Ground State Degeneracy in the Levin-Wen Model for Topological Phases Yuting Hu,1,∗ Spencer D. Stirling,1,2,† and Yong-Shi Wu3,1,‡ 1Department of Physics and Astronomy, University of Utah, Salt Lake City, UT 84112, USA 2Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA 3Department of Physics and Center for Field Theory and Particle Physics, Fudan University, Shanghai 200433, China (Dated: January 26, 2013) We study properties of topological phases by calculating the ground state degeneracy (GSD) of the2dLevin-Wen(LW)model. HereitisexplicitlyshownthattheGSDdependsonlyonthespatial topology of thesystem. Then weshowthat theground stateon asphereisalways non-degenerate. 2 Moreover,westudyanexampleassociatedwithaquantumgroup,andshowthattheGSDonatorus 1 agrees with that of the doubled Chern-Simons theory, consistent with the conjectured equivalence 0 between the LW model associated with a quantum group and thedoubled Chern-Simons theory. 2 PACSnumbers: 05.30.Pr71.10.Hf02.10.Kn02.20.Uw n a J I. INTRODUCTION Simons theory22,23. Like Kitaev’s toric code model1, we 0 expect that the subspace of degenerate ground states in 2 In recent years two-dimensional topological phases the LW model can be used as a fault-tolerant code for ] have received growing attention from the science com- quantum computation. el munity. They represent a novel class of quantum mat- Inthispaperwereporttheresultsofarecentstudyon - ter at zero temperature whose bulk properties are ro- the GSD of the LW model formulated on a (discretized) r t bustagainstweakinteractionsanddisorders. Topological closed oriented surface M. Usually the GSD is exam- .s phases may be divided into two families: doubled (with ined as a topological invariant20,21,23 of the 3-manifold at time-reversal symmetry, or TRS, preserved), and chiral S1 M. In a Hamiltonian approachaccessible to physi- × m ( with TRS broken). Either type may be exploitedto do cists, we will explicitly demonstrate that the GSD in the - fault-tolerant (or topological) quantum computing1–4. LW model depends only on the topology of M on which d Chiral topologicalphases were first discoveredin inte- the system lives and, therefore, is a topological invari- n gerandfractionalquantumHall(IQHandFQH)liquids. ant of the surface M. We also show that the ground o Mathematically, their effective low-energy description is state of any LW Hamiltonian on a sphere is always non- c [ givenbyChern-Simonstheory5or(moregenerally)topo- degenerate. Moreover, we examine the LW model as- logical quantum field theory (TQFT)6. One character- sociated with quantum group SUk(2), which is conjec- 3 istic property of FQH states is ground state degeneracy turedto be equivalenttothe doubledChern-Simonsthe- v (GSD),whichdependsonlyonthespatialtopologyofthe orywith gaugegroupSU(2) atlevel k, andcompute the 1 system7–9 and is closely related to fractionization10–12 GSD on a torus. Indeed we find an agreement with that 7 7 of quasiparticle quantum numbers, including fractional in the corresponding doubled Chern-Simons theory6,24. 5 (braiding)statistics13,14. InsomecasestheGSDhasbeen This supports the above-mentioned conjectured equiva- . computed in refs.12,15. lence betweenthe doubled Chern-Simonstheory andthe 5 LW model, at least in this particular case. 0 Chern-Simons theories are formulated in the contin- 1 uum and have no lattice counterpart. Doubled topolog- The paper is organized as follows. In Section II we 1 ical phases, on the other hand, do admit a discrete de- present the basics of the LW model, easy to read for : scription. The first known example was Kitaev’s toric newcomers. In Section III topological properties of the v i code model1. groundstatesarestudied, andthe topologicalinvariance X More recently, Levin and Wen (LW)16 constructed of their degeneracy is shown explicitly. In section IV we r a discrete model to describe a large class of doubled demonstrate how to calculate the GSD in a generalway. a phases. Their original motivation was to generate InsectionVweprovideexamplesforthecalculationpar- ground states that exhibit the phenomenon of string-net ticularly on a torus. Section VI is devoted to summary condensation17 as a physical mechanism for topological and discussions. The detailed computation of the GSD phases. TheLWmodelisdefinedonatrivalentlattice(or is presented in the appendices. graph)withanexactlysolubleHamiltonian. Theground states in this model can be viewed as the fixed-point states of some renormalization group flow18,19. These II. THE LEVIN-WEN MODEL fixed-point states look the same at all length scales and have no local degrees of freedom. Start with a fixed (connected and directed) trivalent The LW model is believed to be a Hamiltonian ver- graph Γ which discretizes a closed oriented surface M sion of the Turaev-Viro topological quantum field the- (such as a torus). To each edge in the graphwe assign a ory (TQFT) in three dimensional spacetime4,20,21 and, string type j, which runs over a finite set j =0,1,...,N. in particular cases,discretized versionof doubled Chern- Each string type j has a “conjugate” j∗ that describes 2 the effect of reversing the edge direction. For example j may be an irreducible representationof a finite group or (more generally) a quantum group25. Letusassociatetoeachstringtypejaquantumdimen- sion d , which is a positive number for the Hamiltonian j we define later to be hermitian. To each triple of strings Γ(1) ⇒ Γ(2) i,j,k we associate a branching rule δ that equals 1 { } ijk FIG. 1: Given any two trivalent graphs Γ(1) and Γ(2) dis- if the triple is “allowed” to meet at a vertex, 0 if not cretizingthesamesurface, wecanalwaysmutateΓ(1) toΓ(2) (in representation language the tensor product i j k ⊗ ⊗ byacomposition of elementary f moves. Ingeneral Γ(1) and either contains the trivial representation or not). This Γ(2) are not required to be regular lattices. These diagrams data must satisfy (here D = jd2j) happentobethesameas28,butinaslightlydifferentcontext. dkδijkP∗ =didj k X III. GROUND STATES didjδijk∗ =dkD (1) ij X Any ground state Φ (there may be many) must be a j =0istheunique“trivial”stringtype,satisfying0∗ =0 | i simultaneous+1eigenvectorforallprojectorsQˆ andBˆ . and δ0jj∗ =1,δ0ji∗ =0 if i=j. v p 6 Inthissectionwedemonstratethetopologicalproperties The Hilbert space is spanned by all configurations of the ground states on a closed surface with non-trivial of all possible string types j on edges. The Hamilto- topology. nian is a sum of some mutually-commuting projectors H := Qˆ Bˆ (one for each vertex v and each Let us begin with any two arbitrary trivalent graphs − v v− p p Γ(1) andΓ(2) discretizingthesamesurface(e.g.,atorus). plaquettep). HereeachprojectorQˆ =δ withi,j,kon P P v ijk IfwecomparetheLWmodelsbasedonthesetwographs, the edges incoming to the vertex v. Qˆv =1 enforces the respectively, then immediately we see that the Hilbert branching rule on v. Throughout the paper we work on spacesarequite differentfromeachother (they havedif- the subspace of states in which Qˆv = 1 for all vertices. ferent sizes in general). Each projector Bˆp is a sum D−1 sdsBˆps of operators However,wemaymutatebetweenanytwogiventriva- that have matrix elements (on a hexagonalplaquette for lent graphs Γ(1) and Γ(2) by a composition of the follow- P example) ing elementary moves27 (see also Fig 1 ): j7 j12 j7 j12 *j8j9jj1'2' jj3'6' jj5'4' j1j011(cid:12)(cid:12)(cid:12)Bˆps(cid:12)(cid:12)(cid:12)j8j9jj12 jj36 jj54j1j011+ f1. ⇒ , for any edge; (cid:12) (cid:12) =vj1vj2vj3vj(cid:12)4vj5(cid:12)vj6vj1′vj2′vj3′vj4′vj5′vj6′ (2) f2. ⇒ , for any vertex. Gj7j1∗j6 Gj8j2∗j1 Gj9j3∗j2 Gj10j4∗j3Gj11j5∗j4Gj12j6∗j5 s∗j6′j1′∗ s∗j1′j2′∗ s∗j2′j3′∗ s∗j3′j4′∗ s∗j4′j5′∗ s∗j5′j6′∗ Herevj = dj isreal. The symmetrized6j symbols19 G f3. ⇒ , for any triangle structure. are complex numbers that satisfy p symmetry: Gikjlmn =Gmnki∗jl∗ =Gkijlnm∗∗ =(Glj∗∗ki∗∗mn∗)∗ thaStupcopnonseecwtsetawreoggirvaepnhassΓe(q1u)enceΓ(o2f)e.lWemeennotawrycofnmstoruvecst pentagon id: dnGmkpl∗qnGjmipns∗Gjlksr∗∗n =Gjqi∗pkr∗Grmiqls∗∗ a linear transformation (1) → (2) between the two H → H n Hilbertspaces. Thisisdefinedbyassociatinglinearmaps X orthogonality: dnGmkpl∗qnGpl∗km∗n∗i∗ = δdiqδmlqδk∗ip to each elementary f move: i For example, theXsen conditions are known to (b3e) Tˆ1 :(cid:12)(cid:12) jj21 j5 jj43+→Xj5′ vj5vj5′Gjj13jj24jj55′(cid:12)(cid:12) jj12j5' jj43+ (cid:12) (cid:12) dsdaiumtciisebfinleesdio1rn6epoirffescweoenrtrtaeastkpioeonntsdhoienfgsatrreifinpngrietsteeynpgteraostuijopn,tosdpjbaectoe,ablalenidrtrhGee- Tˆ2 :(cid:12)(cid:12)(cid:12)(cid:12)j1j2 j3+→j4Xj5j6 vj4√vjD5vj6Gjj(cid:12)(cid:12)26∗jj34jj15∗(cid:12)(cid:12)j1j4j2j6j5j3+ to be the symmetrized Racah 6j symbols for the group. (cid:12) (cid:12) tInaevt’hsisqucaansteumthdeoLuWblemmooddeell1.caMnorbeegemnaepraplesde2t6sotfodaKtia- Tˆ3 :(cid:12)(cid:12)(cid:12)j1j4j2j6j5j3+→ vj4√vjD5vj6Gjj43∗∗jj62∗jj5∗1∗(cid:12)j1j2 (cid:12)(cid:12)j3+ (4) (cid:12) (cid:12) G,d,δ can be derived from quantum groups (or Hopf (cid:12) (cid:12) {algebra}s)25. We will discuss such a case later using the The m(cid:12)(cid:12)utation transformations betw(cid:12)(cid:12)een (1) and (2) H H quantum group SU (2) (k being the level). are constructed by a composition of these elementary k 3 maps. As a special example, the operator Bˆ = conditions in (3). (Here p and p′ run overthe plaquettes p D−1 d Bˆs is such a transformation. In fact, on onΓ(1) andΓ(2),respectively. AlsonotethattheBˆ ’sare s s p p the particular triangle plaquette p as in (4), we have mutually-commuting projectors, i.e., Bˆ Bˆ = Bˆ , and p p p Bˆp=▽P=Tˆ2Tˆ3, by using the pentagon identity in (3). thus pBˆpistheprojectorthatprojectsontotheground Mutation transformations are unitary on the ground states.) states. To see this, we only need to check that the el- Q These symmetrytransformationslook alittle different ementary maps Tˆ , Tˆ , and Tˆ are unitary. First note 1 2 3 from the usual ones since they may transform between that the following relations hold: Tˆ1† =Tˆ1, Tˆ2† =Tˆ3, and the Hilbert spaces (1) and (2) ontwo different graphs Tˆ† =Tˆ . Weemphasizethatthesearemapsbetweenthe Γ(1) and Γ(2). In gHeneral, Γ(H1) and Γ(2) do not have the 3 2 Hilbert spaces on two different graphs. For example, we same number of vertices and edges. And thus (1) and check Tˆ† =Tˆ by comparing matrix elements (2) have different sizes. However, if we restriHct to the 1 1 H ground-state subspaces (1) and (2), mutation trans- j1 j5 j4 Tˆ† j1j' j4 j1j' j4 Tˆ j1 j5 j4 ∗ formationsare invertibleH. I0n fact, tHhe0y are unitary as we * j2 j3(cid:12) 1(cid:12) j25 j3+≡ * j25 j3(cid:12) 1(cid:12) j2 j3+! have just shown. (cid:12) (cid:12) (cid:12) (cid:12) The tensor equations on the 6j symbols in (3) give (cid:12) (cid:12) (cid:12) (cid:12) ∗ (cid:12)(cid:12) (cid:12)(cid:12) =vj5vj5′ Gjj13(cid:12)(cid:12)jj24jj55(cid:12)(cid:12)′ rsipsaettiaolatospimolopgleyroefstuhlte:tewaochgrmapuhtastiinodnutchesatapurneistearrvyessytmhe- =vj5′vj5G(cid:16)jj42jj13jj55′∗ (cid:17) mtuertersyotfrtahnesfgorrampahtsioanre. dDeusrtirnogyetdh,ewmhuilteattihoenssp,alotciaalltsotrpuocl-- = j1 j5 j4 Tˆ j1j' j4 (5) ogy of the graphs is not changed. Correspondingly, the * j2 j3(cid:12) 1(cid:12) j25 j3+ localinformationofthe groundstates maybe lost,while (cid:12) (cid:12) the topologicalfeature of the ground states is preserved. (cid:12) (cid:12) whereinthe thirdequalitywe used(cid:12)(cid:12)the(cid:12)(cid:12)symmetrycondi- Infact,anytopologicalfeaturecanbespecifiedbyatopo- tion in (3). logicalobservable Oˆ that is invariant under all mutation Similarly, for Tˆ2† =Tˆ3 (or Tˆ3† =Tˆ2), we have OtˆrainssdfoerfimneadtioonnstThˆefrgormapHh(Γ1()1t)oaHnd(2O)ˆ:′Ooˆn′TˆΓ=(2)T)ˆ.Oˆ (where * j1j2 j3(cid:12)(cid:12)Tˆ2†(cid:12)(cid:12)j1j4j2j6j5j3+≡ *j1j4j2j6j5j3(cid:12)(cid:12)Tˆ2(cid:12)(cid:12) j1j2 j3+!∗ aacbtTleeh.riezInesyttmhheemnettoerxpytotlsoreagcnitcsiaofolnrpmwheaatswieoinlblsyipnravoevtsoitdipgeoasltoaegwitchaaeylGtoobScsDheraavrs-- (cid:12) (cid:12) v v v (cid:12) (cid:12) ∗ such an observable. (cid:12)(cid:12) (cid:12)(cid:12) = j4√jD5 j6 (cid:12)(cid:12)Gjj26∗(cid:12)(cid:12)jj34jj15∗ Let us end this section by remarking on uniqueness of =vj4√vjD5vj6G(cid:16)jj43∗∗jj62∗jj5∗1∗ (cid:17) tthoemmuutatatetioΓn(1t)ratonsΓfo(r2m) uastiinongsf.1T,hfe2reanmdafy3bmeomvaens.yEwaacyhs way determines a correspondingtransformationbetween = j1 j3 Tˆ3 j1j4j6j5j3 (6) the Hilbert spaces of ground states, H0(1) and H0(2). It * j2 (cid:12) (cid:12) j2 + turns out that all these transformations are actually the (cid:12)(cid:12) (cid:12)(cid:12) same if the initial and final graphs Γ(1) to Γ(2) are fixed, (cid:12) (cid:12) Now we verify unitary. First, Tˆ†(cid:12)Tˆ (cid:12)= id and Tˆ†Tˆ = i.e., independent of which way we choose to mutate the Tˆ Tˆ = id by the orthogonality c1on1dition in (3)2(n2ote graph Γ(1) to Γ(2). This means that the ground state 3 2 Hilbert spaces on different graphs can be identified (up that, since we have not used any information about the ground states in this argument, Tˆ and Tˆ are unitary toamutationtransformation)andallgraphsareequally 1 2 on the entire Hilbert space). For unitary of Tˆ we check good. 3 Tˆ†Tˆ = Tˆ Tˆ = 1. The last equality only holds on the One consequence of the uniqueness of the mutation 3 3 2 3 tranformation is that the degrees of freedom in the ground states since we have already seen that Tˆ Tˆ = 2 3 ground states do not depend on the specific structure Bˆp=▽ and Bˆp=▽ =1 only on the ground states. of the graph. In this sense, the LW model is the Hamil- As another consequence of the above relations, the tonianversionofsomediscrete TQFT(actually, Turaev- Hamiltonian is hermitian since all Bˆp’s consist of ele- Viro type TQFT, see21). The fact that the degrees of mentaryTˆ1, Tˆ2,andTˆ3 maps. Particularly,onatriangle freedom of the ground states depend only on the topol- plaquette, we have Bˆp†=▽ = (Tˆ2Tˆ3)† = Tˆ3†Tˆ2† = Tˆ2Tˆ3 = ogy of the closed surface M is a typical characteristic of Bˆp=▽. topological phases7–9,12,15. The mutation transformations serve as the symmetry transformations in the ground states. If Φ is a ground | i state then Tˆ Φ is also a ground state, where Tˆ is a IV. GROUND STATE DEGENERACY | i composition of Tˆ’s associated with elementary f moves i from Γ(1) to Γ(2). This is equivalent to the condition In this section we investigate the simplest nontrivial Tˆ( pBˆp) = ( p′Bˆp′)Tˆ, which can be verified by the topologicalobservable,the GSD.Since pBˆp isthepro- Q Q Q 4 jectorthatprojectsontothegroundstates,takingatrace computes GSD=tr( Bˆ ). p p We can show that GSD is a topological invariant. Q Namely, in the previous section we mentioned that, by using (3), pBˆp is invariant under any muta- (a) (b) tion Tˆ between the Hilbert spaces (1) and (2) : Tˆ†( p′Bˆp′)Tˆ =Q pBˆp. Taking a traHce of bothHsides FstIrGuc.t2u:reAsllbytricvoamlepnotsgitriaopnhssocfaenlebmeernetdaurycefd tmootvheesi.r(saim)polnesat leadQs to tr′( p′BˆQp′) = tr( pBˆp), where the traces are sphere: 2 vertices, 3 edges, and 3 plaquettes. (b) on a torus: evaluated on (2) and (1) respectively. 2 vertices, 3 edges, and 1 plaquette. QH H Q TheindependenceoftheGSDonthelocalstructureof the graphs provides a practical algorithm for computing the GSD, since we may always use the simplest graph V. EXAMPLES (see Fig 2 and examples in the next section). Expanding the GSD explicitly in terms of 6j symbols (1)Onasphere. TocalculatetheGSD,weneedtoinput using (2) we obtain the data Gijm,d ,δ and evaluate the trace in (7). { kln j ijm} We start by computing the GSD in the simplest case of GSD= j1 j5 j4 ( Bˆp) j1 j5 j4 a sLpehte’sre.consider the simplest graph as in Fig. 2(a). j1j2j3Xj4j5j6...* j2 j3(cid:12)(cid:12) Yp (cid:12)(cid:12) j2 j3+ We show in Appendix A that the ground state is non- (cid:12) (cid:12) =D−P d d d d(cid:12) ... (cid:12) degenerateonthespherewithoutreferringtoanyspecific s1 s2 s3 s(cid:12)4 (cid:12) structure in the model: GSDsphere =1. In fact, for more s1s2Xs3s4... generalgraphsone canwrite down28 the groundstate as j1′j2′Xj3′j4′j5′...dj1′dj2′dj3′dj4′dj5′...j1j2jX3j4j5...dj1dj2dj3dj4dj5... edpgeBˆspa|0rei luapbetloedabnyosrtmrianlgizatytipoen0f.actor, where in |0i all Gjs2∗1jj51′jj15′Gjs1′∗2jj25jj5′2′Gjs5∗3jj1′2jj2′1 Gjs3∗1jj45′∗j5∗j4′Gjs4′∗2jj5′3′∗jj5∗3Gsj5∗∗4jj34′jj43′ ... QtopWoelognioctailcley tthhaetstahmeeGaSsDthoen2dthpelaonpee)ncdainskbe(wshtuicdhieids (cid:16) (cid:17)(cid:16) (cid:17) (7) using the same technique. This is because the open disk can be obtained by puncturing the sphere in Fig 2(a) at Theformulaneedssomeexplanation. P isthetotalnum- the bottom. Although this destroys the bottom plaque- ber of plaquettes of the graph. Each plaquette p con- tte,wenoticethattheconstraintBˆ =1fromthebottom p tributes a summation over sp together with a factor of plaquette is automatically satisfied as a consequence of dsp. Inthepicturein(7)thetopplaquetteisbeingoper- the same constraint on all other plaquettes. The fact aDted on first by Bˆs1, next the bottom plaquette by Bˆs2, that GSDsphere(= GSDdisk) = 1 indicates the non-chiral p1 p2 topological order in the LW model. third the left plaquette by Bˆs3, and finally the rightpla- p3 (2) Quantum double model. When the data are deter- quette by Bˆs4. Although ordering of the Bˆs operators is mined by representations of a finite group G, the LW p4 p not important (since all Bˆ ’s commute with each other), model is mapped to Kitaev’s quantumdouble model1,26. p it is important to make an ordering choice (for all pla- The ground states corresponds one-to-one to the flat G- quettes on the graph) once and for all. connections1. The GSD is Each edge e contributes a summation over je and je′ Hom(π1(M),G) ttohgreeeth6ejrswyimthbaolsfa.ctor of djedje′. Eachvertex contributes GSDQD =(cid:12) G (cid:12) (8) (cid:12) (cid:12) The indices on the 6j symbols work as follows: since where Hom(π1( ),G) i(cid:12)(cid:12)s the space of ho(cid:12)(cid:12)momorphisms eachvertexbordersthreeplaquetteswhereBˆs’sarebeing from the fundamMental group π (M) to G, and G in the p 1 applied,wepickupa6jsymbolforeachcorner. However, quotient acts on this space by conjugation. orderingis important: because we have anoverallorder- In particular, the GSD (8) on a torus is ing of Bˆs’s, at each vertex we get an induced ordering for the 6jp symbols. Starting with the 6j symbolfurthest GSDtQoDrus = {(a,b)|a,b∈G;aba−1b−1 =e}/∼ (9) left we have no primes on the top row. The bottom two where intheq(cid:12)uotientistheequivalencebyconjug(cid:12)ation, indices pick up primes. All of these variables (primed or ∼ (cid:12) (cid:12) not) are fed into the next 6j symbol and the same rule (a,b) (hah−1,hbh−1) for all h G applies: thebottomtwoindicespickupaprimewiththe ∼ ∈ convention ()′′ =(). The number (9) is alsothe totalnumber of irreducible By the calculation of the GSD, we have characterized representations31 of the quantum double D(G) of the atopologicalpropertyofthe phaseusinglocalquantities group G. On the other hand, the quasiparticles in the living on a graph discretizing M of nontrivial topology. modelareclassified1bythequantumdoubleD(G). Thus 5 the GSD on a torus is equal to the number of particle consistent with the conjecture that the LW model asso- species in this example. ciatedwithquantumgroupistherealizationofadoubled (3) SU (2) structure on a torus. More generally, on a Chern-Simons theory on a lattice or discrete graph. k torus any trivalent graphcan be reduced to the simplest Finally,letusindicatepossibleextensionoftheresults onewithtwoverticesandthreeedges,asinFig2(b). On to more general cases. First, more generally in the LW this graph the GSD consists of six local 6j symbols. model,anextradiscretedegreeoffreedom,labelledbyan index α, may be put on the vertices. Then the branch- GSD=D−1 dsdj1dj2dj3dj1′dj2′dj3′ ing rule δα , when its value is 1, may carry an extra sj1j2Xj3j1′j2′j3′ index α. i(jIkn representation language this implies that Gjs1j3j′∗2jj2′3∗Gjs3′j∗2jj11′j2′Gjs2j1j3′j∗3∗j1′ Gsj2∗j1′j∗3jj3′1∗Gjs3′j2j′∗1′∗j1j∗2∗Gjs1∗j3jj2′∗2∗j3′ gmivueltnipilrereidnueqciubilvealreenptrewsaeynstattoioonbstai,ijnatnhdetkr,ivthiaelrreempraeysebne- (cid:16) (cid:17)(cid:16) (cid:17)(10) tation from the tensor product of i j k. The index ⊗ ⊗ α just labels these different ways.) The 6j symbols ac- Now letus takethe exampleusing the quantumgroup cordingly carry more indices. (For more details see the SU (2). It is known that SU (2) has k+1 irreducible k k firstAppendix in the originalpaper16 of the LW model.) representations,and thus the GSD we calculate is finite. The expression (7) for GSD is expected to be generaliz- We take the string types to be these representations, la- beled as 0,1,...,k, and the data Gijm,d ,δ to be able to these cases. Secondly, the spatial manifold (e.g. { kln j ijm} a torus) on which the graph is defined may carry non- determined by these representations (for more details, see24,29,30). trivialcharge,e.g. labelledbyi¯iintheSUk(2)case. This corresponds to having a so-called fluxon excitation (of In Appendix B we show that in this case (for the LW typei¯i)abovetheoriginalLWgroundstates. Thelowest model on a torus with string types given by irreps of states of this subsector in the LW model coincide with SU (2)) we have GSD = (k+1)2. We argue this both k the ground states for the Hamiltonian obtained by re- analytically and numerically. placing the plaquette projector Bˆ =D−1 d Bˆj with Onthe otherhand, itiswidely believedthatwhenthe p j j p string types in the LW model are irreps from a quantum Bˆp =D−1 jsijBˆpj, where sij is the moduPlar S-matrix. group at level k, then the associated TQFT is given by (See Appendix B.) The GSD in this case is computable P doubled Chern-Simons theory associatedwith the corre- too, but we leave this for a future paper34. sponding Lie group at level k24,32. 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 howthe LWmodeldescribesdoubledtopologicalphases. Along these lines, our result is consistent33 with the Acknowledgments result GSD = k+1 for Chern-Simons SU(2) theory CS at level k on a torus. This can be seen since the Hilbert spaceassociatedto doubledChern-Simons shouldbe the YH thanks Department of Physics, Fudan University tensor product of two copies of Chern-Simons theory at forwarmhospitalityhereceivedduringavisitinsummer level k. 2010. YSW was supported in part by US NSF through ± grant No. PHY-0756958, No. PHY-1068558 and by FQXi. VI. SUMMARY AND DISCUSSIONS Inthispaper,westudiedthe LWmodelthatdescribes 2d topological phases which do not break time-reversal symmetry. By examining the 2d (trivalent) graphs with same topology which are related to each other by a Appendix A: GSD=1 on a sphere given finite set of operations (Pachner moves), we de- velopedtechniques to deal with topologicalproperties of the ground states. Using them, we have been able to Inappendix,wederiveGSD=1onasphereforagen- show explicitly that the GSD is determined only by the eral Levin-Wen model, without referring to any specific topologyofthesurfacethesystemliveson,whichisatyp- structure of the data d,δ,G . All we will use in the ical feature of topological phases. We also demonstrated { } derivation are the general properties in eq. (1) and eq. how to obtainthe GSD fromlocaldatain a generalway. (3). We explicitly showed that the ground state of any LW Hamiltonian on a sphere is non-degenerate. Moreover, The simplest trivalent graph on a sphere has three the LW model associated with quantum group SU (2) plaquettes and three edges, as illustrated in Fig. 2(a). k was studied, and our result for the GSD on a torus is Following the standard procedure as in (7), the GSD is 6 expanded as they are j1 j1 d = sin(jk++12)π GSDsphere = j3 Bˆp2Bˆp3Bˆp1 j3 j sink+π2 * (cid:12) (cid:12) + j1 j1Xj2j3 j2 (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) j2 D = k d2j = 2skin+2 2π (B1) =j1Xj2j3* jj23 (cid:12)(cid:12)D1 Xt dtBˆpt2 The branching ruleXj=is0δrst =1 if k+2 (cid:12) (cid:12) (cid:12) j1 r+s+t is even 1 1 D dsBˆps3D drBˆpr1(cid:12) j3 + ( rr++ss+≥tt,s+2kt≥r,t+r ≥s (B2) Xs Xr (cid:12) j2 ≤ (cid:12) = D1 drvj1vj3vj1′vj3′G(cid:12)(cid:12)rj2∗∗jj31′j∗1∗j3′Gjr2∗jj13′j∗3∗j1′ a6jndsyδmrsbto=l ca0notbheerfwouisned. iTn2h9e,30e.xpHlicoiwtefvoerrm,uwlae fdoor nthoet j1j2Xj3j1′j2′j3′ Xr need the detailed data of the 6j symbol in the following D1 dsvj1′vj2vj1vj2′Gjs3′∗jj1′2′∗∗jj2∗1∗Gjs3′∗∗j1j2j2j′1′ comLeptuutastisotnarotfwtihtehGfoSrDm.ula in (10), and reorder the 6j s X symbols, D1 Xt dtvj2′vj3′vj2vj3Gjt∗1∗jj32′j∗2∗j3′Gjt∗1jj32′∗j3∗j2′ (A1) GSD=D−1 ds vj1vj3vj1′vj3′Gsj2∗∗jj13′∗jj1∗3′Gjs2∗jj3′1∗jj3∗1′ ownhetrheeBˆbpo1ttiosmacbtiunbgbloenptlhaqeuteotpte,buabnbdleBˆpp3laqounettthee, rBˆeps2t sj1j2Xvj3j1′j1′vjj2′2jv3′j1vj(cid:16)2′Gjs3′∗jj1′2′∗∗jj2∗1∗Gjs3′∗∗jj21jj1′2′ (cid:17) plaAqlule6tjtesyomutbsoidlsectahnebtweoelbimubinbaletse.d by using the orthog- (cid:16)vj2′vj3′vj2vj3Gjs1∗∗jj23′∗jj2∗3′Gjs1∗jj23′∗j3∗j2′(cid:17) onality condition in eq. (3) three times, =D−1 (cid:16) ds vj1vj3vj1′vj3′Gsj2∗∗jj(cid:17)13′∗jj1∗3′Gsj2∗j3′jj1∗1′j∗3 r drGrj2∗∗jj31′j∗1∗j3′Gjr2∗jj13′j∗3∗j1′ = d1j2δj1′j2j3′∗δj1j2j3∗ sj1j2Xvjj31′j1′vjj2′2jv3′j1vj(cid:16)2′Gjs3′∗jj1′2′∗∗jj2∗1∗Gjs3′j1j∗2∗j2j′1′∗∗ (cid:17) X s dsGjs3′∗jj1′2′∗∗jj2∗1∗Gjs3′∗∗j1j2j2j′1′ = d1j3′ δj1′j2j3′∗δj1j2′j3′∗ (cid:16)(cid:16)vj2′vj3′vj2vj3Gjs1∗∗jj23′∗jj2∗3′Gjs1∗j2∗j3′j3j2′∗(cid:17)(cid:17) (B3) X Xt dtGjt∗1∗jj32′j∗2∗j3′Gjt∗1jj32′∗j3j2′ = d1j1δj1j2j3∗δj1j2′j3′∗ (A2) swehcLeoernteduteshqecuoasmlyiptmya.mreettrhye fcoornmduitliaonini(nB3(3))wwitahsthuasetdinin(At1h)e. We set j = j∗ for all j and drop all stars, since all irre- and the GSD is a summation in terms of d,δ : ducible representationsof SU (2) areself-dual. Then we { } k find that the summation (B3) has the same form as the GSDsphere = D13 dj1′dj2′dj3δj1j2j3∗δj1′j2j3′∗δj1j2′j3′∗ trace of D−1 sdsBˆps2Bˆps3Bˆps1 on the graph on a sphere j1j2jX3j1′j2′j3′ as in (A1), P (A3) 1 trtorus( d Bˆs) Summing over j1′, j2′, and j3 using (1) finally leads to D s p GSDsphere =1. Xs j1 j1 1 = * j3 (cid:12)D dsBˆps2Bˆps3Bˆps1(cid:12) j3 + Appendix B: GSD on a torus for SUk(2) j1Xj2j3 j2 (cid:12)(cid:12) Xs (cid:12)(cid:12) j2 1 (cid:12) (cid:12) =trsphere( d(cid:12) Bˆs Bˆs Bˆs ) (cid:12) (B4) D s p2 p3 p1 Letus considerthe exampleassociatedwith the quan- s X tum group SU (2) (with the level k an positive integer) k and calculate the GSD on a torus. where Bˆps is defined on the only plaquette p on the torus Therearek+1stringtypes,labeledasj =0,1,2,...,k. (see Fig. 2(b)), while Bˆs Bˆs Bˆs is defined on the same p1 p2 p3 They are the irreducible representations of SU (2). The graph on a sphere as in (A1) (see Fig. 2(a)). k quantumdimensionsd arerequiredtobepositiveforall TheGSDonatorusbecomesatraceonasphere. The j j,inorderthatthe Hamiltonianishermitian. Explicitly, latter is easer to deal with since the ground state on a 7 sphere is non-degenerate. The counting of ground states Applying this rule reduces the summation (B4) to on a torus turns into a problem dealing with excitations on the sphere. 1 Inthefollowingweevaluatethesummationintherep- tr( d Bˆs Bˆs Bˆs ) D s p2 p3 p1 resentation of elementary excitations. let us introduce a s X new set of operators nˆr by a transformation, 1 s s s { p} =tr(D ds ssr1ssr2ssr3nˆrp22nˆrp33nˆrp11) nˆr = s s Bˆs, Bˆs = srsnˆr (B5) Xs r1Xr2r3 r10 r20 r30 p Xs r0 rs p p Xr sr0 p = D1 dssssr1sssr2sssr3δr1r2r3 (B10) Here s is a symmetric matrix (referred to as the mod- r1Xr2r3 Xs r10 r20 r30 rs Then we substitute (B1), (B2) and (B6) in and obtain ular S-matrix for SU (2)), k 1 sin(r+1)(s+1)π s = k+2 (B6) rs √D sink+π2 GSDtorus = k sink+π2δr1+r2+r3,2k SUk(2) sin(r1+1)π sin(r2+1)π sin(r3+1)π and has the properties r1,rX2,r3=0 k+2 k+2 k+2 k r sin π s =s , s =d /√D = k+2 rs sr r0 r sin(r+1)π sin(s+1)π sin(r−s+1)π s s =δ Xr=0Xs=0 k+2 k+2 k+2 rs st rt =(k+1)2. (B11) s X s s s wr ws wt =δ (B7) rst w sw0 (Here we omit a rigorous proof of the last equality.) X WecanalsoverifyGSD=(k+1)2byadirectnumerical Eq. (B5) can be viewed as a finite discrete Fourier computation. Wetaketheapproachin30 toconstructthe (tBra7n)s,fowremsaeteiotnhabte{tnˆwrpe}enar{enˆmrp}utaunadlly{Boˆrpst}h.onBorymparloppreorjteiecs- onnumaepriacraalmdaettearo,ft6hjesKymaubffomlsa.nTvhaerciaobnlsetrAuc(tiinonthdeepseanmdes tors, and they form a resolution of the identity: conventionasin30),whichisspecializedtorootsofunity. We make the following choice: nˆrnˆs =δ nˆr, nˆr =id (B8) p p rs p p r X A=exp(πi/3) at k =1 In particular, nˆ0 = 1 d Bˆs is the operator Bˆ in A=exp(3πi/8) at k =2 (B12) the Hamiltonian.p TheDopesrastorp nˆr projects ontopthe ( A=exp(3πi/5) at k =3 p P states with a quasiparticle (labeled by r type) occupy- ing the plaquette p. Expressed as common eigenvectors of nˆr , the elementary excitations are classified by the By this choice, the quantum dimensions dj take the { p} values as in (B1), and the 6j symbols satisfy the self- configuration of these quasiparticles. consistentconditionsin(3). Usingsuchdataofquantum Particularly, on the graph on a sphere as in (B4), the dimensions d and 6j symbols, We compute the summa- Hilbertspacehasabasisof r ,r ,r ,whereonlythose j 1 2 3 {| i} tion (10) at r , r , and r that satisfy δ = 1 are allowed. Each 1 2 3 r1r2r3 basis vector r ,r ,r is an elementary excitation with 1 2 3 | i thequasiparticleslabeledbyr1,r2,andr3 occupyingthe GSD=4 at k =1 plaquettes p1, p2,andp3. The configurationofquasipar- GSD=9 at k =2 (B13) ticles are globally constrained by δr1r2r3 = 134. There- ( GSD=16 at k =3 fore, tracing opertors nˆr leads to { p} tr(nˆr2nˆr3nˆr1)=δ (B9) which verifies GSD=(k+1)2 in the particular cases. p2 p3 p1 r2r3r1 ∗ Electronic address: [email protected] 3 C. 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