Anyon condensation. Topological symmetry breaking phase transitions and commutative algebra objects in braided tensor categories

Transcription

1 Anyon condenstion Topologicl symmetry breking phse trnsitions nd commuttive lgebr objects in brided tensor ctegories Author: Sebs Eliëns Supervisors: Prof. Dr. F.A. Bis Instituut voor Theoretische Fysic Prof. Dr. E.M. Opdm Korteweg de Vries Instituut Mster s Thesis Theoreticl Physics nd Mthemticl Physics Universiteit vn Amsterdm 31 December 2010

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3 Abstrct This work hs two spects. We discuss topologicl order for plnr systems nd explore grphicl formlism to tret topologicl symmetry breking phse trnsition. In the discussion of topologicl order, we focus on the role of quntum groups nd modulr tensor ctegories. Especilly the grphicl formlism bsed on ctegory theory is treted extensively. We hve incorported topologicl symmetry breking phse trnsitions, induced by bosonic condenste, in this formlism. This llows us to clculte generl opertors for the broken phse using the dt for the originl phse. As n ppliction, we show how to tret topologicl symmetry breking on the level of the topologicl S-mtrix nd illustrte this in two representtive exmples from the su(2) k series. This pproch cn be viewed s n ppliction of the theory of commuttive lgebr objects in modulr tensor ctegories.

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5 Sometimes screm is better thn thesis Mnfred Eigen

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7 Contents 1 Introduction 1 2 Topologicl order in the plne Anyons The frctionl quntum Hll effect Conforml nd topologicl quntum field theory Other pproches to topologicl order Quntum groups in plnr physics Discrete Guge Theory Multi-prticle sttes Topologicl interctions Anti-prticles Generl remrks Chern-Simons theory nd U q [su(2)] Anyons nd tensor ctegories Fusing nd splitting Fusion multiplicities Digrms nd F -symbols Guge freedom Tensor product nd quntum trce Topologicl Hilbert spce Briding Pentgon nd Hexgon reltions Modulrity Verlinde formul Sttes nd mplitudes iii

8 CONTENTS 5 Exmples of nyon models Fiboncci Quntum double Fusion rules Computing the F -symbols Briding The su(2) k theories Bose condenstion in topologiclly ordered phses Topologicl symmetry breking Prticle spectrum nd fusion rules of T Confinement Commuttive lgebr objects s Bose condenstes The condenste The prticle spectrum of T Derived vertices Confinement Breking su(2) Indictors for topologicl order Topologicl entnglement entropy Topologicl S-mtrix s n order prmeter Conclusion nd outlook 97 A Qusi-tringulr Hopf lgebrs 99 B Tensor ctegories: from mth to physics 103 B.1 Objects, morphisms nd functors B.2 Direct sum B.3 Tensor ctegories B.3.1 Unit object B.3.2 Strictness nd coherence B.3.3 Fusion rules B.3.4 Digrms B.4 Duls B.5 Briding B.5.1 Ribbon structure B.6 Ctegories for nyon models C Dt for su(2) References 117 Dnkwoord 123 iv

9 CHAPTER 1 Introduction The theory of phses nd phse trnsitions plys centrl role in our understnding of mny physicl systems. Topologicl phses, or topologiclly ordered phses, pose n interesting problem for theorists in this respect, s they fll outside of the conventionl scheme to understnd phses tht occur in terms of the breking of symmetries. The formlism of topologicl symmetry breking, bsed on the breking of n underlying quntum group symmetry, cn be viewed s n extension of this theory to the context of topologicl phses. We will discuss topologicl symmetry breking nd connect it to the notion of commuttive lgebrs in modulr tensor ctegories. In prticulr, we put topologicl symmetry breking in grphicl form, pplying notions from tensor ctegories, which gives the freedom to clculte generl opertors for the phses under considertion. In condensed mtter physics, fundmentl problem is the determintion of the low-temperture phses or orders of system. Dting bck to Lev Lndu [52, 53], the theory of symmetry breking phse trnsitions forms corner stone in this respect. Put simply, there re two spects to the picture it provides: symmetry breking nd prticle condenstion. This mechnism underlies interesting phenomen, such s superconductivity where the electric U(1) symmetry is broken by condenste of Cooper pirs but lso the formtion of ice (figure 1.1). Group theory provides the right lnguge to discuss mny spects of symmetry breking. The clssifiction of the different phses is essentilly equivlent to the clssifiction of subgroups of the relevnt symmetry group. In the pst few decdes, growing interest hs emerged to study topologicl phses tht re not the product of (group) symmetry breking. Especilly in two dimensions, topologicl phses offer ccess to fundmentlly new physics. They cn hve nyonic qusi-prticle excittions prticles tht re neither bosons nor fermions. These offer route to the fult-tolernt storge nd mnipultion of quntum informtion known s topologicl quntum computtion (TQC), which my some dy be used to relize 1

10 1. INTRODUCTION Figure 1.1: The structure of n ice crystl - If wter freezes, or ny other liquidto-solid trnsition, the molecules order in regulr lttice structure. This breks the trnsltionl nd rottionl symmetry of the fluid stte. Using group theory, one cn see tht there re 230 qulittively different types of crystls corresponding to the 230 spce groups the drem of building quntum computer. The best known physicl reliztion of topologicl phses occurs in the frctionl quntum Hll fluids. These exotic sttes in two-dimensionl electron gses submitted to strong perpendiculr mgnetic field hve quntum numbers tht re conserved due to topologicl properties, not becuse of symmetry. Similr phses might occur in rotting Bose gses, high T c superconductors, nd possibly mny more systems. One cn in fct show tht there is n infinite number of different topologicl phses possible, which suggests world of possibilities if we ever gin enough control to engineer systems tht relize phse of choice. From mthemticl point of view, interesting structures hve entered the theory. The description nd study of topologicl phses involve conforml field theory, topologicl quntum field theory, quntum groups nd tensor ctegories, nd ll these structures re hevily interlinked. They re studied by mthemticins nd theoreticl physicists like nd link topics like string theory, low-dimensionl topology nd knot theory to condensed mtter physics. In this thesis we discuss topologicl ordered plnr systems, nd in prticulr the description using quntum groups nd how these led to tensor ctegories. In fct, we prefer the tensor ctegory viewpoint from which the theory cn be netly summrized by set of F -symbols nd R-symbols. We show how to clculte these in exmples relted to discrete guge theories nd Chern-Simons theory. Our min gol is to discuss how the breking of quntum group symmetry cn be understood from the perspective of tensor ctegories. In the topologicl symmetry breking scheme, key role is plyed by the formtion of bosonic condenste. In this thesis, we rgue tht this notion corresponds to commuttive lgebr object in brided tensor ctegories. Using the formultion of topologicl symmetry breking in terms of tensor ctegories, one cn write down nd clculte digrmmtic expressions for opertors in the broken phse. This llows us to describe phse trnsition on the level of the topologicl S-mtrix, n importnt invrint for the theory. We illustrte this in two representtive exmples. This shows the usefulness of the topologicl S-mtrix, relted to the 2

11 exchnge sttistics of the prticles, s n indictor for topologicl orders. Another importnt indictor for topologicl order is the topologicl entnglement entropy. The quntum dimension of the condenste, or quntum embedding index, ppers s universl quntity chrcterizing the phse trnsitions nd relting the topologicl entnglement entropy of the different phses. This thesis is orgnized s follows. Chpter 2 In this chpter, we discuss some spects of topologicl order in greter detil, nmely nyons nd the frctionl quntum Hll effect. The lrger picture of how conforml field theory, topologicl field theory nd topologicl order re relted is discussed briefly. We lso point out some other developments in the field. Chpter 3 The ppernce of quntum groups s the underlying symmetry for topologicl phses is discussed. Discrete guge theories re tken s representtive exmple. The underlying symmetry is the so-clled quntum double D[H]. It cn be understood physiclly s n lgebr of guge trnsformtions nd flux projections, which gives n intuitive picture of how intercting electric nd mgnetic degrees of freedom cn led to mterilized quntum group symmetry. On more forml level, we discuss the quntum group U q [su(2)] which is importnt for the quntiztion of Chern-Simons theory nd su(2) ( k) Wess-Zumino-Witten models. Chpter 4 Here, we introduce in detil the grphicl formlism for topologicl phses. One needs finite set of chrges, description of fusion rules, so clled F -symbols nd R-symbols to describe the topologicl properties of nyons completely. An importnt quntity is the topologicl S-mtrix. Together with the T -mtrix it forms representtion of the modulr group. As n ppliction nd illustrtion of the grphicl formlism we give proof of the Verlinde formul. Finlly, quntum sttes, density mtrices nd the clcultion of physicl mplitudes is discussed. Chpter 5 The clcultion of topologicl dt for the quntum double D[H] is illustrted nd we present generl formuls for the su(2) k models. Chpter 6 In this chpter, we strt the discussion of topologicl symmetry breking. Topologicl symmetry breking is wy to construct from theory A describing the unbroken phse, theory T nd theory U. The T-theory describes the broken phse, but my 3

12 1. INTRODUCTION include excittions tht brid non-trivilly with the condenste. These excittions cnnot occur in the bulk of the broken phse s the disruption of the condenste costs energy. They get confined to the boundry or s T-hdrons. Projecting out the confined excittions one gets the unconfined theory U-tht describes the bulk of the broken phse. We put this scheme in the context of the grphicl formlism of chpter 4. We discuss how to find well-defined condenste nd how to construct the prticle spectrum upon this. The conditions for confinement re reconsidered nd it is shown tht there is n opertor tht projects out confined excittions. Chpter 7 The quntum dimension of the condenste or quntum embedding index q ppers s universl rtio between the quntum dimensions of the broken phse nd the unbroken phse. This cn physiclly be understood s reltion in terms of the topologicl entnglement entropy. We show how this universlity of q follows from the ssumption of topologicl symmetry breking. Finlly, we give the min ppliction of the grphicl formlism for topologicl symmetry breking. We show tht the symmetry breking phse trnsition cn be described directly on the level of the topologicl S-mtrix. The digrmmtic clcultion gives insight in the nture of topologicl symmetry breking phse trnsitions. The condenste ppers explicitly where in the originl formultion it could lwys be left out. Chpter 8 The lst chpter contins concluding remrks nd nd indictes possible pths for future reserch. 4

13 CHAPTER 2 Topologicl order in the plne In this chpter, we give brief discussion of some essentil topics concerning topologicl phses nd topologicl order in 2+1 dimensions. These re nyons, the FQH effect, some generlities of the mthemticl structures tht re involved. This chpter serves s more elborte introduction nd to sketch bit of history of the topic. Relevnt references re included. 2.1 Anyons In the most mundne spcetime, tht of 3+1 dimensions, prticles fll in precisely two clsses: bosons nd fermions. These cn either be defined by their exchnge properties or by their spins. The importnce of this difference between prticles cn of course not be overstted. It is t the root of the Puli exclusion principle nd Bose-Einstein condenstion, nd determines whether mcroscopic numbers of prticles obey Fermi- Dirc or Bose-Einstein sttistics. According to textbook quntum mechnics, the wve function of multi-prticle stte of fermions is obtined by nti-symmetrising the tensor product of single-prticle wve functions, while for bosons the wve function is symmetrised. This procedure long stood s corner stone of quntum mechnics nd is still procedure in mny pplictions. But one my object tht in quntum theory with indistinguishble prticles, the lbelling of prticle coordintes hs no physicl significnce nd brings unobservble elements into the theory. The consequences of bove observtion were first fully pprecited by Leins nd Myrheim. In their seminl pper [57], they show tht the exchnge properties of prticles re tightly connected to the topology of the configurtion spce of multiprticle systems. In fct, s ws shown in [51], prticle types correspond to unitry representtions of the fundmentl group of this spce. 5

14 2. TOPOLOGICAL ORDER IN THE PLANE Let us discuss this in spcetime picture. The trjectories of prticles cn now conveniently be imgined by their world lines through spcetime. We will consider trjectories tht leve the system in configurtion identicl to the initil configurtion nd tht do not hve intersecting world lines. These correspond to free propgtion of the system without scttering events. If the prticles re distinguishble, the world lines should strt nd end t the sme sptil coordintes. If the prticles re indistinguishble, the world lines re llowed to interchnge sptil coordintes. Figure 2.1: Spcetime trjectories - Two spcetime trjectories for system of three prticles with identicl initil nd finl sttes. If the prticles re of different type, world lines should strt nd end t the sme coordintes in spce (left), while for indistinguishble prticles, permuttions of sptil coordintes re llowed (right). The trjectories fll in distinct topologicl clsses, depending on whether or not they cn be deformed into ech other by smooth deformtions of the world lines (without intermedite intersections). If spce hs three or more dimensions one cn see tht there is no topologiclly different notion of under nd over crossing of world lines, but ny exchnge of coordintes is simply tht: n exchnge. Therefore the exchnge of prticles is governed by the permuttion group S N. However, when the prticles live in plne so spcetime is three-dimensionl such distinction of crossings becomes essentil to incorporte the difference between interchnging the prticles clockwise or counter-clockwise. The relevnt group is not S N but B N, Artins N-strnded brid group [3]. This group is generted by the two different interchnges or hlf-bridings R i =, Ri 1 = i i+1 i i+1 (2.1) which re inverse to ech other. Here, 1 i N 1 lbels the strnd of the brid. These genertors re subject to defining reltions corresponding to topologicl mnipultions of the brid. Pictorilly they re given by the following equtions. 6

15 2.1 Anyons We cn lso write them lgebriclly s R i R j = R j R i for i j 2 R i R i+1 R i = R i+1 R i R i+1 for 1 i N 1 (2.2) The second reltion is the fmous Yng-Bxter eqution [15, 86, 87]. As remrked before, the importnt difference between prticles in the plne nd prticles in three-dimensionl spce is the difference between over nd under crossings of world lines. The reltion to sttistics in higher dimensions represented on the level of groups due the fct tht we cn pss from the brid group B N to the permuttion group S N by implying the reltion R i = Ri 1, or equivlently (R i ) 2 = 1. This one extr reltion mkes huge difference for the properties of the group. While the permuttion group is finite ( S N = N!), the brid group is infinite, even for two strnds. The representtion theory of the brid group is therefore much richer thn tht of the permuttion group. This gives the possibility of highly non-trivil exchnge sttistics in 2+1 dimensions, lso referred to s brid sttistics. Prticles tht obey these exotic brid sttistic were dubbed nyons by Wilczek [80]. Bosons nd fermions, in this context, just correspond to two of n infinite number of possibilities. Since distinguishble nyons cn lso hve nontrivil briding, it is better to think of briding sttistics s kind of topologicl interction. Bosons nd fermions correspond to the two one-dimensionl representtions of S N, even nd odd respectively. In principle, higher dimensionl representtions of S N, known s prsttistics [39], could led to more prticle types, but it hs been shown tht these cn be reduced to the one-dimensionl representtions t the cost of introducing dditionl quntum numbers [31]. One-dimensionl representtions of the brid group simply correspond to choice of phse exp(iθ) ssigned to n elementry interchnge. Since, in this cse the order of the exchnge is unimportnt becuse phses commute, nyons corresponding to one-dimensionl representions re clled Abelin. However, it is no longer true tht higher dimensionl representtions of the brid group cn be reduced. This gives rise to so clled non-abelin nyons, prticles tht implement higher dimensionl brid 7

16 2. TOPOLOGICAL ORDER IN THE PLANE sttistics. These non-abelin nyons re the ones tht cn be used for topologicl quntum computtion s suggested by Kitev in [49]. See [64, 66] for review. 2.2 The frctionl quntum Hll effect Becuse the most prominent system with nyonic excittions is the frctionl Quntum Hll effect, we include brief discussion of some relevnt spects. For greter detil, we refer to the literture. See e.g. [65, 88] for thorough introduction. In 1879, Edwin Herbert Hll discovered phenomenon in electricl conductors tht we know tody s the Hll effect. Under influence of n pplied electric field, the electrons will strt moving in the direction of the field, which leds to current. But the ppliction of perpendiculr mgnetic field will, due to the Lorentz force, led to component to the current which is orthogonl to both the electric nd mgnetic field. This is the Hll effect. As n ideliztion, we my consider two-dimensionl electron gs (2DEG) with coordintes (x, y) = (x 1, x 2 ) tht we submit to n electric field in the x-direction nd perpendiculr mgnetic field B in the z direction. The reltion between the current nd the electric field is given by the resistivity tensor ρ nd the conductivity σ, through the reltions E µ = ρ µν J ν, J µ = σ µν E ν (2.3) Assuming homogeneity of the system, one cn use reltivistic rgument to deduce tht [37] ρ = B ( ) 0 1, σ = nec ( ) 0 1 (2.4) nec 1 0 B 1 0 with n the electron density nd B the strength of the mgnetic field. Remrkbly, since ρ xx = 0 nd σ xx = 0, the system behves s perfect insultor nd perfect conductor in the direction of the electric field simultneously. The Hll resistivity is defined s ρ H = ρ xy. According to the equtions bove, we must hve ρ H = B (2.5) nec The derivtion of this eqution lens criticlly on Lorentz invrince, but not on much more. Therefore, if there is no preferred reference frme, this result is very robust. It should in prticulr hold whether we consider quntum or clssicl electrodynmics. It is therefore striking tht it does not gree with experimentl dt. Experiments revel tht t low tempertures ( 10 mk) nd high mgnetic field ( 10 T ) the dependence of ρ H on B is not liner, but insted plteus develop t precisely quntized vlues ρ H = 1 ν h e 2 (2.6) The number ν is clled the filling frction nd is usully written s ν = N e /N Φ. Here N e is the number of electrons, while N Φ is the number of flux-qunt piercing the 2DEG. To get better understnding of the filling frction, let us elborte on its mening. 8

17 2.2 The frctionl quntum Hll effect f mgnetic field norml to the plne s the in-plne motion into Lndu levels (i1/2) c, where c eb/m* repreon frequency, B the mgnetic field, nd mss of electrons hving chrge e. The ilble sttes in ech Lndu level, d rly proportionl to B. The electron spin the Lndu level into two, ech holding unit re. Thus the energy spectrum of system in mgnetic field is series of ch hving degenercy of eb/h (Ando From nlysing the single-prticle Hmiltonin, one sees tht the energy spectrum forms so clled Lndu levels E n = (n )ω c, ω c = eb mc (2.7) In finite sized system of re A, the vilble number of sttes for ech Lndu level turns out to be [37] N Φ = A B Φ 0 (2.8) Here Φ 0 = hc e is the fundmentl flux quntum, so pprently the number of vilble sttes for ech energy level is the sme s the number of flux-qunt tht pierce the system. Hence, ν = N e /N Φ is the rtio between the number of filled energy sttes the electrons nd the number of vilble energy sttes, which is why it is clled the filling frction. Plteus t integer vlues for the filling frction were first reported in 1980 by von Klitzing t l. [50], nd re referred to s the integer quntum Hll effect (IQHE). Plteus t frctionl filling frctions found two yers lter by Tsui et l. [74] re the hllmrk for the frctionl quntum Hll effect (FQHE). While the IQHE cn essentilly be understood from the single prticle physics, neglecting the Coulomb interction, the FQHE represents n intriguing interply of mny intercting electrons. The Coulomb interction is crucil to explin its fetures. S299 H. L. Stormer, D. C. Tsui, nd A. C. Gossrd: The frctionl quntum Hll effect rture (TLndu/spin splitting) nd in lectron popultion of the 2D system is the Lndu-level filling fctor n/d it turns out, is prmeter of centrl 2D electron physics in high mgnetic 0 is the mgnetic-flux quntum, Figure2.2: def electron density to mgnetic-flux the longitudinl den- V resistnce y /I x ndr xx the= mgnetoresistnce V x /I x re plottedr ginst xx V x /I x The FIG. quntum 1. Composite Hll view effect showing - The Hll the Hll resistnce resistnce R xy R= R H = V y /I x nd xy vrying of twodimensionl the resistnce electron is the system smeofs density the resistivity: n R cm = 2 σ.) t An illustrtion of mgnetic field. (In cinctly, the number of electrons two dimensions, per flux of the physics of 2D electrons the mesurement in B temperture set-up is lso of 85given mk, vs onmgnetic the top field. left. Tken Numbers from identify ref. [73] the t in terms of this filling fctor. filling fctor, which indictes the degree to which the sequence of Lndu levels is filled with electrons. Insted of ris- experiments performed on 2D electron Figure 2.2 shows typicl grph of the plteus in the first Lndu level, the ν = 1/3 tricl resistnce mesurements, lthough ing strictly linerly with mgnetic field, R xy exhibits plteus, being the plteu severl more sophisticted experimentl quntized tht ws tofirst h/(ediscovered. 2 ) concomitnt Note withtht minim theoflongitudinl vnishing resistivity σ xx n successfully employed. In electricl R xx. These re the hllmrks of the integrl (iinteger) two chrcteristic voltges re mesured quntum Hll effect (IQHE) nd frctionl (p/q) quntum B, which, when divided by the pplied Hll effect (FQHE). While the fetures of the IQHE re the e mgnetoresistnce R xx nd the Hll results of the quntiztion conditions for individul electrons in mgnetic field, the FQHE is of mny-prticle origin. The see insert Fig. 1). While the former, mecurrent pth, reduces to the regulr re- insert shows the mesurement9 geometry. Bmgnetic field, I field, the ltter, mesured cross the curishes t B0 nd, in n x current, V x longitudinl voltge, nd V y trnsverse or Hll voltge. From Eisenstein nd Stormer, ordinry

18 2. TOPOLOGICAL ORDER IN THE PLANE drops to zero t the plteus. which mens tht the conductivity tensor is off-digonl s in eqution (2.4). Hence, dissiptionless trnsverse current flows in response to n pplied electric field. In prticulr, when we thred n extr mgnetic flux quntum through the medium, the system will expel net chrge of νe due to the induced mgnetic field. Or, put differently, this will crete qusi-hole of chrge νe nd one flux quntum, illustrting the intimte coupling between chrge nd flux in the quntum Hll effect. It is first indiction tht these systems hrvest elementry qusi-prticle excittions tht re chrge-flux composites, which, due to the internl Ahronov-Bohm interction, cn be nyons. The peculir fct tht these qusi-prticles hve frctionl chrge when ν is non-integer hs been experimentlly verified in [38]. With the ppliction of topologicl quntum computing in mind, most interest lies in systems exhibiting non-abelin nyons. In this respect, series of plteus tht hs been observed in the second Lndu level [81] offers the most promising pltform. Especilly the ν = 5 2 plteu hs received much ttention. See figure 2.3. Figure 2.3: Plteus in the second Lndu level - The resistivity is plotted ginst vrying mgnetic field. The grph shows plteus in the second Lndu level. The filling frction ν = p/q with 2 < ν < 4 is indicted. Tken from ref. [81]. We will leve most of the theoreticl subtleties concerning the quntum Hll effects untouched. In the prgrph below, however, we do wnt to point out the reltion of conforml field theory (CFT) nd topologicl quntum field theory to FQH wve functions, s it reltes to our discussion of topologicl symmetry breking phse trnsitions. 10

19 2.2 The frctionl quntum Hll effect Conforml nd topologicl quntum field theory Over the yers, series of wve functions hs been proposed to cpture the physics of the frctionl quntum Hll effect t different plteus, the Lughlin wve function [55] being the first. It cn ccount for the plteus t filling frction ν = 1/M for odd M, nd reds Ψ L (z) = [ ] i z j ) i<j(z M 1 exp z i 2 (2.9) 4 Here, the z i lbel the complex coordintes for N e electrons. 1 The Lughlin stte describes n electronic system where the electrons crry mgnetic vortices. This ttchment of vortices to electrons is prdigmtic for the vrious quntum Hll sttes tht hve been proposed in the literture. Note tht to ensure fermionic system, i.e. the wve function is nti-symmetric under interchnge of the coordintes, we need M to be odd. Skipping subsequent developments in the history of the FQHE (e.g. the Hldne- Hlprin hierrchy [43, 44, 56] nd the composite fermion pproch by Jin [45]), we wnt to turn our ttention to n observtion first mde by Moore nd Red. In [63] they noted tht the Lughlin wve function could be obtined s correltor in certin rtionl conforml field theory (RCFT). In fct, they rgue tht this reltion should be generl nd conjecture tht every FQHE stte should be relted to conforml field theory, which would give mens to clssify FQHE sttes. Without going into the detils of conforml field theory (see e.g. [33, 36] for introductory texts nd [62, 63] for further informtion), let us reflect for moment on the picture they sketch. They motivte their serch for formultion in terms of RCFT by pointing to the development of Ginzberg-Lndu effective field theories for the FQHE effect where the ction contins Chern-Simons term. Witten showed [83] tht there is connection between Chern-Simon theory in three dimensionl bulk nd Wess-Zumino-Witten theory on the two-dimensionl edge. Moore nd Red rgue tht the long rnge effective field theory should essentilly constitute pure Chern- Simons theory nd tht relted RCFT should come in to ply when we consider two-dimensionl edge of the spcetime. Interestingly, the RCFT cn ccount for wve functions by mens of correltors in (2+0)d interprettion, but lso governs the edge excittion in (1+1)d interprettion. Chern-Simons theory is n exmple of topologicl quntum field theory (TQFT). This is pprent from the fct tht the spcetime metric does not enter the Chern- Simons ction S CS = k d 3 x Tr ɛ µνρ ( µ ν ρ + 2 4π 3 µ ν ρ ). (2.10) A topologicl quntum field theory is generlly quntum field theory for which ll correltion functions re invrint under rbitrry smooth deformtions of the bse mnifold [4, 84]. 1 c The mgnetic length l B = is set to unity, or, equivlently, the complex coordintes re tken eb z = (x + iy)/l B. i 11

20 2. TOPOLOGICAL ORDER IN THE PLANE The picture Moore nd Red proposed for the FQHE is generl picture we hve in mind when we discuss topologicl order. The long rnge physics is governed by n effective field theory tht is TQFT, hence we lso spek of topologicl phse. Often, the physics of time slice nd of gpless edge excittions is controlled by CFTs with the sme topologicl order, lthough this might not be ccurte in ll situtions. Figure 2.4: Hologrphy - A schemtic view of how TQFT nd CFT t lest in certin theories, re relted. A TQFT governs the bulk of the cylinder. Time slices re described by (2+0) dimensionl CFT. An exmple of this re the FQHE wve functions constructed from CFT correltors. Gpless boundry excittions re described by CFT with the sme topologicl order when there is no edge reconstruction. This cn be regrded s n instnce of the hologrphic principle. Moore nd Red proposed the following wve function tht could explin n even denomintor FQH plteu ( ) M [ ] 1 1 Ψ Pf (z) = Pfff exp z i 2 z i z j 4 i<j i (2.11) where Pfff A ij is the squre root of the determinnt of A. The filling frction for this stte is ν = 1/M. Its prticulr benefit lies in the fct tht, for M = 2, it describes fermionic system with ν = 1/2. Fermionic systems with even denomintors cnnot be described by the Lughlin wve function nd sttes derived from this. This Moore- Red stte is likely cndidte for the plteu t ν = 5/2 = , which hs two completely filled Lndu levels nd one level t hlf filling. As it fetures qusi-holes tht re non-abelin nyons nd the ν = 5/2 plteu is experimentlly ccessible, this is promising cndidte for the experimentl verifiction of non-abelin brid sttistics. This experimentl chllenge does not seem to hve been met decisively yet. The Moore-Red stte is constructed from the so clled Ising CFT. Subsequent generliztion of the ides of Moore nd Red ws done by Red nd Rezyi [69]. They constructed series of sttes bsed on CFT relted to su(2) k WZW models (tht include the Ising CFT), tht could explin plteus in the second Lndu level. 12

21 2.3 Other pproches to topologicl order Interestingly, one of these sttes encompsses so clled Fiboncci nyons which re universl for quntum computtion. 2.3 Other pproches to topologicl order The FQHE in 2DEGs is not the only plce to look for (non-abelin) nyons. Systems tht might hve sttes very similr to the FQH sttes discussed bove re p x + ip y superconductors [26,41] nd rotting Bose gses [24,25,68], though the ltter of course would disply bosonic version of the FQHE. We should lso note the existence of two lttice models tht relize topologicl order, t this point. The first is known s the Kitev model. In its most generl form, this model fetures intercting spins plced on the edges of n rbitrry lttice [49]. The spins re lbelled by the elements of group. The Hmiltonin consists of mutully commuting projectors, nd is therefore completely solvble. The projectors cn be recognized s generliztions of either electric or mgnetic interctions. The electric opertors ct on the edges joining t vertex nd ensure tht the ground stte trnsforms trivilly under the ction of the group, which cn be seen s sort of guge trnsformtion. The mgnetic opertors ct on plquettes nd essentilly project out sttes of trivil flux through the plquette. In the simplest cse, where the spins re lbelled by Z 2 nd squre lttice is tken, the ground stte cn be redily identified s loop gs. This lttice model relizes discrete guge theory. 1 For more informtion we refer to the literture. Especilly [17] nd [16] re interesting in the context of this thesis. They study the Kitev model with boundry, which cn be seen s n explicit symmetry breking mechnism nd reltes to the phse trnsitions tht we will discuss in subsequent chpters. These guge theories might lso be relizble in Josephson junction rrys [29]. The other prdigmtic lttice model for topologicl order ws introduced by Levin nd Wen under the nme of string-net condenstes [59]. This system lives on trivlent lttice, where, gin, spins re plced on edges nd interct t vertices. The input is tensor ctegory. We will discuss these structures in detil in chpter 4. The Hmiltonin hs similr structure s in the Kitev model nd is lso completely solvble. Remrkbly, it hs been shown tht the Kitev model cn be mpped to Levin- Wen type model, mking the Kitev model essentilly specil cse of the more generl string-net models. In the next chpter we will discuss the role of quntum groups in (2+1)d. In mny concrete situtions these cn be identified s the underlying symmetry structure. Regrdless of the nme, these re ctully not groups but should be regrded s generliztion. Formlly, they re certin type of lgebrs with dditionl structures (see chpter 3 nd ppendix A). 1 Discrete guge theories re discussed in chpter 3. 13

22 2. TOPOLOGICAL ORDER IN THE PLANE 14

23 CHAPTER 3 Quntum groups in plnr physics In the previous chpter we discussed spects of topologicl order, such s nyonic qusiprticle excittions nd the quntum Hll effect. In the present chpter we will discuss the symmetries underlying topologicl order. Specil to plnr physics, especilly when electric nd mgnetic degrees of freedom strt to interct, is the occurrence of so clled quntum group symmetry. Quntum groups generlize group symmetries in mny wys. This full symmetry is often not mnifest in the Hmiltonin or Lgrngin of the theory. Symmetries of the Hmiltonin re represented on the quntum level s opertors commuting with the Hmiltonin. Since commuting opertors cn be digonlized simultneously, the energy eigensttes cn be lbelled by quntum numbers coming from the symmetry opertors. But it might hppen tht there is lrger set of commuting opertors. This gives n ide of how symmetries not directly pprent from the Hmiltonin or Lgrngin cn mterilize. In this chpter we will first illustrte the occurrence of quntum group symmetry in the context of discrete guge theories. The electric nd mgnetic excittions cn be treted on equl footing, in these theories, when viewed s irreducible representtion of the quntum double of the residul guge group. Then we give more forml tretment of the constituents of quntum groups, or qusi-tringulr Hopf lgebrs s they re lso clled. Finlly, we will discuss the quntum group U q [su(2)] tht is relted to Chern-Simons theory nd the Wess-Zumino- Witten model for conforml field theory. A more forml tretment of the generlities of quntum groups is put forth the ppendix A. 15

24 3. QUANTUM GROUPS IN PLANAR PHYSICS 3.1 Discrete Guge Theory Suppose we strt out with (2+1)d Yng-Mills-Higgs theory with guge group G. We spek of discrete guge theory (DGT) when the continuous guge group G is spontneously broken down to discrete, usully finite, subgroup H. Hence, this cn be regrded s guge theory with residul guge group H. The Higgs mechnism cuses the guge field to cquire mss, mking ll locl interctions freeze out in the low-energy regime, effectively leving only topologicl interctions. In this limit, the theory becomes TQFT [11]. Below, we give quick overview of the spects tht re importnt for our present purpose. See [67] for n elborte exposition on which this pproch is bsed. A DGT hs electric prticles lbelled by the irreps {α} of the residul guge group, which, before we mod out by guge trnsformtions to get the physicl Hilbert spce, crry the corresponding representtion module V α s n internl Hilbert spce. Furthermore, DGT lso hs mgnetic prticles, or fluxes, ssocited to topologicl defects. 1 We think of the fluxes s defined by their effect on chrges in n Ahronov-Bohm type scttering experiment [1]. An electric chrge α tken round flux will feel the influence of the flux s trnsformtion in its internl Hilbert spce V α by the ction of some h H. We denote the flux s ket h lbelled by the trnsformtion h H it induces. However, since flux mesurement followed by guge trnsformtion should give the sme result s pplying guge trnsformtion first nd do the flux mesurement in the trnsformed system we find gh = h g, where h, h is the flux mesured before or fter the guge g trnsformtion respectively. Hence the guge trnsformtions hs to ct on the flux by conjugtion, so the guge invrint lbelling is in fct by the conjugcy clsses of H rther thn just elements. Hence, while electric excittions crry representtion modules s n internl Hilbert spce, the internl spce of fluxes is given by conjugcy clss. Reveling the lgebric structure of the theory puts the foregoing on firmer footing. Define the flux mesurement opertors {P h } h H tht stisfy the projector lgebr P h P h = δ hh P h (3.1) Since guge trnsformtions g H ct on fluxes by conjugtion, we must hve gp h = P ghg 1 g (3.2) The full lgebr of guge trnsformtions nd flux projections is spnned by the combintions {P h g}, h, g H (3.3) tht obey the multipliction rule P h g P h g = δ h,ghg 1P h gg (3.4) 1 Note tht we use electric nd mgnetic s generl terminology for the two distinct types of prticlelike excittions nd it does not imply tht the guge group is U(1). 16

25 3.1 Discrete Guge Theory This is in fct the quntum double D(H) of H which cn obtined from ny finite group by generl construction due to Drinfel d [30]. Note tht the multipliction rule sys tht this lgebr hs unit 1 = h P h e. The full prticle spectrum of DGT with residul guge group H cn be recovered s the set of irreducible representtions of D[H]. 1 Indeed, one finds bck the irreps of H corresponding to electric prticles, s well s the conjugcy clsses, belonging to fluxes. But there is third clss of irreps corresponding to flux-chrge composites or dyons. Physiclly, this is very interesting, becuse due to the Ahronov-Bohm effect, these composites cn hve frctionl spin nd cn be true nyons. The representtion theory of D(H) ws first worked out in [28], but we gin follow [67]. It turns out tht the irreducible representtions of D(H) cn be lbelled uniquely by conjugcy clss A of H, nd n irreducible representtion of the centrlizer of some representing element of A. Thus, when the conjugcy clss {e} of the identity element is tken, or tht of ny other centrl element, we find the irreps of H. But pprently, for generl non-trivil fluxes, not ll trnsformtions cn be implemented strightforwrdly on the chrge prt. To mke the ction of D(H) explicit, we will hve to mke few choices. Pick some order for the elements of A nd write A = { A h 1, A h 2,..., A h k } (3.5) Let C(A) be the centrlizer of A h 1 nd fix set X(A) = { A x 1, A x 2,..., A x k } of representtives for the equivlence clsses H/C(A), such tht A h i = A x A i h A 1 x 1 i. Let us tke A x 1 = e, the unit element of H, for convenience. These choices will effect some of the specifics in wht follows but different choice would led to unitry equivlent representtion of D(H). The internl Hilbert spce Vα A of prticle with flux A nd centrlizer chrge α is now spnned by the quntum sttes A h i, α i = 1,..., k, v j, j = 1,..., dim α (3.6) where we hve chosen bsis { α v j } for the centrlizer representtion α. The ction of n element P h g of D(H) on these bsis sttes, i.e. the effect of guge trnsformtion g followed by flux mesurement P h, is given by Π A α (P h g) A h i, α v j = δh,g A h i g 1 g A h i g 1, Π α ( g) mj α v m (3.7) (where summtion over repeted indices is implied), with g A x 1 g A x i (3.8) nd A x k the element of X(A) ssocited to h k = g A h i g 1. Note tht the element g commutes with A h 1 s it should. Thus when cting on dyons, g is the prt of the 1 We will ssume representtions to be unitry throughout the text 17

26 3. QUANTUM GROUPS IN PLANAR PHYSICS trnsformtion tht slips through the conjugtion of the flux, nd which is subsequently implemented on the chrge. This gives n explicit description of the irreps of D(H) including how the ction of elements P h g cn be worked out. The single-prticle sttes of the theory correspond to these irreps. Before we discuss multi-prticle sttes, let us point out subtlety concerning unitrity of the theory. In order to preserve the inner product of quntum sttes, we should hve Π A α (P h g) = Π A α (P g 1 hg g 1 ) (3.9) Hence it mkes sense to define (P h g) = P g 1 hg g 1, nd more generlly h,g c h,g P h g = h,g c h,g P g 1 hg g 1 (3.10) where on the coefficients c h,g just mens complex conjugtion. Formlly, this turns D[H] into Hopf- -lgebr. Representtions obeying (3.9) re sid to respect -structure. This gives the proper definition of unitry representtion Multi-prticle sttes To let D(H) ct on two-prticle stte, one needs prescription of how n element P h g cts on Vα A Vβ B. Corresponding to intuitive expecttions, the element P h g implements the guge trnsformtion g on both prticles nd then projects out the totl flux. Thus, we ct on stte A h i, α v j B h k, β v l with Π A α (P h g) Π B β (P h g) (3.11) h h =h Formlly, this is nicely cptured by the definition of coproduct : D(H) D(H) D(H) (3.12) by (P h g) = P h g P h g (3.13) h h =h The ction of P h g on two-prticle stte is then defined s the ppliction of Π A α Π B β on (P h g), which indeed gives (3.11). The ction on three-prticle sttes is produced by first pplying to produce n element of D[H] D[H] nd then pply gin on either the left or the right tensor leg. Strting out with the element P h g, either choice produces P h g P h g P h g (3.14) h h h =h s n element of D[H] 3. Then we pply Π A α Π B β ΠC γ on this element to ct on Vα A Vβ B V γ C. Note tht when we describe the trnsformtion in words, it is gin 18

27 3.1 Discrete Guge Theory the ppliction of the guge trnsformtion g on ll three prticles followed by totl flux mesurement projecting out flux h. Extending the ction of P h g on sttes with more thn three prticles is now strightforwrd. The resulting trnsformtion is gin the globl guge trnsformtion g followed by the flux mesurement. On forml level, this is chieved by consecutive ppliction of the coproduct to get n element in D[h] n nd letting ech tensor leg ct on the corresponding prticle. The tensor product representtions of D(H) tht we obtin using the coproduct re in generl not irreducible, but they cn lwys be decomposed into direct sum of irreducible representtions. Thus the spce Vα A Vβ B cn be written s direct sum of subspces tht trnsform irreducibly, leding to the decomposition rules Π A α Π B β = (C,γ) N ABγ αβc ΠC γ (3.15) These so-clled fusion rules ply n importnt role when we strt discussing the generl formlism for nyonic theories in de next chpter. Note tht the representtion Π {e} 0 hs trivil fusion (where 0 denotes the trivil representtion of H). This corresponds to the zero-prticle stte or vcuum of the theory. The mp Π {e} 0 : D[H] C cn lso be clled the counit, nd is then often denoted with ɛ. This is generl constituent of quntum group, nd should stisfy certin comptibility conditions with regrd to the comultipliction Topologicl interctions As remrked before, in the low-energy limit, the excittions of DGT interct solely by Ahronov-Bohm type, topologicl interctions. A very nice feture of DGT theories is tht the precise form of these interctions cn be deduced from intuitive resoning. Suppose we hve two fluxes with vlues h 1 nd h 2. As is illustrted in figure 3.1, the requirement tht the totl flux hs to be conserved leds to the conclusion tht counter-clockwise interchnge results in commuttion of h 2 by h 1. A chrge crossing the h 1 Dirc line tht is depicted in the figure picks up the ction of h 1 in its internl Hilbert spce. Hence we conclude tht the generl briding opertor cts s ( R A (A,α),(B,β) h i, α ) v j B h m, β v n = A h B i h A m h 1 i, Π β ( A hi ) β v A n h i, α v j (3.16) This trnsformtion cn be ccomplished by composing the ction of specil element in D[H] D[H], clled the universl R-mtrix, on Vα A Vβ B, with flipping the tensor legs. The universl R-mtrix for D[H] is R g,h P g e P h g (3.17) Acting on two-prticle stte, R indeed mesures the flux of the left prticle nd implements it on the right prticle. We cn now define R (A,α),(B,β) = τ(π A α Π B β )(R) (3.18) 19

28 3. QUANTUM GROUPS IN PLANAR PHYSICS Figure 3.1: The trnsformtion of fluxes - We strt with two fluxes positioned next two ech other in the plne, s depicted on the left. They crry flux h 1 nd h 2 respectively. Fluxes re mesured through Ahronov-Bohm interction with test chrges tken round pths C 1, C 2. The totl flux h 1 h 2 s is mesured using curve C 12. The grey lines show Dirc strings ttched to the vortices. On the right hnd side, the fluxes re interchnged counter-clockwise. Since the flux h 2 crosses the Dirc string ttched to h 1, its vlue cn chnge to h 2. Becuse the totl flux is conserved we must hve h 2h 1 = h 1 h 2 hence we find tht h 2 = h 1 h 2 h 1 1. So the hlf-briding of fluxes leds to conjugtion. where τ is the flip of tensor legs. It is esy to check tht this gives (3.16) when pplied on two-prticle sttes. The universl R-mtrix hs the following properties. R (P h g) = (P h g)r (3.19) (id )R = R 23 R 12 (3.20) ( id)r = R 12 R 23 (3.21) (3.22) Here R 12 = R 1 nd R 23 = 1 R. These consistency conditions ensure tht the briding is implemented consistently. The first reltion tells us tht the briding commutes with the ction of the quntum double on two-prticle sttes, hence it cts s multipliction by complex number on the irreducible subspces (Schur s lemm). The other two conditions re known s the qusi-tringulrity conditions. They imply the Yng- Bxter eqution for the R-mtrix, R 12 R 23 R 12 = R 23 R 12 R 23 (3.23) On the level of representtions, this leds to the Yng-Bxter eqution for the briding, thus implementing representtion of the brid group B n on the n-prticle Hilbert spce of n identicl prticles. To discuss the spin of generl fluxes, chrges, nd dyons in this context, imgine n excittion (A, α) s composite where the flux A nd the centrlizer chrge α re kept prt by tiny distnce. The world line is now more like ribbon with the flux 20

29 3.1 Discrete Guge Theory nd the centrlizer chrge ttched to opposite edges. A 2π rottion corresponds to full twist of the ribbon, such tht the chrge winds round the flux. This implements the flux on the chrge. This is generlly ccomplished by the letting the element P h h (3.24) h ct, which results in Π A α (P h h) A h i, α v j = A h i, Π α (h 1 ) α v j h (3.25) s cn be seen from working out the rule (3.7). Since the element h 1 commutes by definition with ll elements of the centrlizer C(A), we hve Π α (h 1 ) = e 2πih (A,α) 1 α (3.26) s this follows from Schur s lemm pplied on the unitry irrep α. The phse θ (A,α) = exp(2πih (A,α) ) is the spin of the (A, α)-prticle. The element h P h h from eqution (3.24) is clled the ribbon element of D[H]. Note tht only true dyons cn hve θ (A,α) ± Anti-prticles In DGTs every prticle-type (A, α) hs unique conjugte prticle-type (Ā, ᾱ) with the specil property tht, s pir, they cn fuse to the vcuum, which is formlized in the fusion rule Π A α ΠĀᾱ = Π {e} (3.27) As representtion of D[H], we cn give the structure of (Ā, ᾱ) explicitly. The representtion module of (Ā, ᾱ) is just the dul of the representtion module of (A, α), i.e. = (Vα A ). The ction of P h g on stte A h i, α v j is given by V Āᾱ ΠĀᾱ (P h g): A h i, α v j A h i, α v j Π A α ( Pg 1 h 1 g g 1) (3.28) Indeed, we do find one-dimensionl subspce in Vα A V Āᾱ tht trnsforms like the vcuum under the ction of D[H], nmely the subspce spnned by A h i, α v A j h i, α v j (3.29) i,j We will show tht the subspce spnned by this element trnsforms trivilly, below. Recll tht the vcuum representtion is given by Π {e} 0 (P h g) = ɛ(p h g) = δ h,e (3.30) 21

30 3. QUANTUM GROUPS IN PLANAR PHYSICS Now note tht, in generl, elements of Vα A V Āᾱ cn be regrded s liner opertors on Vα A. In prticulr we hve A h i, α v A j h i, α v j = 1(A,α) (3.31) i,j where 1 (A,α) denotes the identity opertor on V A α. In prticulr it commutes with the representtion mtrices. From this, it is esy to deduce tht, s stte, it trnsforms like the vcuum s climed. Indeed, working out the ction of P h g on 1 (A,α) gives h h =h Π A α (P h g)1 (A,α) Π A α (P g 1 h 1 g g 1 ) = h h =h δ h,h 1ΠA α (P h g 1 g)1 (A,α) = h δ h,e Π A α (P h e)1 (A,α) (3.32) Note the subtle difference between (P h g) nd S(P h g). Agin, it is generl feture tht quntum group symmetry incorportes ntiprticles in nturl wy. For this we need liner mp S from the quntum group to itself, which is clled the ntipode, stisfying S( ) = 1ɛ() = S( ) (3.33) () () for ll elements of the quntum group. Here we hve mde use of the Sweedler nottion () = (3.34) () which generlizes (P h g) = h h P h g P h g (3.35) to rbitrry cses. 3.2 Generl remrks The discussion bove is included to give generl ide of quntum group symmetry. It is possible to give rigorous nd generl definitions of the essentil structures tht ppered bove, nd to define in generl wht we men by quntum group. Let us give the usul mthemticl nomenclture nd some relevnt references. Suppose we strt with n ssocitive lgebr H tht hs unit 1 H. Definition of comultipliction nd counit ɛ stisfying certin xioms gives colgebr structure, which, if it is comptible with the lgebr structure, mkes H bilgebr. If there is n ntipode, this mkes it Hopf lgebr. It cn be shown tht the ntipode, if it exists, is unique, such tht ny bilgebr hs t most one Hopf lgebr structure. The involution or -structure necessry to define unitry representtions mkes it Hopf- -lgebr. 22

31 3.2 Generl remrks The most importnt feture in reltion to plnr physics is the possibility of nontrivil briding. This is given by the existence of universl R-mtrix, which is n element of H H. It hs to stisfy certin consistency reltions which ensure tht the briding of representtions obey the Yng-Bxter eqution. The nme for Hopf lgebr with universl R-mtrix is qusi-tringulr. It is nice to contemplte wht is relly new for quntum groups in comprison with the usul group symmetries bundnt in physics. Let us consider two importnt cses, symmetries given by finite group H nd symmetries given by continuous Lie group G. insted of the group H, we cn lterntively work with the group lgebr C[H] without lossing ny informtion. This is just the complex lgebr generted by the group elements with the multipliction given by the group opertion. C[H] = ( ) Cg, c g g c g g = c g c g gg (3.36) g H g,g g g This is in fct Hopf lgebr with comultipliction, counit nd ntipode It is qusi-tringulr, with universl R-mtrix (g) = g g, ɛ(g) = 1, S(g) = g 1 (3.37) R = e e (3.38) where e is the unit element of H. Since this universl R-mtrix is trivil, briding just mount to flipping the tensor legs. This gives rise to the usul prticle sttistics in higher dimensions. We might cll C[H] tringulr, insted of qusi-tringulr. For Lie groups G the sitution is similr. Insted of the Lie group, one usully works with the Lie lgebr g. We usully llow for norml products xy prt from the brcket [x, y] nd incorporte unit 1 element, which is very nturl if we think of this Lie lgebr s n lgebr of symmetry opertors. Formlly, this leds to the universl enveloping lgebr U[g], which is generted by the unit 1 nd the elements of g subject to the reltions xy yx = [x, y] (3.39) It becomes (qusi-)tringulr Hopf lgebr by defining (1) = 1 1 (x) = x x ɛ(1) = 1 ɛ(x) = 0 S(1) = 1 S(x) = x R = 1 1 (3.40) which cn be seen s the infinitesiml version of the definitions given for C[H]. When we use the term quntum group, we hve qusi-tringulr Hopf lgebrs in mind, with non-trivil R-mtrix leding to interesting briding. Some uthors 23

32 3. QUANTUM GROUPS IN PLANAR PHYSICS seem to prefer to use the term quntum group only for qusi-tringulr Hopf lgebrs tht occur s q-deformtions of semi-simple Lie lgebrs discussed below, or to include qusi-hopf lgebrs, for which the coproduct does not precisely stisfy the xioms for colgebr, but do lmost. For more generl informtion on quntum groups nd precise definitions, we refer to the literture, for exmple [46, 60]. 3.3 Chern-Simons theory nd U q [su(2)] The quntum doubles tht cn be obtined vi the Drinfel d construction form n importnt clss of exmples of quntum groups. The other importnt clss comes from semi-simple Lie lgebr by q-deformtion of the universl enveloping lgebr. We will discuss n importnt exmple, U q [su(2)], tht comes from the Lie lgebr su(2). It plys n importnt role in the quntiztion of Chern-Simons theory with guge group SU(2) [84] nd in the Wess-Zumino-Witten models for CFT [79,82] bsed on the ffine lgebr of su(2) t level k. We follow [72]. One cn view U q [su(2)] s the lgebr generted by the unit 1 nd the three elements H, L + nd L, which stisfy the reltions [H, L ± ] = ±2L ± (3.41) [L +, L ] = qh/2 q H/2 q 1/2 q 1/2 (3.42) where q is forml prmeter tht my be set to ny non-zero complex number. The coproduct is given on the genertors by The counit nd ntipode re There is -structure given by (1) = 1 1 (3.43) (H) = 1 H + H 1 (3.44) (L ± ) = L ± q H/4 + q H/4 L ± (3.45) ɛ(1) = 1, ɛ(h) = 0, ɛ(l ± ) = 0 (3.46) S(1) = 1, S(H) = H, S(L ± ) = q 1/4 L ± (3.47) (L ± ) = L, H = H (3.48) giving rise to the notion of unitry representtions. Without the -structure, it is perhps better to cll this lgebr U q [sl(2)] insted of U q [su(2)], under which nme this quntum group usully ppers in the literture. In the limit q 1, we recover the definitions for the universl enveloping U[su(2)] lgebr of su(2), for instnce [L +, L ] = H (3.49) 24

33 3.3 Chern-Simons theory nd U q [su(2)] Therefore, it is sensible to tlk bout q-deformtion of U[su(2)]. If q is not root of unity, the representtion theory is very similr to tht of U[su(2)]. For every hlf-integer j 1 2Z there is n irreducible highest weight representtion of dimension d = 2j + 1 with highest weight λ = 2j. The representtion modules V λ hve n orthonorml bsis The ction of the genertors is Here we hve used the nottion j, m, m { j, j + 1,..., j 1, j} (3.50) Π λ (H) j, m = 2m j, m (3.51) L ± j, m = [j m] q [j ± m± 1 ] q j, m ± 1 (3.52) [m] q = qm/2 q m/2 q 1/2 q 1/2 (3.53) for the so clled q-numbers tht enter the formul to the commuttion reltion [L +, L ] = [H] q. A forml expression for the universl R-mtrix of U q [su(2)] is R = q H H 4 (1 q 1 ) n ( q n(1 n)/4 q nh/4 (L + ) n) ( q nh/4 (L ) n) (3.54) [n] q! n=0 From this expression, one cn work out the ction on representtions. This produces non-trivil briding in similr fshion s we sw for DGT nd the quntum double. In reltion to Chern-Simons theory nd the WZW models, the most interesting cse is, however, when q is not root of unity. In tht cse, the representtion theory of U q [su(2)] chnges drsticlly. We will not go in to the detils of the mthemticl tricks needed to find the right notion representtions tht describe chrges in these physicl theories. These cn be red, for instnce, in [46,60] or [72], but the right definition ws first discovered in [42]. In this cse, we denote the resulting theory s su(2) k, where k is relted to q s q = exp( 2πi k+2 ). We will come bck to these theories in chpter 5. 25

34 3. QUANTUM GROUPS IN PLANAR PHYSICS 26

35 CHAPTER 4 Anyons nd tensor ctegories As discussed in the previous chpter, quntum group symmetry cn ccount for fusion nd non-trivil briding of excittions in (2+1)-dimensionl systems. But the physicl sttes usully correspond to the unitry irreps rther thn the internl sttes of the representtion modules. Also, in the cse of U q [su(2)] modules for q root of unity, the identifiction of representtions with prticle types is not stright forwrd. One my sk if rigorous formlism exists tht trets these theories on the level of the excittions rther thn the underlying symmetries. The nswer is yes. Mthemticlly, the representtions of quntum group (often) form modulr tensor ctegory, structure tht my be defined independently. In this chpter we introduce mny generlities of these tensor ctegories s they rise in physicl pplictions. These cpture the topologicl properties, including the ddition of quntum numbers (fusion), of mny theories in very generl wy nd llow for grphicl formlism which cn be regrded s kind of topologicl Feynmn digrms. They cn be constructed from quntum groups, but lso rise directly from (rtionl) CFT nd cn be used describe the excittions in Kitev nd Levin-Wen type lttice models directly. They re generlly relted to topologicl quntum field theories. In sted of relying hevily on the lnguge of ctegory theory, we hve chosen route long the lines of [22, 66]. This mens tht we focus on concreteness nd clculbility, in fvour of forml lnguge. For the mthemticlly inclined, this bsiclly mens tht we ssume the ctegory to be strict nd construct everything in terms of simple objects. A pper in the mthemticl physics literture tht trets tensor ctegories in similr fshion is [34] where symmetries of the F -symbols (to be defined below) re derived. Proper mthemticl textbooks re [14, 75]. We hve prticulrly relied on [22] for this chpter. In ppendix B we give more mthemticlly formulted description of the kind of ctegories tht we consider, nd point out how this connects to the formultion below. 27

36 4. ANYONS AND TENSOR CATEGORIES 4.1 Fusing nd splitting For generl theory, the prticle-like excittions fll in different topologicl clsses or sectors. For simplicity, we tret these sectors s elementry excittions nd ssume there is finite number of them. These re the first ingredients of modulr tensor ctegory, which in physicl context hs to be unitry, or, more generlly, brided tensor ctegory. One cn vry the ssumptions on the kind of tensor ctegory in considertion. Ech set of ssumptions gives slightly different mthemticl structure which hs its own nme. To void getting stuck in nomenclture, we will generlly refer to the theory in this chpter. The term nyon models hs lso ppered in the physicl literture for unitry brided tensor ctegories Fusion multiplicities We strt with theory A with finite collection of topologicl sectors C A = {, b, c,... }. We lso refer to these sectors s prticle-types, (nyonic) chrges, (prticle) lbels or sometimes simply prticles or nyons. They obey set of fusion rules tht we write s b = c C A N c b c (4.1) (The domin of the sum will be left implicit from now on.) Here the Nb c re nonnegtive integers clled fusion multiplicities. These determine the possibilities for the totl chrge when two nyons re combined (fused). The totl chrge of two nyons with respective lbels nd b cn be ny of the c with Nb c 0. If N b c 0, we lso spek of the fusion chnnel c of nd b, nd we will sometimes write this s c b. If the stte-spce of pir of prticles is multidimensionl this gives rise to non- Abelin nyons. 1 We sy, therefore, tht the theory is non-abelin if there re chrges nd b tht hve Nb c > 1 (4.2) c The prticle lbels together with the fusion rules (4.1) generte the fusion lgebr of the theory. In this context we sometimes denote the bsis sttes by kets, b, c. The fusion lgebr hs to obey certin conditions which hve cler physicl interprettion. We require tht there is unit, i.e. unique trivil prticle, the vcuum, 0 A tht fuses trivilly with ll lbels: 0 = = 0 for ll. Furthermore, fusion should be n ssocitive opertion, ( b) c = (b c), such tht the totl nyonic chrge is well-defined notion. In terms of the fusion multiplicities, these conditions 1 There is subtlety to the term non-abelin nyons. Even if pir of identicl prticles hve multidimensionl stte spce, their briding might still be Abelin. Hence, on could sy tht this is necessry but not sufficient condition, but we will not bothered with the distinction. 28

37 4.1 Fusing nd splitting cn trnslte to N0 b =N0 b = δ b (4.3) Nb e N ec d = Nf d N f bc (4.4) e f We lso require tht the fusion rules re commuttive, such tht b = b. This should hold for (2+1)-dimensionl systems, since there is no wy to define left nd right unmbiguously. We cn lift this requirement if we consider chrges of (1+1)d system, for exmple when we look t boundry excittions. Finlly, ech chrge C A should hve unique conjugte chrge ā C A or nti-prticle such tht nd ā cn nnihilte ā = 0 + c 0 N c b c (4.5) This does not men tht nd ā cn only fuse to the vcuum, s there might be c 0 with N c ā 0. Note tht ā =. The following reltions for the fusion multiplicities lso follow from these conditions on fusion N 0 b = δ bā (4.6) N c b = N c b = N āb c = N c ā c (4.7) The first line is just the nti-prticle property. The second line cn be derived using commuttivity nd ssocitivity. It is useful to define the fusion mtrices N with (N ) bc = N c b (4.8) The fusion rules give N N b = c N c b N c (4.9) which shows tht the fusion mtrices provide representtion of the fusion lgebr. This is just the representtion of the fusion lgebr tht is obtined by letting it ct on itself vi the fusion product. Associtivity of the fusion rules trnsltes to commuttivity of the fusion mtrices, such tht N N b = N b N (4.10) The explicit digonliztion of the fusion mtrices is very interesting result, tht we come bck to in section Digrms nd F -symbols Specil bout the formlism bsed on ctegory theory in comprison with the quntum group pproch is tht internl sttes re totlly left out of the picture. This mkes 29

38 4. ANYONS AND TENSOR CATEGORIES sense becuse, in physicl settings, to obtin the guge invrint sttes, we still hve to mod out by the quntum group. Opertors on nyons re best described in digrmmtic formlism. 1. They form sttes in certin vector spces. The formlism gives rules to mnipulte the digrms, nd perform digrmmtic clcultions. Mnipulting digrms ccording is wy to rewrite the sme stte, for exmple, by chnging to nother bsis. One lters the representtion, but not the stte (opertor) itself. Ech nyonic chrge lbel is ssocited with directed line. This is the identity opertor for the nyon, or in the lnguge of ctegory theory, the identity morphism. It is often useful to think of it s the world line of the nyon propgting in time, which we will tke s flowing upwrd. Reversing the orienttion of line segment is equivlent to conjugting the chrge lbelling it, so tht = ā (4.11) The second most elementry opertors re directly relted to fusion nd the dul process, splitting of nyons. To every triple of chrge lbels (, b, c), ssign complex vector spce Vb c of dimension N b c. This is clled fusion spce. Sttes of the fusion spce Vb c re denoted by fusion vertices with lbels corresponding to the chrges on the outer legs c µ b, µ = 1,..., N c b (4.12) The dul of the fusion spce is denoted Vc b nd is clled splitting spce. The sttes of the splitting spce Vc b re denoted by splitting vertices like b µ c, µ = 1,..., N c b (4.13) When Nb c = 1 we cn leve the index µ implicit.2 The 0-line cn be inserted nd removed from digrms t will, reflecting the properties of the vcuum, nd is therefore often left out or invisible. When mde explicit, we drw vcuum lines dotted. Hence, we hve 0 = = 0 (4.14) nd similr for the corresponding fusion vertices. In terms of quntum group representtions, splitting vertices correspond to the embedding of n irrep into tensor product representtion, while fusion vertices should 1 In the mthemticl literture this is known s grphicl clculus, which ws introduced by Reshetikhin nd Turev [70]. 2 In subsequent chpters, we will ssume tht ll Nb c = 0, 1. Therefore, digrms will hve no lbels t the vertices, but only t the chrge lines. For completeness, however, we hve left the µ s in tct in this chpter. 30

39 4.1 Fusing nd splitting be thought of s the projection onto n irreducible subspce of tensor product representtion. Generl nyon opertors cn be mde by stcking fusion nd splitting vertices nd tking liner combintions. The intermedite chrges of connected chrge lines hve to gree nd the vertices should be llowed by the fusion rules, otherwise the whole digrm evlutes to zero. This gives digrmmtic encoding of chrge conservtion. As generl nottion we write V 1,..., m for the vector spce of opertors tht tke 1,..., n n nyons with chrges 1,..., n s input nd which produce m nyons of chrges 1,..., m. An importnt exmple is Vd bc. Opertors of this spce cn be mde by stcking two splitting vertices on top of ech other. The choice we hve of connecting the top vertex either on the right or on the left of the bottom vertex leds to two different bses of V bc d. The chnge of bsis in these spces is given by so clled F -symbols 1 [F bc d ] (e,α,β)(f,µ,ν), which re n importnt piece of dt for these models. They re defined by the digrmmtic eqution b c α e β d = f,µ,ν [Fd bc ] (e,α,β)(f,µ,ν) b c µ ν f d (4.15) Like ll other digrmmtic equtions in the formlism, this F -move cn be used loclly, within bigger digrms to perform clcultions. If digrm on the right is not permitted by the fusion rules, we put the corresponding F -symbol to zero. The F -symbols hve to stisfy certin consistency conditions, clled the pentgon reltions, tht we will come bck to lter. There is certin guge freedom in the F -symbols, reflecting the fct tht we cn perform unitry chnge of bsis in the elementry splitting spce Vc b without chnging the theory, which will lso be discussed lter. We could just s well hve introduced the F -symbols in terms of fusion sttes in Vbc d. The F -move for fusion sttes is governed by the djoint of the F -move s introduced bove. Unitrity of the model mounts to [(F bc d ) ] (f,µ,ν)(e,α,β) = [F bc d ] (e,α,β)(f,µ,ν) = [(F bc d ) 1 ] (f,µ,ν)(e,α,β) (4.16) This cn be seen by tking the djoint of the digrmmtic eqution for the F -symbols. In generl, the djoint of digrm in unitry theory is tken by reflecting it in the horizontl plne, such tht top nd bottom interchnge, nd then reversing the orienttion of ll rrows. If one of the upper outer legs of the tree occurring in the F -move eqution is lbelled by the vcuum, it is essentilly just splitting vertex nd the F -move should leve it unchnged. Thus, for exmple [Fd 0bc ] ef = δ eb δ fc, nd similrly if the middle or right upper index equls 0. Note tht when d = 0 there is only one non zero F -symbol for fixed, b, c, nmely [F0 bc ] cā, but this cn be non-trivil. 1 The F -symbols re sometimes clled recoupling coefficients or quntum 6j-symbols.The F -symbols re the nlogue of the 6j-symbols from the theory of ngulr momentum, hence the ltter nme. 31

40 4. ANYONS AND TENSOR CATEGORIES The piring of Vc b nd Vb c or the inner product of fusion/splitting sttes, is gin denoted by stcking the pproprite digrms. We think of the elements of Vc b s kets nd of the elements of Vb c s brs. Composition of opertors is written from bottom to top, in ccordnce with the flow of time. The conservtion of nyonic chrge is lso tken into ccount in the digrms, which gives c µ b c µ = δ cc δ µµ d d b Here, the quntum dimension d is defined for ech prticle lbel s d c c (4.17) d = [F ā ] 00 1 (4.18) This is very importnt quntity in ll tht follows. From the properties of the F - symbol immeditely find tht d 0 = 1. The normliztion fctor ddb d c is inserted in this digrmmtic brcket to mke the digrms invrint under isotopy, i.e. under bending of the chrge lines (end points should be left fixed). For horizontl bending, keeping in tct the flow of chrge in time, isotopy invrince is trivil. Verticl bending however, introduces events of prticle cretion nd nnihiltion. Invrince under this kind of bending is lmost relized by introducing the normliztion of the vertex brcket, but subtle issue remins. Consider the clcultion 0 0 ā = [F ā ] 00 ā 0 0 = d [F ā ] 00 (4.19) which uses the F -move nd (4.17). By the definition of d, the coefficient on the right hs unit norm. However, [F ā ] 00 might hve non-trivil phse fb = d [F ā ] 00 (4.20) As we will see lter, for ā the phse fb = fb ā cn be set to unity by guge trnsformtion. But when is self dul, fb = ±1 is guge invrint quntity known s the Frobenius-Schur indictor. Hence, in the pproprite guge, isotopy invrince is relized up to sign. To ccount for this sign, we introduce cup nd cp digrms with direction given by flg. These re defined s ā ā = fb = 0 ā (4.21) nd = fb ā = ā 0 ā (4.22) 32

41 4.1 Fusing nd splitting Bending line verticlly is now tken to include the introduction of cp/cup pir with oppositely directed flgs. Then we hve the following equlity = = (4.23) With these conventions in plce, isotopy invrince of the digrms is relized. If the cups nd cps re pired up with opposite flgs in digrms, we leve them implicit. This, shows tht loops evlute s the corresponding quntum dimension = d (4.24) when we combine this with reltion (4.17). In ccordnce with the normliztion for the inner product, the identity opertor on pir of nyons cn now be written s b = c,µ d c d d b b c µ µ b (4.25) It is very convenient to define F -symbols for digrms with two-nyon input nd two-nyon output. This is done by the digrmmtic equtions α e c b β d b [Fcd b ] (e,α,β)(f,µ,ν) f µ ν (f,µ,ν) c d = (4.26) Eqution (4.25) gives [F b b ] 0,(c,µ,ν) = d c d d b δ µ,ν (4.27) And more generlly we hve [F b cd ] (e,α,β)(f,µ,ν) = d e d f d d d [F ceb f ] (,α,µ)(d,β,ν) (4.28) These lterntive F -moves cn be used to chnge splitting vertex with one leg bend down into fusion vertex, etceter. This gives equlities like nd 0 c ā µ b b µ b c 0 = µ [F 0c = µ [F b b ] (ā,µ)(c,µ ) c µ c0 ] (ā,µ)(c,µ ) µ c b b (4.29) (4.30) 33

42 4. ANYONS AND TENSOR CATEGORIES By looking t the djoint versions, more equlities my be derived. For the generl spces of opertors V 1,..., m one usully grees on stndrd bsis. The choice generlly mde is the bsis mde by stcking vertices on the left, i.e. 1,..., n consisting of opertors m µ 2 µ 2... e... µ m µ n n (4.31) Any opertor in the theory cn in principle be expressed in terms of the stndrd bsis elements. This is very importnt remrk. It implies, for exmple, tht ny opertor we write down tht cts on n nyon of chrge is, effectively, nothing but multipliction by complex number since the spce V is one-dimensionl. Furthermore, mny equlities cn be deduced by compring coefficients of the corresponding equtions expressed in terms of the stndrd bsis. But for other pplictions, it is sometimes tedious to write everything in terms of the stndrd bsis, nd it is much more convenient to leve the digrms intct Guge freedom As remrked before, there is certin freedom present in ny nyon model. This corresponds to the choice of bses in the Vc b. We cn pply unitry trnsformtions in the spces Vc b without chnging the theory. Denote such unitry trnsformtions by u b c. This gives new bsis sttes µ c b = µ [u b b c ] µ µ µ c The effect of this trnsormtion on the F -symbols is [F bc d ] (e,α,β )(f,µ,ν ) = α,β,µ,ν [(u b e ) 1 ] α α[(u ec d ) 1 ] β β[fd bc (4.32) ] (e,α,β)(f,µ,ν) [u bc f ] µ µ[u f d ] ν ν (4.33) When there re no multiplicities, the trnsformtions u b c re just complex phses. In this cse, which is of most prcticl interest, we hve the freedom to redefine the F -symbols following [F bc d ] ef = ubc f uf d u b e u ec d [F bc d ] ef (4.34) We cn use this freedom to switch to more convenient set of F -symbols. Note tht we do not llow u 0 nd u 0 to differ from unity due to (4.14). 34

43 4.1 Fusing nd splitting From (4.34) it is stright forwrd tht we cn remove the phse fb of [F ā ] 00 for ll ā, by tking u ā 0 = fb nd uā 0 = 1. So there is choice of guge for which ll Frobenius-Schur indictors re equl to unity, except for self-conjugte prticles s climed before. The guge freedom discussed here should not be mistken for guge freedom in the sense of guge theories. We only wish to express tht there is redundncy in the theory Tensor product nd quntum trce The tensor product of two opertors cting on well seprted groups of nyons is mde by juxtposition of the corresponding digrms. For generl opertor we introduce the nottion A 1 A 1... X... A m A n = X V A 1...A m A 1...A n = 1... m 1... n V 1... m 1... m (4.35) where we used cpitl lbels to denote sums over ll nyonic chrges such tht X is defined to ct on ny input of n nyons, giving some m nyon output. The tensor product of two opertors X nd Y is then given by the digrm X Y... = X... Y... (4.36) While it is of course possible to rewrite this in terms of the stndrd bsis, it is usully much more convenient to keep the seprtion explicit. In the mthemticl literture, one often sees eqution (4.24) s the definition of the quntum dimension. It is specil cse of the quntum trce pplied to the identity opertor. The quntum trce of generl opertor X, denoted s Tr X, is formed digrmmticlly by closing the digrm with loops tht mtch the outgoing lines with the incoming lines t the sme position Tr X = Tr A 1... X... A n = A 1... A n X (4.37) A 1 A n The fct tht the quntum dimension is the quntum trce of the identity opertor is nlogues to the fct tht the trce of the identity mtrix gives the dimension of the vector spce it cts on. In cses where we cn spek of n internl Hilbert spce of the 35

44 4. ANYONS AND TENSOR CATEGORIES nyon, s in DGT, the quntum trce is indeed the sme s the trce over this internl spce. From the digrms it is cler tht this definition of trce nd tensor product behve in the pproprite wy. For two opertors X nd Y, we hve Tr(X Y ) = Tr(X) Tr(Y ). We cn lso define the prtil trce, nd trce out subsystem. In this cse only prt of the outer lines is closed with loops, either on the outer left or on the outer right side Topologicl Hilbert spce The topologicl Hilbert spce of collection of nyons with chrges 1,..., n with totl chrge c is the spce V 1,..., n b. It is interesting to study the symptotic behviour of the topologicl Hilbert spce of n nyons of chrge. This is effectively given by the quntum dimension d, s discussed below. This discussion is bsed on [66]. An importnt reltion for the quntum dimension is d d b = b = c,µ d c d d b c µ b µ = c N c b d c (4.38) This shows tht the d form one-dimensionl representtion of the fusion lgebr. Now consider the vector ω = d in the fusion lgebr. Then reltion (4.38) shows tht ω is common eigenvector of the fusion mtrices, nd we hve N ω = d ω (4.39) Since the mtrices N hve non-negtive integer entries nd ω hs only positive coefficients, d must be the lrgest eigenvlue. Note tht this in prticulr mens tht we do not need the F -symbols to deduce d, but tht the fusion rules re sufficient. We cn rephrse the bove observtion s the sttement tht ω is common Perron-Frobenius eigenvector of the N. The significnce for the topologicl Hilbert spce is the following. Suppose we hve collection of n nyons of chrge which hve totl chrge b. Denote the dimension of the topologicl Hilbert spce V b... by N b... We hve N b... = {b i } N b 1 N b 2 b 1... N b b (n 2) (4.40) The mtrix N cn be digonlized s where N = d ω ω +... (4.41) D2 D = ω ω = d 2 (4.42) 36

45 4.2 Briding nd we hve left out sub-leding terms. So for lrge n, we obtin N b... d n d b /D 2 (4.43) Therefore we see tht d controls the rte of growth for the n prticle topologicl Hilbert spce. Corresponding to ω, we hve the chrge line lbelled by ω. Since d = dā, the lbels nd ā pper with identicl weights. Therefore we cn leve out the orienttion of the chrge line, nd write ω = d (4.44) This gives ω = D 2 (4.45) We would like to point out the significnce of the lbel ω when we consider DGT. In the representtion theory of groups, one often considers the group lgebr C[H] of, sy, finite group H s representtion. We cn just let H ct on it vi the group opertion. It is well known fct tht C[H] cn be written s sum of irreducible representtion which occur with multiplicity given by the dimension of the irrep, C[h] α d αv α. In fct, the sme holds for quntum double representtions, D[H] (A,α) d (A,α)V A α. Hence, for these theories ω = D[H]. 4.2 Briding The defining properties of nyonic excittions re their non-trivil spin nd briding properties. These re lso incorported in the formlism by mens of digrmmtic reltions. The effect of two nyons switching plces in the system is governed by the briding opertors, which re written s R b =, R b = R 1 b b = b (4.46) They re defined by set of R-symbols which give the effect on bsis sttes of Vc b. The digrmmtic reltion, or R-moves, re b c µ = ν [R b c ] µν b ν c (4.47) (4.48) The full briding opertor cn thus be represented s = d c [R b b d c,µ,ν d b b c ] µν c ν µ b (4.49) 37

46 4. ANYONS AND TENSOR CATEGORIES The similr equtions hold for the inverse R-move. Unitrity of the R-moves mounts to [(Rc b ) 1 ] µν = [(Rc b ) ] µν = [Rc b ] νµ (4.50) For briding nd fusion to be consistent, the R-moves hve to obey the hexgon equtions, discussed below. These reltions imply tht briding commutes with fusion, which mens tht in digrms lines my be pssed over nd under vertices. They imply the usul Yng-Bxter reltion for brids. Together with trivility of fusion with the vcuum they lso imply R 0 = Rb 0b = 1 (4.51) so tht the vcuum brids trivil, s it should. The topologicl spin θ, lso clled twist fctor, is ssocited with 2π rottion of n nyon of chrge nd defined by θ = θā = d 1 Tr R = fb [Rā 1 ] (4.52) Digrmmticlly, it cn be used to remove twists from digrm: = θ = (4.53) nd = θ = (4.54) Becuse the θ re not necessrily trivil, it is better to think of the lines in digrms s ribbons for which twist is relly non-trivil opertion, s illustrted below. When pplicble, the twist fctor is relted to the (ordinry ngulr momentum) spin or CFT conforml scling dimension h of, s θ = e i2πh (4.55) The effect of double briding or monodromy of two nyons is governed by the monodromy eqution [Rc b ] µλ [Rc b ] λν = θ c δ µ,ν (4.56) θ θ b λ 38

47 4.3 Pentgon nd Hexgon reltions which digrmmticlly gives c µ b = θ c µ θ θ b c b (4.57) It is mtter of topologicl mnipultion to see tht this eqution holds. This is more redily seen when we drw the ribbon version We cn lso write down the full monodromy opertor b = θ c,µ θ b θ c d c d d b c µ µ b b (4.58) which corresponds to one prticle encircling the other, or equivlently 2π rottion of pir of prticles. The monodromy eqution, in ny form, is used frequently in digrmmtic clcultions. 4.3 Pentgon nd Hexgon reltions With the definition of fusion rules, F -symbols, nd R-symbols, the topologicl dt of theory is completely specified nd ll digrms cn be clculted. In order to ensure tht different wys of mnipulting digrm lwys led to the sme nswer, some consistency conditions on the F -symbols nd R-symbols hve to be stisfied. These re the pentgon nd hexgon reltions or equtions. The McLne s coherence theorem from ctegory theory gurntees tht no other consistency checks re necessry [54]. The pentgon equtions re condition on the F -symbols. They cn be understood s the equivlence of two sequences of moves, s shown in figure 4.1, which cn be netly orgnized in pentgon. The resulting eqution is δ [F fcd e ] (g,β,γ)(l,λ,δ) [Fe bl ] (f,α,δ)(k,µ,ν) = [Fg bc h,ρ,σ,ψ ] (f,α,β)(h,ρ,σ) [F hd e ] (g,σ,γ)(k,ψ,ν) [F bcd k ] (h,ρ,ψ)(l,λ,µ) (4.59) 39

48 4. ANYONS AND TENSOR CATEGORIES Figure 4.1: The pentgon reltions - The pentgon reltion sy tht the two possible sequences of mnipulting the bsis sttes of Ve bcd using F -moves give the sme result. The hexgon reltions re similrly understood s the equlity of two sequences of mnipultion. They gurntee consistence of fusion nd briding. The digrmmtic reltions re illustrted in figure 4.2. The resulting equtions re [Re c ] αλ [F cb ] (e,λ,β)(g,γ,ν) [Rg cb ] γµ (4.60) nd λ,γ = d [Fd cb f,σ,δ,ψ ] (e,α,β)(f,σ,δ) [R cf d ] δψ[fd bc ] (f,σ,ψ)(g,µ,ν) (4.61) [(Re c ) 1 ] αλ [F cb ] (e,λ,β)(g,γ,ν) [(Rg cb ) 1 ] γµ (4.62) λ,γ = [Fd cb f,σ,δ,ψ d ] (e,α,β)(f,σ,δ) [(R cf d ) 1 ] δψ [Fd bc ] (f,σ,ψ)(g,µ,ν) (4.63) The pentgon nd hexgon equtions first ppered in this form in the physics literture in [62]. 4.4 Modulrity The consistent definition fusion rules nd unitry F -symbols nd R-symbols gives unitry brided tensor ctegory. But in reltion to topologicl quntum field theories, models obeying n dditionl condition re of prticulr interest. These re known s modulr tensor ctegories. The chrcteristic property is tht the topologicl S-mtrix is non-degenerte. The topologicl S-mtrix, which is defined by S b = 1 D b (4.64) 40

49 4.4 Modulrity Figure 4.2: The hexgon reltions - The hexgon reltions relte fusion to briding. They imply the Yng-Bxter eqution. encodes welth of informtion bout the theory. By pplying the monodromy eqution, we find S b = 1 N c θ c b d c (4.65) D θ θ b c From this expression, we derive tht S b = S b = Sāb nd S 0 = 1 D d. A very useful equlity is b = S b S 0b b (4.66) This holds, becuse both left nd right side re elements of the one-dimensionl vector spce Vb b, hence must be proportionl. The constnt of proportionlity is checked by tking the quntum trce. This is convenient trick to prove equlities of singlenyon opertors. Often, these immeditely imply corresponding result for multi-nyon opertors by decomposing these using (4.25). Modulr tensor ctegories re defined by the non-degenercy of the S-mtrix. When the theory is unitry, s is the cse for physicl theories, the S-mtrix is in fct unitry mtrix. Together with the mtrix T with coefficients T b = e 2πc/24 θ δ b (4.67) it forms representtion of the modulr group SL(2, Z) with defining reltions (ST ) 3 = S 2 nd S 4 = 1. Here c is the topologicl centrl chrge, which is equl to the CFT centrl chrge mod 24 when pplicble. It cn be determined mod 8 from the twist fctors nd quntum dimensions by the reltion ( ) 2πi c exp = 1 d 2 8 D θ (4.68) 41

50 4. ANYONS AND TENSOR CATEGORIES This is ll well known, but since it is n interesting ppliction of the digrmmtic formlism we include derivtion bsed on [14] of the reltions between S nd T below. Let p ± = d 2 θ ±1 (4.69) The following equtions hold ω b = p + θ 1 b b, ω b = p θ b b (4.70) To show these equlities, note tht the left nd right hnd side of these equtions must be proportionl, since they re both single-nyon opertors. Hence we tke the quntum trce on either side to check the proportionlity constnt. For the eqution involving p +, the quntum trce of the right hnd side is evidently p + d b /θ b, while the left hnd side gives d θ DS b = Nb c d d c θ c /θ b (4.71),c = d 2 cθ c d b /θ b c (4.72) = p + d b /θ b (4.73) Here we used tht N c b d = N ā cb d = d b d c by (4.38), the symmetries of the fusion coefficients, nd the fct tht d = dā. As consequence we get ω = p + θ 1 θ 1 b (4.74) b b To see this, pply the ribbon property on the double clockwise brid on the right hnd side, i.e. on the opertor (R b )2, nd compre with (4.70). Define T b = θ δ b, C b = δ b (4.75) Thus T is just T with the phse in front left out, nd C is the chrge conjugtion mtrix. Clerly, C commutes with S nd with T, nd C 2 = 1. But more interestingly 1. (ST ) 3 = p + S 2 C/D 2. (ST 1 ) 3 = p S 2 /D 3. S 2 = p + p C/D 2 42

51 4.4 Modulrity To prove the first reltion, strt with the identity ω = p + θ 1 θc 1 c (4.76) c which follows from (4.74). The right hnd side is equl to p + θ 1 θc 1 S c S 0 while the left hnd side cn be computed s (4.77) d b θ b b b c = b d b θ b S bc S 0b b = D b θ b S bc S b S 0 (4.78) Equting the coefficients gives D b θ bs bc S b = p + θ 1 b S c, which cn be seen to be equivlent to the mtrix eqution DST S = p + T 1 SCT 1 written in components. Multiplying from left by ST nd from the right by T nd dividing out D produces the first sttement. The second sttement is derived similrly. The third sttement follows from the first two. We first show tht S 2 T = T S 2. Since S, T re symmetric, by tking the trnspose of the equlity (ST 1 ) 3 = p + S 2 /D, we find tht (ST ) 3 = (T S) 3, or ST 1 ST 1 S = T 1 ST 1 ST 1 ST. Plugging this D into the following version of the second equlity, ST 1 ST 1 S = T S 2, we find p θ 1 T S 2 = D p T 1 ST 1 ST 1 ST (4.79) = S 1 ( D p ST 1 ST 1 ST 1 ) ST (4.80) = S 1 (S 2 )ST (4.81) = S 2 T (4.82) so S 2 nd T indeed commute. Now multiply the identities T 1 ST 1 ST 1 = p S/D nd T ST ST = p + SC/D (4.83) Pulling fctors of S 2 through T s nd cncelling T T 1 pirs, this leves which proves the third sttement. S 4 = p + p S 2 C/D 2 (4.84) 43

52 4. ANYONS AND TENSOR CATEGORIES This third sttement hs n importnt consequence. It implies tht loops lbelled by ω hve the killing property. This somewht hostile nme comes from the fct tht ll non-trivil chrge lines going through loop lbelled by ω = d get killed in the following wy. ω = p + p δ 0 (4.85) This is gin proved by compring the quntum trce of the left nd right hnd side. The trce of the left hnd side gives D b d b S b = D 2 b S 0b S b = p + p C 0 = p + p δ 0 (4.86) which is clerly the sme s the trce of the right hnd side. Thus loops lbelled by ω give nturl projector in the theory. 1 Tking = 0 we get the following result If we now define p + p = D 2 (4.87) T = ζ 1 T, ζ = (p + /p ) 1 6 (4.88) we find tht S nd T obey the reltions of the modulr group, s ws lluded to before. This lso gives the implicit definition of the topologicl centrl chrge c in terms of the quntum dimensions nd twist fctors of the theory Verlinde formul The fmous Verlinde formul, which first ppered in the context of CFT in [76], expresses the fusion coefficients in terms of the S-mtrix s N c b = x S x S bx S cx S 0x (4.89) Becuse the fusion mtrices by ssocitivity of fusion, bsis of common eigenvectors is expected. The Verlinde formul sys tht this bsis is essentilly given by the columns of the S-mtrix. We will now prove the Verlinde formul using the grphicl formlism. Define Λ () bc = λ() b δ bc, λ () b = S b (4.90) S 0b Recll tht the fusion mtrices re given by (N ) bc = Nb c. We will show tht N S = SΛ () (4.91) 1 This projector hs recently been used to provide spcetime picture for the Levin-Wen models in n interesting work [23] 44

53 4.5 Sttes nd mplitudes by mnipulting the following digrm: b x On the one hnd, we cn pply equlity (4.66) twice to get (4.92) S x S 0x S bx S 0x x (4.93) On the other hnd, we cn first fuse the encircling nyons with chrge nd b nd then pply (4.66), to get Nb d S dx (4.94) S x 0x This gives the equlity d d N d b S dx = S x S bx S 0x (4.95) which is precisely (4.91) in components. When we multiply both sides by S cx, sum over x, nd use tht S 2 = C or x S dxs cx = δ dc, we find (4.89). 4.5 Sttes nd mplitudes To incorporte quntum sttes of nyons in the formlism, we hve to keep trck of the cretion history. Generl sttes for system with nyons of chrge 1,..., n cn be written down s Ψ = 1, 2, 3, 4,c ψ 1, 2, 3, 4,c (d 1 d 2 d 3 d 4 d c ) 1/ c (4.96) where we hve suppressed the Greek indices lbelling the vertices. The normliztion fctor is included such tht Ψ Ψ = 1 iff ψ 1, 2, 3, 4,c 2 = 1 (4.97) 1, 2, 3, 4,c where Ψ is of course defined by conjugting the digrm Ψ = 1, 2, 3, 4,c ψ 1, 2, 3, 4,c c (d 1 d 2 d 3 d 4 d c ) 1/4 (4.98) nd the brcket is formed by stcking the digrms on top of ech other. Amplitudes for nyon opertors cn now be clculted in the usul wy. We sndwich the opertor 45

54 4. ANYONS AND TENSOR CATEGORIES between the ket nd br of the stte nd clculte the vlue of the digrm. As n exmple, let us consider the monodromy opertor. Suppose we crete two nyon pirs ā nd b b. This gives the tensor product stte Ψ = 1 ā b b d d b (4.99) The mplitudes for the monodromy of the inner b-pir gives rise to the monodromy mtrix M b = 1 = N c θ c d c b (4.100) d d b θ c θ b d d b b which, s the digrm shows, is closely relted to the S-mtrix. Interferometry experiments re possibly ble to mesure mtrix elements of the monodromy mtrix in the ner future for interesting systems, such s in the quntum Hll effect. Since the mtrix is highly constrined by the vrious S-mtrix reltions, mesuring only few mtrix elements might mke it possible to construct the full mtrix nd discover the topologicl order. See [18 21] for theoreticl tretment of interferometry experiments in lnguge similr to the presenttion here. One cn lso dopt density mtrix formlism which cn hndle systems of nontrivil totl chrge. By writing ρ = Ψ Ψ digrmmticlly, we find tht the density mtrix corresponding to pure stte is of the form ρ = 1, 2, 3, 4,c 1, 2, 3, 4,c ρ ( c)( c ) (d 1 d 2 d 3 d 4 ) 1/ c c (4.101) Generl density mtrices, corresponding to mixed sttes or sttes with non-trivil nyonic chrge, cn be constructed from these by trcing out nyons of the system using the quntum trce. For exmple, generl two-nyon density mtrix is of the form,b,,b,c ρ,b,,b,c (d d b d d b d2 c) 1/2 b c b (4.102) which cn be obtined from the pure stte density mtrix,b,c,b,c ρ (,b,c,)(,b,c ) (d d b d c d d b d c ) 1/2 b c b c (4.103) by trcing out the right nyon. 46

55 4.5 Sttes nd mplitudes Let us end this chpter by discussing the full brid opertor B or B-move. It is defined by b c = b c [Bd bc ] ef e f (4.104) f d Reinstlling the Greek indices, we find tht [Bd bc ] (e,α,β)(f,µ,ν) = [Fd cb ] (e,α,β)(g,γ,δ) [Rg cb ] γη [(Fd bc ) 1 ] (g,η,δ)(f,µ,ν) (4.105) g,γ,δ,η This opertor cptures the effect of briding on generl nyonic sttes. The monodromy mtrix is for exmple recovered s the (0, 0) of the pproprite squre of this opertor, M b = f,µ,ν d [Bāb b ] 0,(f,µ,ν) [Bāb b ] (f,µ,ν),0 (4.106) 47

56 4. ANYONS AND TENSOR CATEGORIES 48

57 CHAPTER 5 Exmples of nyon models In principle, the unitry brided tensor ctegories discussed in chpter 4 cn be constructed in three step procedure: First, fix set of prticle lbels nd fusion rules. Next, solve the pentgon reltion to find consistent set of F -symbols. Finlly, solve the hexgon reltions to find R-symbols. If no solution to the pentgon reltions exists, the declred fusion rules re inconsistent with locl quntum mechnics nd no model with these fusion rules exists. When we do find solution to the pentgon reltions, but not to the hexgon reltions, there is no wy to define consistent briding for the nyons which excludes the model with these fusion rules s n effective theory for bulk nyons but, s we will see, might describe boundry effects of medium crrying nyons. If there re multiple solutions for the pentgon/hexgon reltions tht re not relted by guge trnsformtion, there re pprently non-equivlent nyon models tht obey the sme fusion rules. A theorem know s Ocnenu rigidity ensures tht, for given set of fusion rules, there cn only be finite number of non-equivlent nyon models. Although the procedure outlined bove cn in principle be used to find ll nyon models, in prctice it is not fesible to find mny interesting models this wy. The reson is tht the number of equtions in the pentgon reltions, s well s the number of vribles, grows very rpidly with the number of chrges in the theory. For smll numbers of chrges, some clssifiction results hve been obtined by direct ttck of the pentgon/hexgon reltions. For exmple, ll models with up to four chrges were clssified in [71]. Also, in [22], solutions to the pentgon nd hexgon reltions were found explicitly for number of interesting fusion rules by solving them directly. Most known models, however, were constructed vi completely different route, nmely s representtion ctegories of quntum group tht we discussed before. In this chpter we will show how to obtin the fusion rules, F -symbols nd R-symbols for the quntum double. We will lso give this dt for the su(2) k theories relted to 49

58 5. EXAMPLES OF ANYON MODELS U q [su(2)], Chern-Simons theory nd the WZW-models nd we flesh out the detils for some representtive exmples tht will be used lter. But we will strt by discussing the well know Fiboncci nyons, for which the pentgon nd hexgon equtions cn be solved by hnd. 5.1 Fiboncci Suppose we hve only one non-trivil chrge besides the vcuum, which we lbel by 1. The two possibilities for the one non-trivil fusion rule re 1 1 = 0 nd 1 1 = The first gives n Abelin theory tht is not very interesting, so we tke the second rule. This fixes the theory completely, s we will see, nd gives rise to the so-clled Fiboncci nyons. The discussion below is bsed on [66]. The F -symbols we need to find re [F0 111 ] ef nd [F1 111 ] ef. The mtrix [F0 111 ] ef, hs only one non-zero component, which we cn set to unity. This gives [F ] ef = δ e1 δ f1 (5.1) This leves us with the tsk to find the 2 2 mtrix F with F ef = [F0 111 ] ef. Since we re looking for unitry theory, we serch n F of the form ( ) z w F = w (5.2) z with z 2 + w 2 = 1. Without multiplicities in the theory, the pentgon reltions simplify to [Fe fcd ] gl [Fe bl ] fk = [Fg bc ] fh [Fe hd ] gk [Fk bcd ] hl (5.3) h Now tke = b = c = d = e = f = k = 1 nd g = l = 0. The pentgon eqution for this combintion of lbels is [F1 111 ] 00 = [F1 111 ] 01 [F1 111 ] 10 (5.4) which gives z = w 2. This leds us to the generl solution ( φ F = 1 e iθ ) φ 1 e iθ φ 1 φ 1 (5.5) where φ = , i.e the golden rtio. The phse e iθ cn be set to unity with guge trnsformtion. All pentgon reltions re now indeed stisfied. Next we will solve the hexgon reltions. Without multiplicities, these tke the following form. (R c Re c [Fd cb ] eg Rg bc = f e ) 1 [Fd cb ] eg (Rg bc ) 1 = f [F cb d ] ef R fc d [F bc d ] fg (5.6) [Fd cb ] ef (R fc d ) 1 [Fd bc ] fg (5.7) 50

59 5.2 Quntum double Since the vcuum brids trivilly, we only need to find the R-symbols with the two upper indices equl to 1, so only R 0 = R0 11 nd R 1 = R1 11 corresponding to pir of Fiboncci nyons hving trivil or non-trivil totl chrge respectively. We find R 0 = e 4πi/5, R 1 = e 3πi/5 (5.8) We cn red off the quntum dimension d 1 = from F, nd find tht then nontrivil prticle hs spin θ 1 = fb 1 (R0 11) = e 2πi 2 5. Now let N0 n = N 0 1 n be the dimension of the topologicl Hilbert spce of n Fiboncci nyons with trivil totl chrge. It is esy to see tht N 1 0 = 0, N 2 0 = 1, N 3 0 = 1, N 4 0 = 2, N 5 0 = 3 (5.9) when looking t the number of fusion sttes explicitly. In generl, the N0 n obey simple recursion reltion. If the first two prticles hve trivil totl chrge, the n 2 other prticles cn fuse in N0 n 2 different wys. When the chrge of the first two prticles is 1, the other n 2 chrges cn combine with this chrge in N0 n 1 wys. So the mulitplicities obey the recursion reltion N n 0 = N n N n 2 0 (5.10) which is well known to produce the Fiboncci numbers. Indeed, one recognises the first of these in the list 5.9 This is why these nyons re given their specific nme. 5.2 Quntum double Using the fcts bout the quntum double D[H] nd its representtions from chpter 3, it is possible to deduce fusion rules, F -symbols nd R-symbols, nd thus to obtin description of the fusion briding properties of the excittions in discrete guge theories in the lnguge of the previous chpter. The connection with the grphicl formlism nd the representtions of the quntum double is s follows. Digrms correspond to opertors tht commute with the ction of D[H] in the internl spce, i.e. opertors tht re guge invrint opertors nd commute with flux projections. By Schurs lemm, these must be proportionl to the identity for irreps. The identity opertor of the crrier spce Vα A is represented by chrge line lbelled by = (A, α) directed upwrds. The splitting vertex lbelled by = (A, α), b = (B, β) on top nd c = (C, γ) on the bottom is identified with the inclusion of Vγ C in Vα A Vβ B, up to scle fctor. The dul fusion vertex, on the other hnd, is ssocited with the projection onto Vγ C s irreducible subspce of Vα A Vβ B. If there re multiple copies of Vγ C present in Vα A Vβ B, we cn lbel them with indices µ. Note tht unitry trnsformtion on the µ s in µ (Vγ C ) µ leves the representtions in tct. The quntum dimension nd quntum trce correspond to the usul dimension nd trce 51

60 5. EXAMPLES OF ANYON MODELS of the modules V A α. identity, µ c b inclusion, c µ b projection (5.11) Throughout this section, it will be very convenient to condense the nottion, s the discussion bove lredy shows. We gree to write, b, c,... for the prticle-lbels (A, α), (B, β), (C, γ),.... To void confusing the internl spce Vα A with the fusion/splitting spces, we will lso refer to it s, nd similr for the other prticle lbels. The representtion mps re denoted Π = Π A α, etceter. We cn lbel the bsis sttes of by indices i = 1,..., d. We denote the bsis sttes of therefore s {, i }, i = 1,..., d (5.12) leving the prticle lbel explicit, in sted of h j, α v k s we did before. The coordinte functions of the representtion mtrices re denoted Π ij. The fusion product b mens the tensor product representtion Fusion rules The fusion coefficients Nb c, corresponding to the decomposition b = Π Π b = c Π c c N c b c (5.13) cn be obtined by using certin orthogonlity reltions for the chrcters of representtion. These re well known in the cse of groups, but generlize to representtions of the quntum double. It mens tht for unitry representtions, we hve 1 H Tr (Π (P h g)) Tr h,g ( Π b (P h g) ) = δ b (5.14) In fct, more generl orthogonlity reltions for the coordinte functions ij. They obey Π ij(p h g) Π b kl (P h g) = δ,b δ i,k δ j,l (5.15) h,g H This follows from Woronowiczes theory of pseudogroups [85] nd generlize the well known Schur orthogonlity reltions for groups. We will use them hevily in the clcultion of F -symbols. From the orthogonlity of chrcters, we see tht the fusion multiplicities my be obtined from the representtions mtrices s N c b = 1 H = h,g Tr h,g h h =h ( Π Π b ( (P h g)) i,j Next, we show how to clculte F -symbols. ) Tr (Π c (P h g) ) (5.16) ( ) Π ij(p h g)π b ij(p h g) Π c kk (P h g) k (5.17) 52

61 5.2 Quntum double Computing the F -symbols Consider the digrmmtic eqution for the F -moves: b c e = b c [Fd bc ] ef f f d d (5.18) We see now tht the F -symbols, relte two different wys to embed the irrep d in the triple tensor product b c. We cn either first embed d in e c nd then embed e in b, or we cn strt with embedding d in f nd then embedding f in b c. The two different pths re relted by trnsformtion inside the subspce of chrge d in b c. This trnsformtion is wht the F -moves describe. The F -symbols re conveniently clculted using Clebsch-Gordn coefficients, which describe how the irreps re precisely embedded in tensor product representtions. When there re no fusion multiplicities,nb c = 0, 1, we cn clculte the Clebsch-Gordn symbols using generliztion of the projection opertor technique ( technique well known from the theory of group representtions). We will therefore ssume tht the theory is multiplicity free. This is the cse, for exmple, when H = D 2, the group of unit quternions described in [27], which we took s n exmple to test our clcultions. When there re no fusion multiplicities, there is unique bsis { c, k }, N c b = 1, k = 1,..., d c (5.19) in b corresponding to the decomposition in irreps. stndrd inner product bsis Of course, there is lso the, i b, j, i = 1,..., d ; j = 1,..., d b (5.20) The Clebsch-Gordn coefficients, sometimes lso clled 3j-symbols, precisely give the reltion between these two. They re defined by c, k = ( ) b c, i b, j (5.21) i j k i,j The inverse of this reltion is written s, i b, j = c,k ( ) c b c, k (5.22) k i j To clculte the ctul coefficients, we will use projectors P ij defined by P ij = d Π H ij(p h g) Π(P h g) (5.23) h,g H where the lst Π(P h g) stnds for the pproprite representtion of the P h g element of the quntum double, depending on wht it is cting on. These projectors, which cn be pplied in ny representtion, ct s P ij b, k = δ,b δ j,k, i (5.24) 53

62 5. EXAMPLES OF ANYON MODELS s consequence of the orthogonlity reltion (5.15). Applying the projector to direct product of two sttes nd using eqution (5.22), gives ( ) P c c b lk, i b, j = c, l k i j = i,j ( c b k i j By using the definition (5.23), this is seen to be equl to P c lk, i b, j = d c H = d c H h,g H h,g H Π c lk (P h g)π(p h g) Π A α, i Π B β, j Π c lk (P h g) h h =h ) ( ) b c i j, i b, j. (5.25) l Π i i (P h g)πb j j (P h g), i b, j, (5.26) where we hve inserted the definition of the comultipliction to ct on the product stte. Equting expressions (5.25) nd (5.26), we obtin ( ) ( ) c b b c k i j i j l Unitrity mounts to = d c H h,g H Π c lk (P h g) ( ) c b = k i j Now pick some triple (i, j, k) such tht d c H h,g H Π c kk (P h g) h h =h h h =h Π i i (P h g)πb βj j(p h g) (5.27) ( ) b c. (5.28) i j k Π ii(p h g)π b jj(p h g) (5.29) is non-zero. From eqution (5.27) with i = i, j = j nd k = k nd from the unitrity condition (5.28) it follows tht this number is rel nd positive. This fixes one of the Clebsch-Gordn symbols, by ( ) b c = d c i j k H h,g H h h =h Π c kk (P h g)π ii(p h g)π b jj(p h g) We cn use eqution (5.27) to clculte ll others, which results in ( ) b c d c i j k = H h,g H h h =h Πc k k (P h g)π i i (P h g)πb j j (P h g) ( ) 1/2 h,g H h h =h Πc kk (P h g)π ii (P h g)πb jj (P h g) 1/2 (5.30) 54

63 5.2 Quntum double From the Clebsch-Gordn symbols it is strightforwrd to clculte the F -symbols. The definitions imply the reltion [ ] ( ) ( ) ( ) ( ) Fd bc b c f f d b e e c d = (5.31) j k n m k l i j m m k l f;n e,f It follows from (5.28) tht ( ) ( ) b c b c i j k i j k = δ cc δ kk (5.32) i,j Together, these give the following expression for the F -symbols in terms of the Clebsch- Gordn coefficients [ Fd bc ]e,f = ( ) ( ) ( ) ( ) b c f f d b e e c d (5.33) j k n m k l i j m m k l i,j,k,m,n (Note tht l cn be chosen freely.) In principle, one cn now run computer progrm to clculte the Clebsch-Gordn coefficients nd subsequently the F -symbols from the bove description Briding Recll tht the briding for the quntum double is defined through the ction of the universl R-mtrix R = P g e P h g (5.34) h,g H nd subsequent flipping of the tensor legs. Becuse (P h g)r = R (P h g), the effect of briding on irreducible subspces c of b b is in principle just multipliction by complex number Rc b by Schur s lemm. But if some c occurs in b with multiplicity greter thn one, it cn in fct ct s unitry trnsformtion [Rc b ] µν in the subspce of totl chrge c. We cn derive the R-symbols by composing the briding briding with the inclusion of c. Denote the representtion mtrix of the ction of R on stte, i b, j s R,b (ij),(i j ), i.e. R,b (ij),(i j ) Π ii Πb jj (R) (5.35) For the simple cse where there re no multiplicities of c, we get R b c = d d b i,i =1 j,j =1 ( b c i j l ) ( ) b c j i R,b l (ij),(i j ) (5.36) where gin l cn be chosen freely. If multiplicities of c do occur, we cn still obtin the [Rc b ] µν by first including the µ th copy of c nd next project onto the ν th copy. This cn be done by introducing more 55

64 5. EXAMPLES OF ANYON MODELS generl Clebsch-Gordn coefficients tht lso crry n index µ tht keeps trck of the copy of c. The formul then becomes [R b c ] µν = d d b i,i =1 j,j =1 5.3 The su(2) k theories ( b c, µ i j l ) ( c, ν b l j i ) R,b (ij),(i j ) (5.37) The dt for su(2) k theories, which were given this nme becuse they pper in the WZW-models bsed on the ffine lgebr of su(2) t level k, cn be derived from the representtion theory of U q [su(2)] where q is root of unity nd relted to k by s q = e i 2π k+2. As for generl vlues of q, similr structure for the representtions ppers s in the representtion theory of su(2), with integers replced by the q-numbers [n] q = qn/2 q n/2. But, when considering the pproprite ctegory of representtions, q 1/2 q 1/2 the highest vlues for the highest weights get truncted. There re only finite number of irreducible representtions corresponding to the finite set of chrge lbels. We will stte the generl formuls s they ppered in [72], where the uthors refer to [47] for the clcultion. The chrges of su(2) k re lbelled by integers = 0, 1,..., k tht re relted to the highest weights j s = 2j ( nottion lso frequently used). The fusion rules re given by truncted version of the usul ddition rules for SU(2)-spin b = c N c b c (5.38) = b + ( b + 2) + + min{ + b, 2k b} (5.39) i.e. Nb c = 1 when b c min{ + b, 2k b} nd + b + c = 0 (mod 2), nd zero otherwise. For the F -symbols, one hs the generl formul { [Fd bc ] ef = ( 1) (+b+c+d)/2 b e [e + 1] q [f + 2] q c d f } (5.40) where { } b e = (, b, e) (e, c, d) (b, c, f) (, f, d) c d f { ( 1) z [z + 1] q! z [z +b+e 2 ] q![z e+c+d 2 ] q![z b+c+f 2 ] q![z +f+d 2 ] q! 1 [ +b+c+d 2 z] q![ +e+c+f 2 z] q![ b+e+d+f 2 z] q! } (5.41) 56

65 5.3 The su(2) k theories with (, b, c) = [ +b+c 2 ] q![ b+c 2 ] q![ +b c 2 ] q! [ +b+c 2 + 1] q, nd [n] q! = n [m] q (5.42) The sum over z should run over ll integers for which the q-fctorils re well defined, i.e. such tht no of the rguments become less thn zero. This depends on the level k. The expression for is only well defined for dmissible triples (, b, c), by which we men tht + b + c = 0 (mod 2) nd b c + b (we will tke it to be zero for other triples, implementing consistency with the fusion rules). Note tht is invrint under permuttions of the input. The R-symbols re given by the generl eqution This gives topologicl spins The quntum dimensions re m=1 R b c = ( 1) c b q 1 8 (c(c+2) (+2) b(b+2)) (5.43) (+2) 2πi θ = e 4(k+2) (5.44) ( ) sin (+1)π k+2 d = ( ) (5.45) sin π k+2 For lter reference, we will work out the vlues of these functions for some representtive vlues of k. k=2 The simplest non-trivil cse occurs for k = 2 which is closely relted to the Ising model. The chrges re denoted 0, 1, 2, with non-trivil fusion rules The relevnt q-numbers nd fctorils re 1 1 = 0 + 2, 1 2 = 1, 2 2 = 0 (5.46) n [n] q [n] q! (5.47) This gives quntum dimensions nd spins su(2) 2 0 d 0 = 1 h 0 = 0 1 d 1 = 2 h 1 = d 2 = 1 h 2 = 1 2 (5.48) Using the generl formul, we cn clculte the F -symbols. Let us clculte [F1 111 ] 00. The relevnt s re [0] q![0] q![1] q! (1, 1, 0) = (1, 0, 1) = (0, 1, 1) = = 1 [2] q! 2 1/4 (5.49) 57

66 5. EXAMPLES OF ANYON MODELS This gives { } = 1 2 z ( 1) z [z + 1] q! ([z 1] q!) 4 [2 z] q![1 z] q! = 1 [2] q! 2 ([0] q!) 4 [1] q![0] q! = 1/ 2 (5.50) which produces By similr clcultions we find [F ] ef = [F ] 00 = 1/ 2 (5.51) ( ) 1/ 2 1/ 2 1/ 2 1/, e, f = 0, 2 (5.52) 2 k=4 For su(2) 4 hs been very importnt exmple while working on this thesis. Since the primry gol hs been to study Bose condenstion, it is of prticulr interest becuse it is one of the most simple models with non-trivil boson. The chrges of su(2) 4 re lbelled by 0, 1, 2, 4 with the fusion rules given by the generl formul, for exmple The relevnt q-numbers nd q-fctorils re 2 2 = , 1 3 = (5.53) n [n] q [n] q! (5.54) This gives spins nd quntum dimensions su(2) 4 0 d 0 = 1 h 0 = 0 1 d 1 = 3 h 1 = d 2 = 2 h 2 = d 3 = 3 h 3 = d 4 = 1 h 4 = 1 (5.55) In chpter 6 we need some specific vlues for the F -symbols, hence we clculte these here. These re [F4 242 ] 22, [F0 242 ] 22, [F2 242 ] 22, or ctully [F22 22] 40, [F22 22] 42 nd [F22 22] 44. The relevnt s re (2, 2, 2) = 1/ 6 (2, 2, 0) = 1/ 2, (2, 2, 4) = 1/ 2 (5.56) 58

67 5.3 The su(2) k theories We get [F ] 22 = 2 (2, 4, 2) (2, 2, 4) (4, 2, 2) (2, 2, 4) (5.57) ( 1) z [z + 1] q! ([z 4] z q!) 4 [6 z] q![4 z] q![6 z] q! (5.58) = [5] q! ([2] q!) 2 (5.59) = 1 (5.60) Similrly, [F ] 22 = 2 (2, 4, 2) (2, 2, 0) (4, 2, 2) (2, 2, 0) (5.61) ( 1) z [z + 1] q! ([z 4] z q!) 2 ([z 2] q!) 2 ([4 z] q!) 3 (5.62) = [5] q! ([2] q!) 2 (5.63) = 1 (5.64) nd finlly, [F ] 22 = 2 (2, 4, 2) (2, 2, 2) (4, 2, 2) (2, 2, 2) (5.65) ( 1) z [z + 1] q! ([z 4] z q!) 2 ([z 3] q!) 2 ([5 z] q!) 2 ([4 z] q!) (5.66) = [5] q! ([0] q!) 3 ([1] q!) 4 (5.67) = 1 (5.68) By [F b cd ] (e,α,β)(f,µ,ν) = d e d f d d d [F ceb f ] (,α,µ)(d,β,ν) (5.69) this gives [F22 22 ] 40 = 1 22, [F22 ] 42 = 1, [F22 22 ] 44 = (5.70) k=10 Also su(2) k ws of prticulr interest to us. While hving resonbly smll number of chrges eleven it is much more intricte exmple thn su(2) 4. There is non-trivil bosonic chrge, lbelled 6 nd the result of condenstion ws known. Hence it presented fine test cse for the theory of subsequent chpters. 59

68 5. EXAMPLES OF ANYON MODELS We list the quntum dimensions nd spins here. su(2) 10 0 d 0 = 1 h 0 = 0 1 d 1 = s + 3 h 1 = d 2 = h 2 = d 3 = h 3 = d 4 = h 4 = d 5 = h 5 = d 6 = h 6 = 1 7 d 7 = h 7 = d 8 = h 8 = d 9 = h 9 = d 10 = 1 h 10 = 5 2 (5.71) The fusion rules nd F -symbols cn be clculted using the generl formule. Clculting the F -symbols by hnd becomes very cumbersome, hence we implemented the formule in Mthemtic to do clcultions on su(2) 10 nd generl su(2) k models. 60

69 CHAPTER 6 Bose condenstion in topologiclly ordered phses In this chpter, we strt the study of phse trnsitions between topologiclly ordered phses, which is the min gol of this thesis. The theory developed in the previous chpters is used to extend the formlism for topologicl symmetry breking phse trnsitions. This should llow one to construct full digrmmtic formlism for topologicl symmetry breking, such tht rbitrry digrms for the symmetry-broken phse cn be clculted, using dt from the originl theory only. We strt with short review of the topologicl symmetry breking pproch, which is bsed on quntum group symmetry breking by bosonic condenste. The min ide is tht mechnism of symmetry breking, very similr to spontneous symmetry breking known from the ordinry theory of phse trnsitions, cn ccount for mny trnsitions between topologicl phses, only the symmetry is given by quntum group. This ws first explored in the context of discrete guge theories in [5, 8]. Lter, the formlism ws refined nd pplied to nemtic phses in liquid crystls in [7, 12, 61]. These erlier pproches hd the drwbck tht they were not pplicble to tret cses most interesting for the FQHE, where non-integer quntum dimensions occur. In [9], generl scheme ws lid out tht cn lso hndle these cses. An importnt feture of this generl scheme is tht the explicit description of the quntum group disppers nd everything is done on the level of the representtions, or prticle lbels. In mthemticl terminology, on might sy tht the representtion ctegory of the quntum group is put centerstge while the quntum group itself is put to the bckground. It could therefore be expected tht there is connection with the mthemticl literture on (modulr) tensor ctegories. The generliztion of the notion of subgroup to the pproprite ctegory theoreticl context is commuttive lgebr, s rgued in [48]. The results of [9] nd [48] re in perfect greement, lthough the lnguge is very different in mny respects. This motivted the study of the mthemticl literture to see wht physiclly meningful 61

70 6. BOSE CONDENSATION IN TOPOLOGICALLY ORDERED PHASES results could be extrcted. 6.1 Topologicl symmetry breking We outline the formlism of topologicl symmetry breking below. This is described in detil in [9]. The topologicl symmetry breking scheme hs two steps, or three stges. We strt with (2+1) dimensionl system exhibiting topologicl order with underlying quntum group symmetry lbelling the topologicl sectors. This is the unbroken phse described by the quntum group or theory A (we use the nottion A for both the quntum group nd its ctegory of representtions, if no confusion is likely to rise). Figure 6.1: Topologicl symmetry breking scheme - Topologicl symmetry breking hs two steps, or three stges. The first step is the restriction of the quntum group A to the intermedite lgebr T A. The second step is the projection onto the Hopf quotient U of T. Figure from [9]. The first step of the topologicl symmetry breking scheme is the formtion of bosonic condenste. For this to hppen, the prticle spectrum of the A-theory should of course include bosonic excittion. The formtion of condenste reduces the symmetry A to some sublgebr T of A which cn be thought of s the stbilizer of the condenste. This T-lgebr represents the second stge of the symmetry breking scheme. The precise definition of T hs been discussed for quntum double theories nd generliztions thereof [5, 7, 8, 12, 61]. Since quntum groups re not groups, some subtle issues occur. Which of the structures of A must be inherited by T? It is priori uncler if T hs the full structure of quntum group. In fct, llowing the freedom tht T does not give consistent briding gives rise to one of the min benefits of the quntum group pproch, the description of excittions on the boundry of the broken phse or interfce between two phses. When T does not llow for the definition of universl R-mtrix there is no consistent briding of the excittions of the T-theory. This is cused by excittions tht 62

71 6.1 Topologicl symmetry breking do not brid trivilly with the condenste nd thereby loclly destroy it. These excittions pull strings in the condenste with energy proportionl to their length. In order to minimize the energy, the excittions tht pull strings will be confined, either like T-hdrons or on the boundry of the system. The intermedite T-lgebr thus provides nturl description of the boundry excittions. If we think of geometry where droplet of the system in the broken phse is surrounded by region in the unbroken phse, s in figure 6.2, the T-theory gives description of the interfce between the two phses. In the second step of the scheme, we project out the confined excittions to obtin description of the bulk theory. On the level of the quntum group this corresponds to the surjective Hopf mp from T to some Hopf quotient U. To describe bulk nyons, U should give rise to consistent briding nd ll other fetures of the tensor ctegories we described in chpter 4. Figure 6.2: Two-phse system - The figure system in the unbroken phse I described by A tht contins droplet in the symmetry broken phse II described by U. The excittions on I/II interfce re nturlly described by the intermedite lgebr T Prticle spectrum nd fusion rules of T Let us describe the formlism in some more detil. To look for boson b in the spectrum of A, which we lbel by Greek index for future convenience, we must first know wht we men by bosonic excittion in this context. An obvious requirement is trivil spin, expressed by θ b = e 2πih b = 1 h b Z (6.1) But in order to form stble condenste, there must be multi-prticle stte of b s tht is invrint under briding. Unlike the higher dimensionl cse, this is not gurnteed 63