A new equivalence between fused RSOS and loop models Lukasz Fidkowski1,2 1Microsoft Station Q, University of California, Santa Barbara 93106-6105 2 Department of Physics, Stanford University, Stanford, CA 94305 (Dated: February 6, 2008) 7 0 We consider the topological theories of [1] and [2] and study ground state amplitudes of string 0 net configurations which consist of large chunks G of (trivalent) regular lattice. We evaluate these 2 amplitudes in two different ways: first we use the Turaev-Viro prescription to write the amplitude n as a sum over labelings of the faces of G, and second we use the local rules that constrain the a amplitude(theF-matrix)toresolvesubgraphsincreativeways. InthecaseoftheDoubledFibonacci J theory this second way allows us to produce loop models. In particular, we show that the hard 3 hexagonmodelisequivalenttoan anisotropicloop model. Many otherinterestingequivalencescan 2 presumably be obtained. ] h I. INTRODUCTION II. TURAEV-VIRO EVALUATION c e m Fromnowonwewillbe workingwithplanarnets that In [1] and [2] two dimensional models realizing non- - abelian topological phases of matter are introduced. are located in a bounded region of the plane. Think- t ing of these as graphs on the sphere, we call the region a These models are formulated in terms of string nets, t of G (i.e. face of G, or vertex of the dual graph Gˆ) s which are trivalent graphs labeled by the particle types t. 0,...,N 1 of the theory. States in Hilbert space are that contains the point at infinity the region at infinity. a − One way to evaluate G makes use of the fact that it is specified by assigning an amplitude to each string net h i m justtheexpectationvalueof“Wilsonnets”intheChern- configuration. The ground state amplitudes of these - models turn out to satisfy certain local rules - specifi- Simonstheorydescribingtheinfraredlimitofourmodel. d The work of Turaev and Viro, also called the “shadow cally d-isotopy and the F-matrix constraints (see fig. 1). n method”4, shows that: o c [ a) 1 hGi=Xw(Lf) (1) v n Lf 3 d 7 n Let us explain the meaning of this formula. It will 5 be necessary to first introduce some more notation. The 1 sum in (1) is over the labelings L of the faces of G by 0 f 7 b) the particle types of 0,...,N 1 of the theory, with the − 0 caveatthatwefixthelabeloftheregionatinfinitytobe0 / (thetrivialparticletype). Theweightw(L )isdefinedas t f a m F follows. LetLf(F)betheparticletypethatlabelsfaceF. m n mn Recallthatwealsohavealabelingoftheedges,whichwe - n callLe(E). Now,foreachedgeE wecandefineΘ(E)and d for each vertex V we can define the tetrahedral symbol n o FIG. 1: Local relations for ground state amplitudes. a) d- Tet(V). These are given as the quantum evaluations of c isotopy invariance. Here dn is an invariant associated with the appropriate theta and tetrahedral graphs involving v: each particle type n and called its quantum dimension b) theedgesandfacesadjacenttotheappropriatevertexor i F-matrix relations. In both a) and b), by each graph we edge (see fig. 2). X implicitly mean the ground state amplitude of that graph Notice that the tetrahedraland theta symbols are not r always well defined - there is a condition on which par- a ticle types are allowed to come together at a trivalent We will call the ground state amplitude assigned to vertex. If all tetrahedral and theta symbols are well de- a net G its quantum evaluation and denote it G . One fined, the labeling is called admissible; quick inspection canshowthatonaplane(orequivalentlyasphehre)i G is of fig. 2 shows that it is sufficient to check that all the uniquely determined once we stipulate empty graphhi= theta graphs are well defined. We note that this is pre- 1. This is done by repeatedly using Fh -matrix andid- ciselythesamesituationasinafusedRSOSmodelwhose isotopy moves to reduce a graph down to the empty net degrees of freedom live on the faces of G3. (thisisnottrueonasurfaceofnontrivialtopology,where The weightofaninadmissible labeling isdefined to be one has multiple ground states). 0; that of an admissible one is: 2 III. LOOP MODELS E F (E) = 1 F L f(F1 ) L (E) To illustrate the second way of computing the quan- 2 tum evaluation we specialize to the case of the doubled L f(F2 ) Fibonaccitheory,whichhastwoparticletypes: 0(trivial) and1(nontrivial). Wecansticktojustdrawingnontriv- ialedges,andhencereducetoconsideringonlyunlabeled string nets - the restrictionon the particle types allowed EF32 VEE12F3 Tet(V) = L f(F2 L)e (E 3) Le (E1 ) L f(F3 ) tnisootfuounsuievsaealtethnatevvFeer-rtmteixcaetjsru.isxTtabhneedceosdmseienssoctetohopefytchoreenlsastetrciaooinnnsdttoaofprpherasovoailncvhge F L (E ) 1 e 2 all the trivalent vertices in a regular lattice. We start L (F ) with a string net that looks like a large chunk of the f 1 configuration in fig. 3. FIG. 2: Definition of thethetaand tetrahedral symbols w(Lf)= Y dLf(F) Y Θ(E)−1 Y Tet(V) facesF edgesE verticesV (2) We have thus expressed the quantum evaluation as the partition function of an RSOS-like model with local Boltzmannweights. Beforeweworkoutthe implications andspecificexamplesofthisrepresentation,wemotivate further the origin of (1). Take the 3 dimensional ball B with boundary S2 and an associated topological quantum field theory. We can use the Turaev-Viro formalism to evaluate the expecta- tion value of any (labeled) string net G on S2 as follows FIG. 3: A distinctive trivalent lattice (for simplicity, we describe the procedure for unoriented theories). First, we pick a cellulation of B compatible Wenowresolvethetrivalentverticesusingtheidentity with G - that is we subdivide B into cells. Compatibil- in fig. 4 b) (note: τ =(1+√5)/2). ity with G is just the condition that the intersection of the boundary S2 with the 2-skeletonof the cellulation is a) G. ForeachlabelingLofthe 2-skeletoncompatiblewith the labeling of the edges of G (this means that we label the 2-cells which contain an edge of G with the label on = 1/2 -1/2 that edge) we define a weight w(L) in a fashion analo- gousto(2): each2-cellcontributesafactorofaquantum dimension, each 1-cellan inversetheta symbol, and each 0-cell a tetrahedral symbol. The Turaev-Viro prescrip- b) tion gives the formula for the expectation value of G as a sum over admissible labelings, with the caveatthat we hold one pre-chosen region fixed with the label 0 (this -1 = turns out to be just a normalization condition). Equation (1) follows as a special case of this formula. We obtain it by taking a specific cellulation of B. The FIG.4: Local identitiesthatareusefulforresolvingtrivalent cellulation we choose is as follows: let M be a smaller vertices sphere, concentric with ∂B = S2 and with radius r < 1. The string net G lives on ∂B. By “dropping” G Wethushavetwopossibilitiesforresolvingeachcircle, down to M we sweep out a 2-dimensional surface (with and get an expression for the quantum evaluation as a singularities); taking its union with M we obtain the 2- sum over fully packed loops, with a factor of τ for each skeleton of our cellulation. We then see that the 2-cells loop. One such sample configuration is shown in fig. 5. inthiscellulationcorrespondtoeitheredgesofGorfaces Thefactorofτ−1 outfrontcontributesonlytoanoverall of G, and (1) follows from (2). vertex fugacity. 3 onal lattice (fig. 6). We use the identity in fig. 4 a) to resolve the circled portions of the lattice, and obtain the amplitude as a sum over loop configurations such as that in fig. 7. This time the weighting of the loops isn’t quite topological because of the coefficients in fig. 4 a). However, all the weights are positive, because each mi- nus sign comes in an even number of times. We obtain an anisotropic model of loops. FIG. 5: The quantum evaluation is expressed as a sum over resolutions such as thisone Onthe other hand, the Turaev-Viroprescriptiongives the quantum evaluation as a sum over labelings of the faces of the lattice with 0’s and 1’s. In order for a la- FIG. 7: Sample loop configuration from resolving thehexag- beling to be admissible we just can’t have two adjacent onal lattice regions labeled with a 0. It is useful to think of the re- gionslabeledwitha 0asbeingoccupiedbyaparticle,so The Turaev-Viro prescription, on the other hand, thattheconstraintisthatwe’renotallowedtohavepar- shows that the corresponding RSOS model is just the ticles on adjacent regions. The local Boltzmann weight critical hard hexagon model. We believe this is general turns out to simply be determined by a particle fugacity and all models obtained in this way will be critical. So (the fugacities of the octagons and circular regions are wehaveanequivalencebetweenthecriticalhardhexagon different), with an overall normalization constant whose model and ananisotropic model ofloops. Note also that logarithm scales with the number of regions. This is all wecanusetheidentityinfig. 4a)toresolveallthetriva- analyzedindetailforadifferentlatticeinAppendix Aof lent vertices of any graph that admits a dimer covering. [2]. Wehavethusshownthattwodifferentmodels-afused RSOS model and a fully packed loop model - have the IV. CONCLUSIONS same partition function. Presumably we can extend this toacorrespondencebetweentheoperatorsofthe models We found an equivalence between some fused RSOS and show that the two are actually equivalent. Equiva- modelsandloopmodels. Webelievethemodelsinvolved lences of this type have been recently studied3, and our will always be critical5. This equivalence is probably resultseemstobe alongsimilarlinesasthatof[3]. How- closelyrelatedtothatof[3]. Inprinciplewecanbemore ever,wehaven’tcarefullystudiedthepreciseconnection. general: in a generaltopologicaltheory with a given lat- tice string net, we can be find creative ways of applying localrelationsandobtainnontrivialmodels. Itwouldbe interesting to explore this equivalence further, and also extend it past simple equality of partition functions. In particular, one would like to study the conformal field theories describing the critical points. V. ACKNOWLEDGEMENTS IwouldliketothankPaulFendley andMichaelFreed- man for useful discussions and especially Kevin Walker for originally suggesting the shadow method as a way FIG.6: Thehexagonallattice. Thecircledportionsaretobe of computing quantum evaluations. This research was resolved supported partly by the NSF under grant no. PHY Another example comes from considering the hexag- 0244728. − 4 1 M.LevinandX.G.Wen,Phys.Rev.B.71,045110 (2005). Recoupling Theory and Invariants of 3-Manifolds, (Prince- 2 L. Fidkowski, M. Freedman, C. Nayak, K. Walker, and Z. ton University Press, Princeton, NJ 1994). Wang, cond-mat/0610583. 5 E. Witten, Gauge Theories and Integrable Lattice Models, 3 Paul Fendley,cond-mat/0609435. Nucl.Phys. B 322, 629 (1989). 4 Louis H. Kauffman, and Sostenes L. Lins, Temperley-Lieb