Ground-state degeneracy in the Levin-Wen model for topological phases

We study the properties of topological phases by calculating the ground-state degeneracy (GSD) of the two-dimensional Levin-Wen (LW) model. Here it is explicitly shown that the GSD depends only on the spatial topology of the system. Then we show that the ground state on a sphere is always nondegenerate. Moreover, we study an example associated with a quantum group, and show that the GSD on a torus agrees with that of the doubled Chern-Simons theory, which is consistent with the conjectured equivalence between the LW model associated with a quantum group and the doubled Chern-Simons theory.

PHYSICAL REVIEW B 85, 075107 (2012) Ground-state degeneracy in the Levin-Wen model for topological phases Yuting Hu,1,* Spencer D. Stirling,1,2,† and Yong-Shi Wu1,3,‡ 1Department of Physics and Astronomy, University of Utah, Salt Lake City, Utah 84112, USA 2Department of Mathematics, University of Utah, Salt Lake City, Utah 84112, USA 3Department of Physics and Center for Field Theory and Particle Physics, Fudan University, Shanghai 200433, China (Received 30 August 2011; published 7 February 2012) We study the properties of topological phases by calculating the ground-state degeneracy (GSD) of the two-dimensional Levin-Wen (LW) model. Here it is explicitly shown that the GSD depends only on the spatial topology of the system. Then we show that the ground state on a sphere is always nondegenerate. Moreover, we study an example associated with a quantum group, and show that the GSD on a torus agrees with that of the doubled Chern-Simons theory, which is consistent with the conjectured equivalence between the LW model associated with a quantum group and the doubled Chern-Simons theory. DOI: 10.1103/PhysRevB.85.075107 PACS number(s): 71.10.Hf, 05.30.Pr, 02.10.Kn, 02.20.Uw I. INTRODUCTION In recent years, two-dimensional (2D) topological phases have received growing attention from the science community. They represent a novel class of quantum matter at zero temperature whose bulk properties are robust against weak interactions and disorders. Topological phases may be divided into two families: doubled (with time-reversal symmetry, or TRS, preserved) and chiral (with TRS broken). Either type may be exploited to do fault-tolerant (or topological) quantum computing.1-4 Chiral topological phases were first discovered in inte-ger and fractional quantum Hall (IQH and FQH) liquids. Mathematically, their effective low-energy description is given by Chern-Simons theory5 or (more generally) topological quantum field theory (TQFT).6 One characteristic property of FQH states is ground-state degeneracy (GSD), which depends only on the spatial topology of the system7-9 and is closely related to fractionization10-12 of quasiparticle quantum numbers, including fractional (braiding) statistics.13,14 In some cases, the GSD has been computed, as in Refs. 15 and 12. Chern-Simons theories are formulated in the continuum and have no lattice counterpart. Doubled topological phases, on the other hand, do admit a discrete description. An early known example was Kitaev's toric code model.1 More recently,Levin andWen (LW)16 constructed a discrete model to describe a large class of doubled phases. Their original motivation was to generate ground states that exhibit the phenomenon of string-net condensation17 as a physical mechanism for topological phases. The LW model is defined on a trivalent lattice (or graph) with an exactly soluble Hamiltonian. The ground states in this model can be viewed as the fixed-point states of some renormalization group flow.18,19 These fixed-point states look the same at all length scales and have no local degrees of freedom. The LW model is believed to be a Hamiltonian version of the Turaev-Viro topological quantum field theory (TQFT) in three-dimensional spacetime4,20,21 and, in particular cases, a discretized version of the doubled Chern-Simons theory.22,23 Like Kitaev's toric code model,1 we expect that the subspace of degenerate ground states in the LW model can be used as a fault-tolerant code for quantum computation. In this paper, we report the results of a recent study on the GSD of the LW model formulated on a (discretized) closed oriented surface M. Usually the GSD is examined as a topological invariant20,21,23 of the 3-manifold S1 ×M. In a Hamiltonian approach accessible to physicists, we will explicitly demonstrate that the GSD in the LWmodel depends only on the topology of M on which the system lives and, therefore, is a topological invariant of the surface M. We also show that the ground state of any LW Hamiltonian on a sphere is always nondegenerate. Moreover, we examine the LW model associated with quantum group SUk(2), which is conjectured to be equivalent to the doubled Chern-Simons theory with gauge group SU(2) at level k, and compute the GSD on a torus. Indeed we find an agreement with that in the corresponding doubled Chern-Simons theory.6,24 This supports the above-mentioned conjectured equivalence between the doubled Chern-Simons theory and the LWmodel, at least in this particular case. The paper is organized as follows. In Sec. II, we present the basics of the LWmodel, which is easy to read for newcomers. In Sec. III, topological properties of the ground states are studied, and the topological invariance of their degeneracy is shown explicitly. In Sec. IV, we demonstrate how to calculate the GSD in a general way. In Sec. V, we provide examples for the calculation, particularly on a torus. Section VI is devoted to summary and discussions. The detailed computation of the GSD is presented in the appendices. II. THE LEVIN-WEN MODEL Start with a fixed (connected and directed) trivalent graph which discretizes a closed oriented surface M (such as a torus). To each edge in the graph, we assign a string type j , which runs over a finite set j = 0,1, . . . ,N. Each string type j has a "conjugate" j ∗ that describes the effect of reversing the edge direction. For example, j may be an irreducible representation of a finite group or (more generally) a quantum group.25 Let us associate to each string type j a quantum dimension dj , which is a positive number for the Hamiltonian that we define later to be Hermitian. To each triple of strings {i,j,k}, we associate a branching rule δij k that equals 1 if the triple is "allowed" to meet at a vertex, and 0 if not (in representa-tion language, the tensor product i ⊗ j ⊗ k either contains the trivial representation or not). This data must satisfy 1098-0121/2012/85(7)/075107(8) 075107-1 ©2012 American Physical Society YUTING HU, SPENCER D. STIRLING, AND YONG-SHI WU PHYSICAL REVIEW B 85, 075107 (2012) (here, D = j d2 j ) k dkδij k∗ = didj , (1) ij didj δij k∗ = dkD, where j = 0 is the unique "trivial" string type, satisfying 0∗ = 0 and δ0jj∗ = 1,δ0ji∗ = 0 if i = j . The Hilbert space is spanned by all configurations of all possible string types j on edges. The Hamiltonian is a sum of some mutually commuting projectors, H := − v ˆQ v − p ˆB p (one for each vertex v and each plaquette p). Here, each projector ˆQ v = δij k , with i,j,k on the edges incoming to the vertex v. ˆQ v = 1 enforces the branching rule on v. Throughout the paper, we work on the subspace of states in which ˆQ v = 1 for all vertices. Each projector ˆB p is a sum D −1 s ds ˆB s p of operators that have matrix elements (on a hexagonal plaquette, for example) j1 ' j2 ' j3 ' j4 ' j5 ' j6 ' j7 j8 j9 j10 j11 j12 ˆB s p j1 j2 j3 j4 j5 j6 j7 j8 j9 j10 j11 j12 = vj1vj2vj3vj4vj5vj6vj 1vj 2vj 3vj 4vj 5vj 6G j7j ∗ 1 j6 s∗j 6j ∗ 1 ×G j8j ∗ 2 j1 s∗j 1j ∗ 2 G j9j ∗ 3 j2 s∗j 2j ∗ 3 G j10j ∗ 4 j3 s∗j 3j ∗ 4 G j11j ∗ 5 j4 s∗j 4j ∗ 5 G j12j ∗ 6 j5 s∗j 5j ∗ 6 . (2) Here, vj = dj is real. The symmetrized 6j symbols19 G are complex numbers that satisfy symmetry: Gijm kln = Gmij nk∗l∗ = Gklm ∗ ijn∗ = Gj ∗ i ∗ m ∗ l∗k∗n ∗ ; pentagon id: n dnGmlq kp∗nGjip mns∗Gjs ∗ n lkr∗ = Gjip q∗kr∗Griq ∗ mls∗ ; and orthogonality: n dnGmlq kp∗nGl ∗ m ∗ i ∗ pk∗n = δiq di δmlqδk∗ip. (3) For example, these conditions are known to be satisfied16 if we take the string types j to all be irreducible representations of a finite group, dj to be the dimension of corresponding representation space, and G to be the symmetrized Racah 6j symbols for the group. In this case, the LW model can be mapped26 to Kitaev's quantum double model.1 More general sets of data {G,d,δ} can be derived from quantum groups (or Hopf algebras).25 We will discuss such a case later using the quantum group SUk(2) (with k being the level). III. GROUND STATES Any ground state | (there may be many) must be a simultaneous +1 eigenvector for all projectors ˆQ v and ˆB p. In this section, we demonstrate the topological properties of the ground states on a closed surface with nontrivial topology. Let us begin with any two arbitrary trivalent graphs (1) and (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 Hilbert spaces are quite different from each other (they have different sizes in general). Γ(1) ⇒ Γ(2) FIG. 1. Given any two trivalent graphs (1) and (2) discretizing the same surface, we can always mutate (1) to (2) by a composition of elementary f moves. In general, (1) and (2) are not required to be regular lattices. These diagrams happen to be the same as Ref. 28, but in a slightly different context. However, we may mutate between any two given trivalent graphs (1) and (2) by a composition of the following elementary moves27 (see also Fig. 1 ): f1. ⇒ , for any edge; f2. ⇒ , for any vertex. f3. ⇒ , for any triangle structure. Suppose we are given a sequence of elementary f moves that connects two graphs, (1) → (2). We now construct a linear transformation H(1) → H(2) between the two Hilbert spaces. This is defined by associating linear maps to each elementary f move: ˆ T1 : j1 j2 j3 j5 j4 → j5 vj5vj5Gj1j2j5 j3j4j5 j1 j2 j5 ' j3 j4 ˆ T2 : j1 j2 j3 → j4j5j6 vj4vj5√ vj6 D Gj2j3j1 j∗ 6 j4j∗ 5 j1 j4 j2 j5 j j3 6 ˆ T3 : j1 j4 j2 j5 j j3 6 → vj4vj5√ vj6 D Gj ∗ 3 j ∗ 2 j ∗ 1 j∗ 4 j6j∗ 5 j1 j2 j3 (4) The mutation transformations between H(1) and H(2) are constructed by a composition of these elementary maps. As a special example, the operator ˆB p = D −1 s ds ˆB s p is such a transformation. In fact, on the particular triangle plaquette p as in (4), we have ˆB p= = Tˆ2 Tˆ3 by using the pentagon identity (id) in (3). Mutation transformations are unitary on the ground states. To see this, we only need to check that the elementary maps Tˆ1, Tˆ2, and Tˆ3 are unitary. First note that the following relations hold: Tˆ † 1 = Tˆ1, Tˆ † 2 = Tˆ3, and Tˆ † 3 = Tˆ2. We emphasize that these are maps between the Hilbert spaces on two different 075107-2 GROUND-STATE DEGENERACY IN THE LEVIN-WEN . . . PHYSICAL REVIEW B 85, 075107 (2012) graphs. For example, we check Tˆ † 1 = Tˆ1 by comparing matrix elements, j1 j2 j3 j5 j4 ˆ T † 1 j1 j2 j5 ' j3 j4 ≡ j1 j2 j5 ' j3 j4 ˆ T1 j1 j2 j3 j5 j4 ∗ =vj5vj5 Gj1j2j5 j3j4j5 ∗ =vj5vj5Gj4j1j5 j2j3j∗ 5 = j1 j2 j3 j5 j4 ˆ T1 j1 j2 j5 ' j3 j4 (5) where in the third equality we used the symmetry condition in (3). Similarly, for Tˆ † 2 = Tˆ3 (or Tˆ † 3 = Tˆ2), we have j1 j2 j3 ˆ T † 2 j1 j4 j2 j5 j j3 6 ≡ j1 j4 j2 j5 j j3 6 ˆ T2 j1 j2 j3 ∗ =vj4vj5√ vj6 D Gj2j3j1 j∗ 6 j4j∗ 5 ∗ =vj4vj5√ vj6 D Gj ∗ 3 j ∗ 2 j ∗ 1 j∗ 4 j6j∗ 5 = j1 j2 j3 ˆ T3 j1 j4 j2 j5 j j3 6 (6) Now we verify unitarity. First, Tˆ † 1 Tˆ1 = id and Tˆ † 2 Tˆ2 = Tˆ3 Tˆ2 = id by the orthogonality condition in (3) (note that since we have not used any information about the ground states in this argument, Tˆ1 and Tˆ2 are unitary on the entire Hilbert space). For the unitarity of Tˆ3, we check Tˆ † 3 Tˆ3 = Tˆ2 Tˆ3 = 1. The last equality only holds on the ground states since we have already seen that Tˆ2 Tˆ3 = ˆB p=, and ˆB p= = 1 only on the ground states. As another consequence of the above relations, the Hamil-tonian is Hermitian since all ˆB p's consist of elementary Tˆ1, Tˆ2, and Tˆ3 maps. Particularly, on a triangle plaquette, we have ˆB † p= = (Tˆ2 Tˆ3)† = ˆ T † 3 Tˆ † 2 = Tˆ2 Tˆ3 = ˆB p=. The mutation transformations serve as the symmetry transformations in the ground states. If | is a ground state, then Tˆ | is also a ground state, where Tˆ is a composition of Tˆi 's associated with elementary f moves from (1) to (2). This is equivalent to the condition Tˆ ( p ˆB p) = ( p ˆB p )Tˆ , which can be verified by the conditions in (3). (Here, p and p run over the plaquettes on (1) and (2), respectively. Also note that the ˆB p's are mutually commuting projectors, i.e., ˆB p ˆB p = ˆB p, and thus p ˆB p is the projector that projects onto the ground states.) These symmetry transformations look a little different from the usual ones since they may transform between the Hilbert spaces, H(1) and H(2), on two different graphs, (1) and (2). In general, (1) and (2) do not have the same number of vertices and edges. And thus H(1) and H(2) have different sizes. However, if we restrict to the ground-state subspaces H(1) 0 and H(2) 0 , then mutation transformations are invertible. In fact, they are unitary, as we have just shown. The tensor equations on the 6j symbols in (3) give rise to a simple result: each mutation that preserves the spatial topology of the two graphs induces a unitary symmetry transformation. During the mutations, local structures of the graphs are destroyed, while the spatial topology of the graphs is not changed. Correspondingly, the local information of the ground states may be lost, while the topological feature of the ground states is preserved. In fact, any topological feature can be specified by a topological observable ˆO that is invariant under all mutation transformations Tˆ from H(1) to H(2): ˆO ˆ T = ˆ T ˆO (where ˆO is defined on the graph (1), and ˆO is defined on the graph (2)). The symmetry transformations provide a way to charac-terize the topological phase by a topological observable. In the next section, we will investigate the GSD as such an observable. Let us end this section by remarking on the uniqueness of the mutation transformations. There may be many ways to mutate (1) to (2) using f1, f2, and f3 moves. Each way determines a corresponding transformation between the Hilbert spaces of the ground states, H(1) 0 and H(2) 0 . It turns out that all of these transformations are actually the same if the initial and final graphs, (1) to (2), are fixed, i.e., independent of which way we choose to mutate the graph (1) to (2). This means that the ground-state Hilbert spaces on different graphs can be identified (up to a mutation transformation), and all graphs are equally good. One consequence of the uniqueness of the mutation transformation is that the degrees of freedom in the ground states do not depend on the specific structure of the graph. In this sense, the LW model is the Hamiltonian version of some discrete TQFT (actually, Turaev-Viro-type TQFT; see Ref. 21). The fact that the degrees of freedom of the ground states depend only on the topology of the closed surface M is a typical characteristic of topological phases.7-9,12,15 IV. GROUND-STATE DEGENERACY In this section we investigate the simplest nontrivial topological observable, namely, the GSD. Since p ˆB p is the projector that projects onto the ground states, taking a trace computes GSD = tr( p ˆB p). We can show that GSD is a topological invariant. Namely, in the previous section we mentioned that by using (3), p ˆB p is invariant under any mutation Tˆ between the Hilbert spaces H(1) andH(2): Tˆ †( p ˆB p )Tˆ = p ˆB p. Taking a trace of both sides leads to tr( p ˆB p ) = tr( p ˆB p), where the traces are evaluated on H(2) and H(1), respectively. The independence of the GSD on the local structure of the graphs provides a practical algorithm for computing the GSD, since we may always use the simplest graph (see Fig. 2 and examples in the next section). 075107-3 YUTING HU, SPENCER D. STIRLING, AND YONG-SHI WU PHYSICAL REVIEW B 85, 075107 (2012) Expanding the GSD explicitly in terms of 6j symbols using (2), we obtain GSD = j1j2j3j4j5j6... j1 j2 j3 j5 j4 (p ˆB p) j1 j2 j3 j5 j4 = D −P s1s2s3s4··· ds1ds2ds3ds4 · · · × j 1j 2j 3j 4j 5... dj 1dj 2dj 3dj 4dj 5 · · · j1j2j3j4j5··· dj1dj2dj3dj4dj5 . . . × Gj2j5j1 s ∗ 1 j 1j 5 G j 1j2j 5 s ∗ 2 j5j 2 G j5j 1j 2 s ∗ 3 j2j1 G j3j4j ∗ 5 s ∗ 1 j ∗ 5 j 4 G j 4j ∗ 5 j3 s ∗ 2 j 3j ∗ 5 G j ∗ 5 j 3j 4 s ∗ 4 j4j3 . . . (7) The formula needs some explanation. P is the total number of plaquettes of the graph. Each plaquette p contributes a summation over sp together with a factor of dsp D . In the picture in (7), the top plaquette is being operated on first by ˆB s1 p1, next the bottom plaquette by ˆB s2 p2 , third the left plaquette by ˆB s3 p3 , and finally the right plaquette by ˆB s4 p4 . Although ordering of the ˆB s p operators is not important (since all ˆB p's commute with each other), it is important to make an ordering choice (for all plaquettes on the graph) once and for all. Each edge e contributes a summation over je and j e together with a factor of djedj e . Each vertex contributes three 6j symbols. The indices on the 6j symbols work as follows: since each vertex borders three plaquettes where ˆB s p's are being applied, we pick up a 6j symbol for each corner. However, ordering is important: because we have an overall ordering of ˆB s p's, at each vertex we get an induced ordering for the 6j symbols. Starting with the 6j symbol furthest left, we have no primes on the top row. The bottom two indices pick up primes. All of these variables (primed or not) are fed into the next 6j symbol and the same rule applies: the bottom two indices pick up a prime with the convention () = (). By the calculation of the GSD, we have characterized a topological property of the phase using local quantities living on a graph discretizing M of nontrivial topology. V. EXAMPLES (1) On a sphere. To calculate the GSD, we need to input the data {Gijm kln ,dj ,δijm} and evaluate the trace in (7). We start by computing the GSD in the simplest case of a sphere. (a) (b) FIG. 2. All trivalent graphs can be reduced to their simplest structures by compositions of elementary f moves. (a) On a sphere: two vertices, three edges, and three plaquettes. (b) On a torus: two vertices, three edges, and one plaquette. Let us consider the simplest graph, as in Fig. 2(a). We show in Appendix A that the ground state is nondegenerate on the sphere without referring to any specific structure in the model: GSDsphere = 1. In fact, for more general graphs, one can write28 the ground state as p ˆB p|0 up to a normalization factor, where in |0 all edges are labeled by string type 0. We notice that the GSD on the open disk (which is topologically the same as the 2D plane) can be studied using the same technique. This is because the open disk can be obtained by puncturing the sphere in Fig. 2(a) at the bottom. Although this destroys the bottom plaquette, we notice that the constraint ˆB p = 1 from the bottom plaquette is automatically satisfied as a consequence of the same constraint on all other plaquettes. The fact that GSDsphere (=GSDdisk) = 1 indicates the nonchiral topological order in the LW model. (2) Quantum double model. When the data are determined by representations of a finite group G, the LW model is mapped to Kitaev's quantum double model.1,26 The ground states corresponds one to one to the flat G connections.1 The GSD is GSDQD = Hom[π1(M),G] G , (8) where Hom[π1(M),G] is the space of homomorphisms from the fundamental group π1(M) to G, and G in the quotient acts on this space by conjugation. In particular, the GSD (8) on a torus is GSDtorus QD = |{(a,b)|a,b ∈ G; aba −1b −1 = e}/ ∼ |, (9) where ∼ in the quotient is the equivalence by conjugation, (a,b) ∼ (hah −1,hbh −1) for all h ∈ G. The number (9) is also the total number of irreducible representations31 of the quantum double D(G) of the group G. On the other hand, the quasiparticles in the model are classified1 by the quantum double D(G). Thus, the GSD on a torus is equal to the number of particle species in this example. (3) SUk(2) structure on a torus. More generally, on a torus, any trivalent graph can be reduced to the simplest one with two vertices and three edges, as in Fig. 2(b). On this graph, the 075107-4 GROUND-STATE DEGENERACY IN THE LEVIN-WEN . . . PHYSICAL REVIEW B 85, 075107 (2012) GSD consists of six local 6j symbols. GSD = D −1 sj1j2j3j 1j 2j 3 dsdj1dj2dj3dj 1dj 2dj 3 × G j1j2j ∗ 3 sj ∗ 3 j 2 G j ∗ 3 j1j 2 sj2j 1 G j2j ∗ 3 j 1 sj1j ∗ 3 G j ∗ 2 j3j ∗ 1 sj ∗ 1 j 3 G j 3j ∗ 1 j ∗ 2 sj ∗ 2 j ∗ 1 G j ∗ 1 j ∗ 2 j 3 sj3j ∗ 2 . (10) Now let us take the example using the quantum group SUk(2). It is known that SUk(2) has k + 1 irreducible rep-resentations, and thus the GSD we calculate is finite. We take the string types to be these representations, labeled as 0,1, . . . ,k, and the data {Gijm kln ,dj ,δijm} to be determined by these representations (for more details, see Refs. 24,29, and 30). In Appendix B, we show that in this case [for the LWmodel on a torus with string types given by irreducible representations (irreps) of SUk(2)], we have GSD = (k + 1)2. We argue this both analytically and numerically. On the other hand, it is widely believed that when the string types in the LW model are irreps from a quantum group at level k, then the associated TQFT is given by the doubled Chern-Simons theory associated with the corresponding Lie group at level±k.24,32 This equivalence tells us that in this case, the LW model can be viewed as a Hamiltonian realization of the doubled Chern-Simons theory on a lattice, and it provides an explicit picture of how the LW model describes doubled topological phases. Along these lines, our result is consistent33 with the result GSDCS = k + 1 for Chern-Simons SU(2) theory at level k on a torus. This can be seen since the Hilbert space associated to doubled Chern-Simons should be the tensor product of two copies of Chern-Simons theory at level ±k. VI. SUMMARY AND DISCUSSIONS In this paper, we studied the LW model that describes 2D topological phases, which do not break time-reversal symmetry. By examining the 2D (trivalent) graphs with the same topology, which are related to each other by a given finite set of operations (Pachner moves), we developed techniques to deal with topological properties of the ground states. Using them, we have been able to show explicitly that the GSD is determined only by the topology of the surface the system lives on, which is a typical feature of topological phases. We also demonstrated how to obtain the GSD from local data in a general way. We explicitly showed that the ground state of any LW Hamiltonian on a sphere is nondegenerate. Moreover, the LW model associated with quantum group SUk(2) was studied, and our result for the GSD on a torus is consistent with the conjecture that the LW model associated with the quantum group is the realization of a doubled Chern-Simons theory on a lattice or discrete graph. Finally, let us indicate a possible extension of the results to more general cases. First, more generally in the LW model, an extra discrete degree of freedom, labeled by an index α, may be put on the vertices. Then the branching rule δα ij k , when its value is 1, may carry an extra index α. (In representation language, this implies that given irreducible representations i,j and k, there may be multiple inequivalent ways to obtain the trivial representation from the tensor product of i ⊗ j ⊗ k. The index α just labels these different ways.) The 6j symbols accordingly carry more indices. (For more details, see the first appendix in the original paper16 of the LW model.) The expression (7) for GSD is expected to be generalizable to these cases. Second, the spatial manifold (e.g., a torus) on which the graph is defined may carry nontrivial charge, e.g., labeled by i¯i in the SUk(2) case. This corresponds to having a so-called fluxon excitation (of type i¯i) above the original LW ground states. The lowest states of this subsector in the LW model coincide with the ground states for the Hamiltonian obtained by replacing the plaquette projector ˆB p = D −1 j dj ˆB j p with ˆB p = D −1 j sij ˆB j p, where sij is the modular S matrix (see Appendix B). The GSD in this case is computable too, but we leave this for a future paper.34 ACKNOWLEDGMENTS Y.H. thanks the Department of Physics, Fudan University for warm hospitality he received during a visit in sum-mer 2010. Y.S.W. was supported in part by the US NSF through Grants No. PHY-0756958 and No. PHY-1068558, and by FQXi. APPENDIX A: GSD = 1 ON A SPHERE In this appendix, we derive GSD = 1 on a sphere for a general Levin-Wen model, without referring to any specific structure of the data {d,δ,G}. All we will use in the derivation are the general properties in Eqs. (1) and (3). The simplest trivalent graph on a sphere has three pla-quettes and three edges, as illustrated in Fig. 2(a). Following the standard procedure as in (7), the GSD is expanded as GSDsphere = j1j2j3 j1 j2 j3 ˆB p2 ˆB p3 ˆB p1 j1 j2 j3 = j1j2j3 j1 j2 j3 1 D t dtˆB tp 2 1 D s dsˆB s p3 1 D r dr ˆB r p1 j1 j2 × j3 = j1j2j3j 1j 2j 3 1 D r drvj1vj3vj 1vj 3G j ∗ 2 j3j ∗ 1 r∗j 1 ∗ j 3 G j2j1j ∗ 3 r∗j 3 ∗ j 1 × 1 D s dsvj 1vj2vj1vj 2G j 3j 1 ∗ j ∗ 2 s∗j 2 ∗ j ∗ 1 G j 3 ∗ j2j 1 s∗j1j 2 × 1 D t dtvj 2vj 3vj2vj3G j ∗ 1 j 2 ∗ j 3 t∗j3j ∗ 2 G j1j 3 ∗ j 2 t∗j2j ∗ 3 , (A1) where ˆB p1 is acting on the top bubble plaquette, ˆB p2 on the bottom bubble plaquette, and ˆB p3 on the rest of the plaquettes outside the two bubbles. 075107-5 YUTING HU, SPENCER D. STIRLING, AND YONG-SHI WU PHYSICAL REVIEW B 85, 075107 (2012) All 6j symbols can be eliminated by using the orthogonality condition in Eq. (3) three times, r drG j ∗ 2 j3j ∗ 1 r∗j 1 ∗ j 3 G j2j1j ∗ 3 r∗j 3 ∗ j 1 = 1 dj2 δj 1j2j 3 ∗δj1j2j ∗ 3 , s dsG j 3j 1 ∗ j ∗ 2 s∗j 2 ∗ j ∗ 1 G j 3 ∗ j2j 1 s∗j1j 2 = 1 dj 3 δj 1j2j 3 ∗δj1j 2j 3 ∗ , (A2) t dtG j ∗ 1 j 2 ∗ j 3 t∗j3j ∗ 2 G j1j 3 ∗ j 2 t∗j2j3 = 1 dj1 δj1j2j ∗ 3 δj1j 2j 3 ∗ , and the GSD is a summation in terms of {d,δ}: GSDsphere = 1 D3 j1j2j3j 1j 2j 3 dj 1dj 2dj3δj1j2j ∗ 3 δj 1j2j 3 ∗δj1j 2j 3 ∗ . (A3) Summing over j 1, j 2, and j3 using (1) finally leads to GSDsphere = 1. APPENDIX B: GSD ON A TORUS FOR SUk(2) Let us consider the example associated with the quantum group SUk(2) (with the level k as a positive integer) and calculate the GSD on a torus. There are k + 1 string types, labeled as j = 0,1,2, . . . ,k. They are the irreducible representations of SUk(2). The quantum dimensions dj are required to be positive for all j in order that the Hamiltonian is Hermitian. Explicitly, they are dj = sin (j+1)π k+2 sin π k+2 , (B1) D = k j=0 d2 j = k + 2 2 sin2 π k+2 . The branching rule is δrst = 1 if ⎧⎪⎨ ⎪⎩ r + s + t is even, r + s t, s + t r, t + r s, r + s + t 2k, (B2) and δrst = 0 otherwise. The explicit formula for the 6j symbol can be found in Refs. 29 and 30. However, we do not need the detailed data of the 6j symbol in the following computation of the GSD. Let us start with the formula (10), and reorder the 6j symbols, GSD = D −1 sj1j2j3j 1j 2j 3 ds vj1vj3vj 1vj 3G j ∗ 2 j3j ∗ 1 s∗j ∗ 1 j 3 G j2j ∗ 3 j 1 s∗j1j ∗ 3 × vj 1vj2vj1vj 2G j 3j ∗ 1 j ∗ 2 s∗j ∗ 2 j ∗ 1 G j ∗ 3 j1j 2 s∗j2j 1 × vj 2vj 3vj2vj3G j ∗ 1 j ∗ 2 j 3 s∗j3j ∗ 2 G j1j2j ∗ 3 s∗j ∗ 3 j 2 , = D −1 sj1j2j3j 1j 2j 3 ds vj1vj3vj 1vj 3G j ∗ 2 j3j ∗ 1 s∗j ∗ 1 j 3 G j ∗ 2 j ∗ 1 j3 sj 3j 1 ∗ × vj 1vj2vj1vj 2G j 3j ∗ 1 j ∗ 2 s∗j ∗ 2 j ∗ 1 G j 3j ∗ 2 j 1 ∗ sj ∗ 1 j 2 ∗ × vj 2vj 3vj2vj3G j ∗ 1 j ∗ 2 j 3 s∗j3j ∗ 2 G j ∗ 1 j 3j 2 ∗ sj ∗ 2 j3 , (B3) where the symmetry condition in (3) was used in the second 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 ir-reducible representations of SUk(2) are self-dual. Then we find that the summation (B3) has the same form as the trace of D −1 s ds ˆB s p2 ˆB s p3 ˆB s p1 on the graph on a sphere as in (A1), trtorus 1 D s ds ˆB s p = j1j2j3 j1 j2 j3 1 D s ds ˆB s p2 ˆB s p3 ˆB s p1 j1 j2 j3 = trsphere 1 D s ds ˆB s p2 ˆB s p3 ˆB s p1 , (B4) where ˆB s p is defined on the only plaquette p on the torus [see Fig. 2(b)], while ˆB s p1 ˆB s p2 ˆB s p3 is defined on the same graph on a sphere as in (A1) [see Fig. 2(a)]. The GSD on a torus becomes a trace on a sphere. The latter is easier to deal with since the ground state on a sphere is nondegenerate. The counting of ground states on a torus turns into a problem dealing with excitations on the sphere. In the following, we evaluate the summation in the representation of elementary excitations. Let us introduce a new set of operators {ˆn r p } by a transformation, ˆnr p = s sr0srs ˆB s p, ˆB s p = r srs sr0 ˆn r p. (B5) Here, srs is a symmetric matrix [referred to as the modular S matrix for SUk(2)], srs = 1 √ D sin (r+1)(s+1)π k+2 sin π k+2 , (B6) and has the properties srs = ssr, sr0 = dr/ √ D, s srssst = δrt , (B7) w swr swsswt sw0 = δrst . Equation (B5) can be viewed as a finite discrete Fourier transformation between {ˆn r p } and {ˆB s p }. By properties (B7), we see that {ˆn r p } are mutually orthonormal projectors, and they form a resolution of the identity ˆn r p ˆn s p = δrs ˆn r p, r ˆn r p = id. (B8) 075107-6 GROUND-STATE DEGENERACY IN THE LEVIN-WEN . . . PHYSICAL REVIEW B 85, 075107 (2012) In particular, ˆn 0 p = 1 D s ds ˆB s p is the operator ˆB p in the Hamiltonian. The operator ˆn r p projects onto the states with a quasiparticle (labeled by r type) occupying the plaquette p. Expressed as common eigenvectors of {ˆn r p }, the elementary excitations are classified by the configuration of these quasi-particles. Particularly, on the graph on a sphere as in (B4), the Hilbert space has a basis of {|r1,r2,r3}, where only those r1, r2, and r3 that satisfy δr1r2r3 = 1 are allowed. Each basis vector |r1,r2,r3 is an elementary excitation with the quasiparticles labeled by r1, r2, and r3 occupying the plaquettes p1, p2, and p3. The configuration of quasiparticles is globally constrained by δr1r2r3 = 1.34 Therefore, tracing operators {ˆn r p } leads to tr ˆn r2 p2ˆn r3 p3ˆn r1 p1 = δr2r3r1 . (B9) The application of this rule reduces the summation (B4) to tr 1 D s ds ˆB s p2 ˆB s p3 ˆB s p1 = tr 1 D s ds r1r2r3 ssr1 ssr2 ssr3 sr10sr20sr30 ˆn r2 p2ˆn r3 p3ˆn r1 p1 = r1r2r3 1 D s ds ssr1 ssr2 ssr3 sr10sr20sr30 δr1r2r3 . (B10) Then we substitute (B1), (B2), and (B6) in and obtain GSDtorus SUk (2) = k r1,r2,r3=0 sin π k+2 δr1+r2+r3,2k sin (r1+1)π k+2 sin (r2+1)π k+2 sin (r3+1)π k+2 = k r=0 r s=0 sin π k+2 sin (r+1)π k+2 sin (s+1)π k+2 sin (r−s+1)π k+2 = (k + 1)2. (B11) (Here we omit a rigorous proof of the last equality.) We can also verify GSD = (k + 1)2 by a direct numerical computation. We take the approach in Ref. 30 to construct the numerical data of 6j symbols. The construction depends on a parameter, i.e., the Kauffman variable A (in the same convention as in Ref. 30), which is specialized to roots of unity. We make the following choice: ⎧⎪⎨ ⎪⎩ A = exp(πi/3) at k = 1, A = exp(3πi/8) at k = 2, A = exp(3πi/5) at k = 3. (B12) By this choice, the quantum dimensions dj take the values as in (B1), and the 6j symbols satisfy the self-consistent conditions in (3). 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