, δ } :GSD sphere = 1 D X j j j j ′ j ′ j ′ d j ′ d j ′ d j δ j j j ∗ δ j ′ j j ′ ∗ δ j j ′ j ′ ∗ (A3)Summing over j ′ , j ′ , and j using (1) finally leads toGSD sphere = 1. Appendix B: GSD on a torus for SU k (2) Let us consider the example associated with the quan-tum group SU k (2) (with the level k an positive integer)and calculate the GSD on a torus.There are k +1 string types, labeled as j = 0 , , , ..., k .They are the irreducible representations of SU k (2). Thequantum dimensions d j are required to be positive for all j , in order that the Hamiltonian is hermitian. Explicitly, they are d j = sin ( j +1) πk +2 sin πk +2 D = k X j =0 d j = k + 22 sin πk +2 (B1)The branching rule is δ rst = 1 if ( r + s + t is even r + s ≥ t, s + t ≥ r, t + r ≥ sr + s + t ≤ k (B2)and δ rst = 0 otherwise. The explicit formula for the6 j symbol can be found in . However, we do notneed the detailed data of the 6 j symbol in the followingcomputation of the GSD.Let us start with formula in (10), and reorder the 6 j symbols,GSD = D − X sj j j j ′ j ′ j ′ d s (cid:16) v j v j v j ′ v j ′ G j ∗ j j ∗ s ∗ j ′∗ j ′ G j j ′∗ j ′ s ∗ j j ∗ (cid:17)(cid:16) v j ′ v j v j v j ′ G j ′ j ′∗ j ∗ s ∗ j ′∗ j ∗ G j ′∗ j j ′ s ∗ j j ′ (cid:17)(cid:16) v j ′ v j ′ v j v j G j ∗ j ′∗ j ′ s ∗ j j ∗ G j j j ∗ s ∗ j ′∗ j ′ (cid:17) = D − X sj j j j ′ j ′ j ′ d s (cid:16) v j v j v j ′ v j ′ G j ∗ j j ∗ s ∗ j ′∗ j ′ G j ∗ j ∗ j sj ′ j ′ ∗ (cid:17)(cid:16) v j ′ v j v j v j ′ G j ′ j ′∗ j ∗ s ∗ j ′∗ j ∗ G j ′ j ∗ j ′ ∗ sj ∗ j ′ ∗ (cid:17)(cid:16) v j ′ v j ′ v j v j G j ∗ j ′∗ j ′ s ∗ j j ∗ G j ∗ j ′ j ′ ∗ sj ∗ j (cid:17) (B3)where the symmetry condition in (3) was used in thesecond equality.Let us compare the formula in (B3) with that in (A1).We set j = j ∗ for all j and drop all stars, since all irre-ducible representations of SU k (2) are self-dual. Then wefind that the summation (B3) has the same form as thetrace of D − P s d s ˆ B sp ˆ B sp ˆ B sp on the graph on a sphereas in (A1),tr torus ( 1 D X s d s ˆ B sp )= X j j j * j j j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D X s d s ˆ B sp ˆ B sp ˆ B sp (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j j j + =tr sphere ( 1 D X s d s ˆ B sp ˆ B sp ˆ B sp ) (B4)where ˆ B sp is defined on the only plaquette p on the torus(see Fig. 2(b)), while ˆ B sp ˆ B sp ˆ B sp is defined on the samegraph on a sphere as in (A1) (see Fig. 2(a)).The GSD on a torus becomes a trace on a sphere. Thelatter is easer to deal with since the ground state on asphere is non-degenerate. The counting of ground stateson a torus turns into a problem dealing with excitationson the sphere.In the following we evaluate the summation in the rep-resentation of elementary excitations. let us introduce anew set of operators { ˆ n rp } by a transformation,ˆ n rp = X s s r s rs ˆ B sp , ˆ B sp = X r s rs s r ˆ n rp (B5)Here s rs is a symmetric matrix (referred to as the mod-ular S -matrix for SU k (2)), s rs = 1 √ D sin ( r +1)( s +1) πk +2 sin πk +2 (B6)and has the properties s rs = s sr , s r = d r / √ D X s s rs s st = δ rt X w s wr s ws s wt s w = δ rst (B7)Eq. (B5) can be viewed as a finite discrete Fouriertransformation between { ˆ n rp } and { ˆ B sp } . By properties(B7), we see that { ˆ n rp } are mutually orthonormal projec-tors, and they form a resolution of the identity:ˆ n rp ˆ n sp = δ rs ˆ n rp , X r ˆ n rp = id (B8)In particular, ˆ n p = D P s d s ˆ B sp is the operator ˆ B p inthe Hamiltonian. The operator ˆ n rp projects onto thestates with a quasiparticle (labeled by r type) occupy-ing the plaquette p . Expressed as common eigenvectorsof { ˆ n rp } , the elementary excitations are classified by theconfiguration of these quasiparticles.Particularly, on the graph on a sphere as in (B4), theHilbert space has a basis of {| r , r , r i} , where only those r , r , and r that satisfy δ r r r = 1 are allowed. Eachbasis vector | r , r , r i is an elementary excitation withthe quasiparticles labeled by r , r , and r occupying theplaquettes p , p , and p . The configuration of quasipar-ticles are globally constrained by δ r r r = 1 . There-fore, tracing opertors { ˆ n rp } leads totr(ˆ n r p ˆ n r p ˆ n r p ) = δ r r r (B9) Applying this rule reduces the summation (B4) totr( 1 D X s d s ˆ B sp ˆ B sp ˆ B sp )=tr( 1 D X s d s X r r r s sr s sr s sr s r s r s r ˆ n r p ˆ n r p ˆ n r p )= X r r r D X s d s s sr s sr s sr s r s r s r δ r r r (B10)Then we substitute (B1), (B2) and (B6) in and obtainGSD torus SU k (2) = k X r ,r ,r =0 sin πk +2 δ r + r + r , k sin ( r +1) πk +2 sin ( r +1) πk +2 sin ( r +1) πk +2 = k X r =0 r X s =0 sin πk +2 sin ( r +1) πk +2 sin ( s +1) πk +2 sin ( r − s +1) πk +2 =( k + 1) . (B11)(Here we omit a rigorous proof of the last equality.)We can also verify GSD = ( k +1) by a direct numericalcomputation. We take the approach in to construct thenumerical data of 6 j symbols. The construction dependson a parameter, the Kauffman variable A (in the sameconvention as in ), which is specialized to roots of unity.We make the following choice: ( A = exp( πi/
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