Topological Quantum Computation

Transcription

1 Topologicl Quntum Computtion Shwn X. Cui August 15, 2018 Abstrct This is the note for the 10-lecture course Topologicl Quntum Computtion (TQC) I tught t Stnford University during Spring The course is imed t bsic introduction to TQC, with focus on the mthemticl side of the theory. Topics include toric code, quntum double model of finite groups, non-abelin nyons, brid groups, modulr tensor ctegories, Jones polynomil, etc. In prticulr, it contins creful tretment of ribbon opertors in the quntum double model. Ech section below covers the mteril for one lecture. Plese be wre tht the note hs not been proofred, so it my contin typos/errors. Use it t your cution. Contents 1 Topologicl quntum computtion: n overview Some Bsics of Quntum Computtion (QC) Error Correcting Code Topologicl Quntum Computing (TQC) Toric Code Ground Sttes Excittions Quntum Double of Finite Groups Representtions of Finite Groups Quntum Double Representtions of DG Kitev s Quntum Double Model The Hmiltonin of the Model Ground Stte Spce of the Model (Optionl) Excittions of the Model (Summry) Emil: cuixsh@gmil.com 1

2 5 Ribbon Opertors 25 6 Briding nd Fusion 32 7 Quntum Computing with Kitev s Model 39 8 Unitry Modulr Tensor Ctegory(UMTC) 46 9 Unitry Modulr Tensor Ctegory(UMTC) II SU(2) k nd Jones Polynomil Jones Polynomil SU(2) k A Appendix 69 A.1 Homework A.2 Homework Topologicl quntum computtion: n overview 1.1 Some Bsics of Quntum Computtion (QC) Let s strt with some bsics ingredients in quntum computtion. By qubit is ment 2-dimensionl Hilbert spce C 2 with preferred orthonorml bsis { 0, 1 }. In generl, one cn lso tlk bout qudit, which is the d-dimensionl Hilbert spce C d for some integer d > 2. A (1-qubit) quntum stte is non-zero vector ψ C 2. Usully, we normlize ψ so tht it hs norm ψ ψ = 1. The spce of n-qubits is represented by the n-fold tensor product (C 2 ) n, nd n n-qubit stte is non-zero vector in (C 2 ) n. Quntum gtes re opertors tht trnsform quntum sttes. An n-qubit quntum gte is unitry opertor U U(2 n ). The following re some 1-qubit gtes. They re the Puli mtrices. X = ( ) Z = ( ) Y = ixz = ( 0 i i 0 A mesurement is probbilistic opertion on sttes. Given normlized 1-qubit stte ψ = 0 + b 1, mesuring it (with respect to the stndrd bsis) results in the stte 0 with probbility 2 nd the stte 1 with probbility b 2. More generlly, we cn mesure sttes with respect to n observble O, which is Hermitin opertor cting on certin n-qubits, tht is, O = O. The eigenvlues of O re ll rel nd we hve the spectrl decomposition: ) O = i λ i P i, (1) where P i is the projector onto the λ i -eigenspce. Then if we mesure n n-qubit stte ψ with respect to O, the probbility to get the outcome λ i is ψ P i ψ, nd when the outcome 2

3 is λ i, the resulting stte becomes P i ψ. With this nottion, the mesurement with respect to the stndrd bsis is equivlent to the mesurement with respect to the observble Z. It is direct to see tht the P i s in Eqution 1 stisfy P i = P i, P i P j = δ i,j P i, P i = Id. (2) A set of opertors {P i } stisfying Eqution 2 is clled complete set of projectors. Given n rbitrry complete set of projectors {P i }, one cn define n observble O by Eqution 1 by choosing some mutully distinct rel numbers {λ i }. Thus n equivlent formultion of mesurement is in terms of projectors. Tht is, given complete set of projectors P = {P i }, mesurement of ψ with respect to P will project ψ to P i ψ with probbility ψ P i ψ. The process of quntum computtion is illustrted s follows: ψ 0 M? ψ 1 ψ 2 ψ 3 U 0 U 1 U 2 U 3 U 4 U 5 U 6 i preprtion gtes mesurement In the circuit bove, the flow proceeds from left to right with time. Ech wire represents qubit, ech box represents quntum gte, nd M mens the mesurement of the first qubit with respect to the stndrd bsis. In ctul physicl systems, the qubits will lwys interct with the environment which introduces noises (or errors). Another type of errors hppens when pplying quntum gtes. Any gte cn only be designed to certin ccurcy. There re two pproches to del with errors. The softwre pproch is by using error correcting codes, nd the hrdwre pproch is topologicl quntum computing (TQC). 1.2 Error Correcting Code We will not go into much detil on error correcting code, but will only sketch the generl ide. For references on this spect, see for instnce Chpter 10 [16]. The key to error correction is distinguishing the notion of logicl qubits with physicl qubits. We encode logicl qubit into subspce of multiple physicl qubits. Under certin ssumptions, there is wy to detect errors nd ccording to different syndromes we cn even correct them. For instnce, ( toy exmple), consider the encoding of logicl qubit into three physicl qubits: C 2 C 2 C 2 C L := L := 111 3

4 outcome mesuring Z 1 Z 2 mesuring Z 2 Z 3 no error: b error X 1 : b error X 2 : b error X 3 : b Tble 1: Outcomes of mesurements Thus generl logicl stte is ψ L = 0 L + b 1 L. Assume ech time the error will only hppen to one of the three physicl qubits (we do not know which, priori) nd the only type of error is bit flip, nmely, the opertor X, then we cn detect nd correct n error s follows. Let Z i be the Puli Z cting on the i-th qubit. It is cler tht Z 1 Z 2 hs eigenvlue 1 on the subspce spn{ 00, 11 } nd eigenvlue 1 on spn{ 01, 10 }. Hence mesurement of Z 1 Z 2 serves to check if the two qubits re ligned with the sme direction. To detect if there is ny error, we mke two mesurements, one with respect to Z 1 Z 2 nd the other with respect to Z 2 Z 3. The outcomes of the mesurements for ech possible error re listed in Tble 1. From the tble, we see tht the outcomes of mesurements re different for ech possible error (nd the cse with no error). Hence, bsed on the mesurement outcome, we know to which qubit, if ny, the error hppens, nd we cn simply correct it by pplying the X opertor to tht qubit. Of course, this is only toy model. More complicted encoding is required in order to correct other types of errors nd lso errors involving more thn one qubit. For instnce, the following nine-qubit code corrects rbitrry single qubit errors. See Chpter 10.2 [16]. C 2 (C 2 ) L := ( ) L := ( ) Topologicl Quntum Computing (TQC) TQC is n pproch to relizing quntum computing with non-abelin nyons/qusi-prticles in certin two dimensionl quntum systems. The informtion is encoded in non-locl degrees of the system mking it fult-tolernt to locl errors. The process of informtion is chieved by briding of nyons, which effects unitry trnsformtion cting s quntum gtes. Mesurement of sttes is performed through fusing nyons. In terms of computtionl power, TQC is equivlent to the stndrd circuit model, but the former hs the dvntge of being utomticlly fult-tolernt to locl errors. The quntum medium to crry out TQC is two dimensionl topologicl phse of mtter (TPM), the content of the 2016 Nobel Prize in physics. Roughly, TPM is quntum system with certin robust properties tht depends on the topology of the underlying mteril. TPMs cn be found in certin frctionl quntum Hll sttes. As concrete exmple, think of plne of electrons subject to strong mgnetic field in the verticl direction where the temperture is lowered to close to bsolute zero degree. See Figure 1. An importnt feture 4

5 B Figure 1: A collection of electrons subject to the plne. The nyons re point-like excittions. of TPM is tht it hrbors nyons/qusi-prticles. They re low energy, point-like excittions. They cn be loclly moved, but cn not be loclled destroyed or creted. Ech nyon hs topologicl chrge or type. The sttistics for exchnging two nyons re more generl thn bosons/fermions. Consider n nyons seprted in spce nd ech of them hs type. Denote the spce of sttes with such configurtion by V. There is n energy gp E > 0 between such configurtion nd other spectrl. Hence the system will evolve within itself s long s the locl perturbtions re smll enough. If dim V = 1, we cll the nyon type Abelin. Otherwise, it is clled non-abelin. For non-abelin type, we cn encode informtion in V, which is not ccessible by locl errors. When two nyons re swpped, the system undergoes unitry trnsformtion. See Figure 2. The world lines of the swp produce brid digrm. We cll such process briding of nyons. The unitry trnsformtion only depends on the isotopy clsses of the brid digrm, which mens the only wy the system evolves is by briding nyons. These unitry trnsformtions ct s quntum gtes nd they form representtion of the n-strnd brid group, which is defined by the presenttion: B n = σ 1,, σ n 1 σ i σ i+1 σ i = σ i+1 σ i σ i+1, σ i σ j = σ j σ i, i j > 1. If we bring two nyons close to ech other nd fuse them, some other nyon will emerge, nd there re different possible outcomes. Abstrctly, we denote the fusion by: b = c, d, This corresponds to projection to different superselection sectors, nd cn be used to perform mesurement. The generl process of TQC is illustrted in Figure 3. We strt by creting some nyons from the vccum to initilize the stte. Then we brid nyons to trnsform the stte. Finlly we fuse the nyons bck to vccum. The study of TQC is closely relted to number of subjects, such s topologicl phse of mtter, topologicl quntum field theories, modulr tensor ctegories, knot invrints, etc. We will touch some of the reltions lter in the course. 5

6 time Figure 2: Briding of two nyons vcuum fusion U ψ 0 briding ψ 0 cretion vcuum Figure 3: The process of TQC. 6

7 X Xv X X Z Z p Z Z 2 Toric Code Figure 4: A L L squre lttice on the torus Toric code [14] is n exctly solvble lttice model defined on closed surfce, i.e., surfce without boundry. It hs gpped Hmiltonin whose low energy excittions re nyons. Although ll of the nyon types in toric code re Abelin nd hence re not useful for TQC, the toric code is beutiful theory tht illustrtes mny concepts of topologicl phses. Thus it is worth studying the model to some detils. Lter, we will generlize it to Kitev s quntum double model for ny finite groups, where non-abelin nyons cn emerge. Recll tht the Puli mtrices re defined s follows. X = ( ) Z = ( ) Y = ixz = ( ) 0 i i 0 Let L be squre lttice of size L L with periodic boundries both in the horizontl direction nd in the verticl direction. See Figure 4. Nmely, we identify the two horizontl boundries s well the two verticl ones. In nother word, L is lttice on the torus. Remrk 2.1. The Hmiltonin model to be introduced below cn be defined on lttices of ny shpe on ny closed surfce. Here we simply choose the squre lttice on the torus s n illustrtive exmple. Denote by V = {vertices}, E = {edges}, F = {fces or plquettes}. To ech edge e E ssocite qubit H e = C 2. The totl Hilbert spce H tot is defined s H tot := e E H e. To define the Hmiltonin, we first introduce some locl opertors. To ech v V nd p F, define A v := X e ( ) ( ) Id e, B p := Z e Id e, e str(v) e E str(v) e p e E p 7

8 where str(v) is the set of edges djcent to v, p is the set of edges on the boundry of p, nd X e (resp. Z e ) is the opertor X (resp. Z) cting on the qubit H e. Grphiclly, A v nd B p re shown in Figure 4. The following properties re esy to check. For ny v, v V nd p, p F, we hve A 2 v = Id, B 2 p = Id (3) A v A v = A v A v, B p B p = B p B p, A v B p = B p A v. (4) This mens tht ll the A v s nd B p s mutully commute with other, nd hence they cn be digonlized simultneously. Moreover, ech A v nd ech B p hs eigenvlue either 0 or 1. The Hmiltonin is defined s H := v V (1 A v ) + p F (1 B p ). (5) Since ll the terms in the Hmiltonin re Hermitin opertors tht commute with ech other, the Hmiltonin cn be solved exctly. Moreover, s it will be shown below, the Hmiltonin is frustrtion free, nmely, the ground sttes re chieved s the common eigensttes of ll A v s nd B p s with eigenvlue one. Denote the ground sttes mnifold by V gs, then V gs = { ψ H tot : A v ψ = ψ, B v ψ = ψ, for ll v V, p F }. (6) 2.1 Ground Sttes Now we compute the degenercy of the ground sttes. Let L be the dul lttice of L s shown in Figure 5, where solid lines represent the lttice L nd dshed lines L. Then vertex v, edge e, nd plquette p in L correspond to the plquette v, edge e, nd vertex p, respectively, in L. A bsis element of H tot is n ssignment {x e : e E} of 0 or 1 to ech edge. Given bsis element, we construct grph G in L s follows. Initilly, G hs the sme set of vertices s those of L with no edges. For ech edge e E, if x e = 1, then we dd the dul edge e to G. See Figure 6. By this procedure, we hve one-to-one correspondence between bsis elements of H tot nd grphs in L. Denote by G the bsis element corresponding to the grph G. Note tht G is n eigenstte of B p for ech p, nd it hs eigenvlue 1 if nd only if the vertex of G inside p hs even degree. Thus, n equivlent condition for G to be n eigenstte with eigenvlue 1 for ll B p s is tht G hs even degree t ll vertices. Tht is, G is collection of loops in L, or in nother word, multi-loop. See Figure 7 for locl configurtion of such G. The ground sttes mnifold V gs is subspce of the spce spnned by bsis elements corresponding to multi-loops. Now we look t the ction of A v on multi-loop element G. At ech edge djcent to v, A v flips 0 nd 1. Denote by v the plquette in L dul to v. Then A v G corresponds to the grph G which is obtined from G by replcing the edges of G tht belong to v with their complement in v. See Figure 8. Hence G is obtined by deforming G through the dul plquette v, nd we sy G nd G re homologous. In generl, for rbitrry two 8

9 Figure 5: The lttice L (solid gry lines) nd the dul lttice L (dshed gry lines). Figure 6: An exmple of grph G in the dul lttice. The bsis element G hs component 1 on edges whose dul belongs to G nd hs component 0 otherwise. Figure 7: A locl picture of multi-loop round ech dul vertex. 9

10 v A v v G G Figure 8: The ction of A v on multi-loop. multi-loops G 1 nd G 2, they re clled homologous to ech other if one is obtined from the other by sequence of deformtions through dul plquettes. This defines n equivlence reltion on the set of multi-loops. For ech equivlence clss [G], define the stte [G] := G. It is direct to check tht G [G] A v [G] = [G], for ny v V, since A v simply permutes the terms in the summtion of [G]. Hence [G] is ground stte. Apprently, the sttes [G] corresponding to different equivlence clsses of multi-loops re linerly independent. It is not hrd to see tht they ctully spn ll of V gs. In summry, bsis of V gs is given by { [G] : [G] equivlence clss of multi-loops}, nd the dimension of V gs is the number of equivlence clsses. In the lnguge of homology, multi-loops re 1-cycles (with Z 2 coefficients). Two multiloops re homologous or equivlent if they belong to the sme homology clss. Equivlence clsses of multi-loops re homology clsses 1-cycles, nd thus the dimension of V gs is equl to H 1 (torus; Z 2 ), the number of elements in the first homology of the torus with Z 2 coefficients. One cn lso use the notion of cohomology, but the two notions re equivlent by Poincre dulity. On the torus, there re four equivlence clsses of multi-loops. They re represented by {[ ], [m], [l], [d]}, where (See Figure 9) is the empty set or ny contrctible loop, m is the horizontl loop, l is the verticl loop, nd d is the digonl loop. Note tht d is in the sme equivlence clss s the multi-loop which is the union of m nd l. Also note tht ll the loops re on the dul lttice. Thus on the torus, the ground stte degenercy is four. On the sphere, the ground stte degenercy is 1 since ll multi-loops re contrctible. 2.2 Excittions We hve seen tht ground stte ψ is defined by the following constrints: A v ψ = ψ, v V, (7) B p ψ = ψ, p F. (8) An elementry excittion or 1-prticle excittion is stte ψ for which t most one constrint in Eqution 7 is violted. More generlly, n n-excittion corresponds to ψ for which t most n constrints re violted. 10

11 l d m Figure 9: A representtive for ech equivlence clss of multi-loops. 0 t 1 t t t 0 t 1 t Figure 10: Connected pths (strings) in the lttice nd dul lttice. The A v s nd B v s re not independent opertors, since they stisfy A v = Id, B p = Id. (9) v V Therefore, it is impossible to hve exctly one A v or one B p violted, which mens singleprticle excittions do not exist, or rther, the only single-prticle excittion is the vcuum. Now we consider two-prticle excittions. Let t be connected pth in the lttice L nd denote the two end points of t by 0 t nd 1 t. Define S Z (t) := Z e. (10) p F e t E See Figure 10. Clerly S Z (t) commutes with ll B p s nd ll A v s except the two vertex opertors t the end points of t where they nti-commute: S Z (t)a i t = A i ts Z (t), i = 0, 1. Let E be ground stte nd let ψ Z (t) = S Z (t) E. Then ψ Z (t) violtes exctly two constrints, one t 0 t nd one t 1 t, since A i t ψ Z (t) = A i ts Z (t) E = S Z (t)a i t E = ψ Z (t). Hence we hve pir of prticles locted t the vertices 0 t nd 1 t. We cll them qusiprticles of Z-type or electric chrges. These qusi-prticles hve energy 4. The opertor 11

12 S Z (t) is clled string opertor of Z-type. While the opertor itself depends on the string t, the stte ψ Z (t) is unchnged s we deform t keeping the end points fixed. Hence, only the isotopy clss of strings mtter to the stte ψ Z (t). Similrly, we cn consider pth t in the dul lttice L. The two end points, 0 t nd 1 t correspond to two plquettes in the originl lttice. See Figure 10. Define the string opertor of X-type by S X (t ) = X e. (11) e E (t ) Then S X (t) commutes with ll A v s nd ll B p s except t the two plquettes ( 0 t ) nd ( 1 t ), where they nti-commute. Let ψ X (t ) = S X (t ) E. Then ψ X (t ) represents pir of qusi-prticles t the plquettes ( 0 t ) nd ( 1 t ). They re clled qusi-prticles of X-type or mgnetic chrges, nd they hve energy 4. Agin the ψ X (t ) only depends on the isotopy clss of the pth t. Thus, there re two types of qusi-prticles. The electric chrges live on vertices while mgnetic chrges live on plquette. There is nother type of qusi-prticle which is the composite of n electric chrge nd mgnetic chrge. Note tht this composite hs energy 8, nd it occupies plquette nd vertex on it. If we think of the vcuum s prticulr type of qusi-prticle, we hve in totl four types of qusi-prticles. (Note tht the degenercy of the ground sttes on the torus is lso four; this is not coincidence.) We could hve rbitrrily even number of qusi-prticles of ech type. To obtin this configurtion, we simply connect ech pir of qusi-prticles of the sme type by string. The spce of ech configurtion hs dimension exctly 4. However, the degenercy rises purely due to the degenercy of the ground sttes on the torus. If the lttice is on sphere, then there will no degenercy. Hence, ll the qusi-prticles re Abelin. The only topologicl degree of freedom is the chrge configurtion nd the only locl degrees re phse fctors. Nonetheless, let s tke look t their briding sttistics. Consider pir of electric chrges nd pir of mgnetic chrges ψ = S Z (t 1 )S X (t 2 ) E (See Figure 11). If we move n electric chrge round mgnetic chrge, then resulting stte would be ψ = S Z (t 1 t)s X (t 2 ) E. Thus, ψ = S Z (t 1 )S Z (t)s X (t 2 ) E = S Z (t 1 )S X (t 2 )S Z (t) E = ψ, where we hve used the fct tht S Z (t) cts on the ground stte by identity for ny contrctible loop t. So by drgging n electric chrge round mgnetic chrge, the stte chnges by globl phse (ctully only minus sign). This is the nture of Abelin qusiprticles. In the cse of non-abelin qusi-prticles, s we will see lter, the briding process cn produce nontrivil trnsformtions in some higher dimensionl spce. Finlly, to fuse two prticles of the sme type, we simply drw nother string connecting them to form closed loop. The result would lwys be the vcuum. This implies ny qusi-prticle is its own ntiprticle. 12

13 t t 1 t 2 Figure 11: Drgging n electric chrge round mgnetic chrge. 3 Quntum Double of Finite Groups 3.1 Representtions of Finite Groups We strt with some bsics on representtion theory of finite groups. Let G be finite group whose identity element is denoted by e. A unitry representtion of G is pir (V, χ), where V is finite dimensionl Hilbert spce, nd χ is group morphism χ : G U(V ), where U(V ) is the group of unitry trnsformtions on V. For g G, v V, when no confusion rises, we usully write χ(g)(v) simply s g.v nd cll it the ction of g on v. We lso sy the group G cts on the spce V by the representtion χ. Sometimes, we lso cll χ itself the representtion, nd denote the corresponding Hilbert spce by V χ. Finitely, denote by χ := dim V. Let { j : j = 1,, χ } be n orthonorml bsis of V, nd denote the mtrix elements of g by Γ χ ij (g), nmely, χ g. j = Γ χ ij (g) i. i=1 In the following, when tlking bout representtions, we lwys ssume bsis for the Hilbert spce hs been chosen nd hence ech group element corresponds to unitry mtrix. We cll representtion (V, χ) irreducible (or n irrep, for short), if V does not contin proper subspce except 0 which is invrint under the ction of G. Every group hs trivil irrep which is one dimensionl nd ll elements of the group ct by the identity opertor. Denote this trivil irrep by 1. Let s look t some exmples. Let Z d = {0, 1,, d 1} be the cyclic group of d elements. It hs genertor := [1] with order d, nmely, d = e. Let S 3 be the group of permuttions on three elements. It hs two genertors µ = (123) nd σ = (23) with the reltions µ 3 = σ 2 = (µσ) 2 = e. When defining representtion, it suffices to specify the mtrices of genertors of group nd verify the reltions of the genertors re preserved. Z d hs d inequivlent irreps, ll of which re dimensionl. (More generlly, ll irreps of finite Abelin group re one dimensionl.) For ech i = 0, 1,, d 1, there is n irrep, denoted by [ωd i ], mpping the genertor to the 1 1 mtrix (ωi d ), where ω d := e 2π 1 d. 13

14 dimension mtrix of µ mtrix of σ [+] 1 (1) (1) [ ] 1 ( (1) ) ( ( 1) ) ω [2] 2 0 ω Tble 2: Irreps of S 3. S 3 hs three irreps, which we denote by {[+], [ ], [2]}, the mtrices of the genertors µ nd σ for ech irrep re listed in Tble 2. From the tble, we see tht [+] is the trivil irrep 1 nd [ ] is clled the sign irrep s it mps permuttion to its signture. It is fct tht every finite group G hs only finitely mny inequivlent irreps. Denote by Irr(G) the set of ll inequivlent irreps. For ech irrep χ Irr(G), ssume n orthonorml bsis for V χ hs been chosen nd the mtrix elements of g G re given by Γ χ ij (g). The following reltion is well known nd importnt in representtion theory. Shur Orthogonlity Reltion: for ny two irreps χ, χ Irr(G), nd i, j = 1,, χ, i, j = 1,, χ, we hve g G Γ χ ij (g)γχ i j (g) = G χ δ χ,χ δ i,i δ j,j. (12) The orthogonlity reltion hs mny implictions. For instnce, tke χ to be the trivil irrep 1 nd i = j = 1, then Γ χ i j (g) = 1 for ll g, nd hence we hve Γ χ ij (g) = G δ χ,1δ i,1 δ j,1. (13) 3.2 Quntum Double g G The quntum double of finite group G is n lgebr DG with two set of genertors {A g : g G} nd {B h : h G}. The multipliction of the genertors is given s follows: A g1 A g2 = A g1 g 2 B h1 B h2 = δ h1,h 2 B h2 A g B h = B ghḡ A g, (14) where, nd throughout the context, ḡ mens g 1. The identity element with respect to multipliction is given by both B h nd A e. From the multipliction rules, we see tht h G the A-type genertors (i.e., the A g ) spn sublgebr isomorphic to the group lgebr C[G] nd the B-type genertors (i.e., the B h ) spn sublgebr isomorphic to the dul of C[G], tht is, the lgebr of functions on G. The quntum double hs bsis given by {D (h,g) := B h A g : h, g G}. (15) DG hs more structures thn being n lgebr. In fct, it is qusi-tringulr Hopf lgebr with the mps (, ɛ, S, R) defined s follows. 14

15 is the co-multipliction : DG DG DG, (A g ) = A g A g, (B h ) = h=h 1 h 2 B h2 B h1. (16) is n lgebr morphism, hence the imge of ny element under is known. For instnce, (D (h,g) ) = (B h ) (A g ) = (B h2 B h1 )(A g A g ) h=h 1 h 2 = B h2 A g B h1 A g = D (h2,g) D (h1,g). h=h 1 h 2 h=h 1 h 2 ɛ is the counit ɛ : DG C which is lso n lgebr morphism. On genertors, it is defined by: S is the ntipode S : DG DG, ɛ(a g ) = 1,, ɛ(b h ) = δ h,e. (17) S(A g ) = Aḡ, S(B h ) = B h. (18) But note tht S is n nti-lgebr morphism, tht is S(b) = S(b)S(). S(D (h,g) ) = S(A g )S(B h ) = D (ḡ hg,ḡ). Clerly, S 2 = Id. R is the universl R-mtrix R DG DG which is given by Hence R = g A g B g = h,g D (h,g) D (g,e). (19) As n element of DG DG, R is invertible, nd its inverse is given by R 1 = g Aḡ B g. (20) 3.3 Representtions of DG Irreducible representtions of DG re closely relted with those of G. We give complete chrcteriztion of irreps of DG. Let G cts on itself by conjugtion, nmely, for g, x G, the ction of g sends x to gxḡ. An orbit under this ction is clled conjugcy clss. Clerly, G is prtitioned into severl conjugcy clsses: G = i C i, where C i is conjugcy clss. For ny element r G, the conjugcy clss contining r is given by C(r) = {grḡ : g G}. The stbilizer (or centrlizer) Z(r) of r is the subgroup which fixes r by conjugtion, tht is, Z(r) = {g G : gr = rg}. For ech c C(r), rbitrrily choose q c G such tht q c r q c = c. The choice of q c is not unique, nd ny other q c stisfying the required condition is equl to q c z for some z Z(r). In prticulr, we lwys ssume q r = e. It is direct to check tht {q c : c C(r)} is coset representtive for Z(r), nmely, G = c C q c Z(r). 15

16 Lemm 3.1. For ny g G, c C, we hve q gcḡ gq c Z(r). Proof. ( q gcḡ gq c )r( q gcḡ gq c ) 1 = q gcḡ gq c r q c ḡq gcḡ = q gcḡ (gcḡ)q gcḡ = r. An irrep of DG is chrcterized by pir (C, χ), where C is conjugcy clss of G, nd (ssuming n element r C hs been rbitrrily chosen nd fixed,) χ is n irrep of Z(r). The Hilbert spce corresponding to (C, χ) is given by V (C,χ) = C[C] V χ. Thus bsis is given by The ction of DG on V (C,χ) is given by { c j : c C, j = 1,, χ }. B h c j = δ h,c c j A g c j = gcḡ χ ( q gcḡ gq c ) j = i Γ χ ij ( q gcḡgq c ) gcḡ i. (21) It is stright forwrd to check the bove equtions indeed defines representtion of DG nd it is in fct irreducible. In defining the ctions, we need to rbitrrily choose the q c s, but the resulting representtions for different choices of q c s turn out to be isomorphic. Also, the second prt in the pir (C, χ) depends on the choice of n element r C. For different r s, the corresponding centrlizer Z(r) s re isomorphic, we will not obtin ny new representtions of DG. The ction of A g in Eqution 21 is simplified in certin cses. If χ = 1 is the trivil irrep of Z(r), then V (C,χ) C[C], nd A g c = gcḡ. (22) If, on the other hnd, C is the trivil conjugcy clss {e}, then Z(r) = Z(e) = G. Hence χ is n irrep of G, nd V (C,χ) V χ, nd A g j = χ(g) j. (23) If both C nd χ re trivil, then the representtion is one dimensionl, nd the sclr corresponding to the genertors of DG re given by A g 1, B h δ h,e. (24) Note tht this is exctly the counit mp ɛ in Eqution 17. The conjugcy clsses nd centrlizer of certin elements of S 3 re given in Tble 3. With the chosen elements r nd q c s for ech conjugcy clss s shown in Tble 3, we derive ll irreps of D(S 3 ). See Tble 4, where bsis for ech irrep nd the ction of A µ nd A σ re 16

17 {c : c C i } r C i {q c : c C i } Z(r) C 1 {e = (1)} e {e} S 3 C 2 {(12), σ = (23), (13)} σ {(13), e, (12)} {e, σ} Z 2 C 3 {µ = (123), (132)} µ {e, σ} {e, µ, µ 2 } Z 3 Tble 3: Conjugcy clsses C 1, C 2, C 3 of S 3. Here for ech C i, we rbitrrily choose r C i nd rbitrrily choose q c s for c C i. Note tht the order of the elements in {q c : c C i } is the sme s tht in {c : c C i }. For instnce, q (12) = (13). (C, χ) bsis dimension mtrix of A µ mtrix of A σ [+] + 1 (1) (1) C 1 [ ] 1 ( (1) ) ( ( 1) ) ω [2] 2 +, ω [1] (12), (23), (13) C [ 1] (12),, (23),, (13), ( 1 ) 0 1 ( 0 ) [1] (123), (132) C 3 ( ) ( ) ω [ω] (123), ω, (132), ω 2 0 ω ) ( 1 0 ) ( ω [ ω] (123), ω, (132), ω 2 0 ω 1 0 Tble 4: Irreps of DS 3 where ω = ω 3. 17

18 given. Since µ nd σ generte S 3, we cn deduce the ction of ll A g s. The ction of B h is simple in ll cses, which we ignore in the tble. If V is representtion of DG, V is lso representtion. Given co-vector φ V, the ction is given by (D (h,g).φ)(v) := φ ( S(D (h,g) )v ), v V. If we choose ny bsis { j : j = 1,, dim V } for V nd let the dul bsis be { j : j = 1,, dim V }, then D (h,g) j = dim V i=1 j S(D (h,g) ) i i. Given two representtions V, W of DG, V W is representtion whose ction is given by the co-multipliction, nmely, for v V, w W, D (h,g).(v w) := (D (h,g) )v w. Explicitly, A g.(v w) = A g v A g w, B h.(v w) = h=h 1 h 2 h 2 v h 1 w. W V becomes representtion in the sme wy. It is esy to check the nive swp mp Flip V,W between V W nd W V is not covrint under the ction of DG if G is non-abelin. (Check the ction of B h before nd fter the swp.) It turns out the correct covrint swp is the composition of the nive one with the ction of the universl R- mtrix. Recll tht R DG DG is n invertible element. We let the first fctor of R ct on V nd the second fctor ct on W, then R cn be viewed s mp from V W to V W. Define c V,W := Flip V,W R : V W R V W Flip V,W W V. (25) It is direct to check c V,W commutes with the ction of DG. The explicit formul for c V,W is given by c V,W (v w) = g B g w A g v. (26) Then c W,V c V,W (v w) = g,h B h A g v A h B g w. Thus in generl c W,V c V,W Id V W. The inverse of c V,W is given by c 1 V,W = R 1 Flip : W V Flip W,V V W R 1 V W. (27) 18

19 4 Kitev s Quntum Double Model Now we study Kitev s quntum double model bsed on finite group G [14]. When the group is tken to be Z 2, the model reduces to the well-known toric code. For non-abelin groups, the model produces qusi-prticles which re non-abelin. Generliztion of the model from finite groups to C Hopf lgebrs ws pointed out in [14], nd ws explicitly studied in [6]. The model ws lter further generlized to C wek Hopf lgebrs (quntum groupoids) in [7]. For detiled exposition of the Kitev s model, we recommend [4]. 4.1 The Hmiltonin of the Model Let Σ be ny oriented surfce without boundry such s the sphere or the torus, nd let L be n rbitrry lttice on Σ. For convenience, we still ssume L is squre lttice, but this ssumption is not essentil. As before, V, E, nd F denote the set of vertices, edges, nd plquettes, respectively. Now we rbitrrily fix n orienttion on ech edge. Associte to ech edge the Hilbert spce C[G] with the orthonorml bsis { g : g G}. The totl Hilbert spce H tot is the tensor product of the C[G] s over ll edges. Throughout the context, by site is ment pir s = (v, p), where v V, p F nd v p. See Figure 12. For site s = (v, p), we connect v to the center of p by red segment. For ech site s = (v, p) nd g, h G, we define the locl opertors A g (s) nd B h (s) s shown in Figure 13. A g (s) cts on edges which re djcent to v. For ech such edge, the ction is multipliction on the left by g if the edge is pointed wy from v, nd multipliction on the right by ḡ otherwise. Note tht A g (s) does not depend on the plquette, but only on v, hence we lso write A g (s) = A g (v). The ction of B h (s) is described s follows. Given bsis element in H tot, one strts from v, trvels long the boundry of p in the counterclockwise direction, nd multiply the group elements in the order s they re met. But if one edge is oriented opposite to the trveling direction, then one multiplies the inverse of the group element on tht edge insted of the group element itself. Then B h (s) cts s identity if h equls the product just obtined, nd s 0 otherwise. In nother word, B h (s) projects to the subspce spnned by those bsis elements for which the product long p strting from v is equl to h. Note tht B h (s) does depend on both v nd p. The following identities re esily verified. A g1 (s)a g2 (s) = A g1 g 2 (s), B h1 (s)b h2 (s) = δ h1,h 2 B h2 (s), (28) A g (s)b h (s) = B ghḡ A g (s), A e (s) = h G B h (s) = Id (29) Hence the opertors {A g (s), B h (s)} define representtion of the quntum double DG. Nmely, A g nd B h from DG ct on H tot s A g (s) nd B h (s), respectively. We denote by D(s) the lgebr generted by the A g s nd B h s, nd cll it the lgebr of locl opertors t site s. (30) 19

20 p v Figure 12: A lttice L on the surfce Σ whose edges re rbitrrily oriented. g 4 g 1 g 3 v g 2 p A g(s) g 4 ḡ p gg 1 gg 3 v gg 2 h 3 p h 4 h 2 v h 1 B h (s) h 3 p δ h,h1h 2 h3h 4 h 4 h 2 v h 1 Figure 13: The definition of locl opertors A g (s) nd B h (s) 20

21 Now we introduce the vertex nd plquette opertors. A(v) := 1 A g (v), B(p) := B e (v, p). (31) G g G Note tht while in generl B h (v, p) depends both on v nd p, B e (v, p) only depends on p since the property tht product of group elements equls the identity is cyclic. It is direct to check tht A(v) nd B(p) re both projectors nd they mutully commute with ech other. The Hmiltonin is defined by H = v V (1 A(v)) + p F (1 B(p)), (32) nd the ground stte is given by It is direct to check tht V gs = { ψ H tot : A(v) ψ = ψ, B(p) ψ = ψ }. (33) A g (s)a(v) = A(v), B h (s)b(p) = δ h,e B(p). Hence we hve ψ V gs if nd only if A g (s) ψ = ψ, B h (s) ψ = δ h,e ψ. (34) Note tht Eqution 34 mens tht D(s) cts on V gs by the trivil representtion. Thus, V gs is the subspce corresponding to trivil representtion of DG t every site s. 4.2 Ground Stte Spce of the Model (Optionl) In this subsection, we compute the ground stte degenercy. Denote by π 1 (Σ) the fundmentl group of Σ nd by Hom(π 1 (Σ), G) the set of ll group morphisms from π 1 (Σ) to G. There is n ction G on Hom(π 1 (Σ), G) by conjugtion. Nmely, for g G, φ Hom(π 1 (Σ), G), g.φ := gφ( )ḡ. Proposition 4.1. The dimension of V gs (Σ) is equl to the number of orbits in Hom(π 1 (Σ), G) under the G-ction. Proof. A bsis element of the totl Hilbert spce is n ssignment of group element to ech edge. Let g be ny bsis element where g = {g α G : α E}. Let γ be ny oriented pth in the lttice. Denote by g γ the group element obtined by multiplying the group elements long the pth γ. But if one edge in the pth is oriented opposite to the pth, then multiply the inverse of the group element on tht edge insted. Then the constrint B(p) g = g is equivlent to the condition tht g p = e, where p is the boundry of p oriented counterclockwise. Note tht the condition g p = e is independent of the choice of strting vertex on p. Hence the subspce fixed by ll B p s is spnned the following set: S = { g : g p = e, p} = { g : g γ = e, for ny contrctible closed γ}. 21

22 For ny h G, we cll the opertor A h (v) guge trnsformtion t the vertex v. For two bsis elements g, g S, we cll g nd g guge equivlent or g g if g cn be obtined from g by pplying some guge trnsformtions t severl vertices. Guge equivlence defines n equivlence reltion on S nd denote by [S] the set of equivlence clsses. For ech [g] [S], define [g] := g. (35) g g It is direct to check { [g] : [g] [S]} forms bsis of V gs (Σ). Now we build correspondence between [S] nd orbits in Hom(π 1 (Σ), G). Choose ny vertex v 0 s the bse point nd choose mximl spnning tree T contining v 0. By definition, mximl spnning tree is subgrph of the lttice L (with plquettes ignored) which contins ll vertices of L nd does not contin ny loop. Thus ny mximl spnning tree contins exctly N := V 1 edges. Define mp Φ : S Hom(π 1 (Σ, v 0 ), G) (36) s follows. Let γ be ny closed pth strting nd ending t v 0. For ny g S, define Φ( g )([γ]) := g γ, nmely, Φ( g ) mps closed pth γ to the product of the group elements on it. The fct tht g γ0 = e for ny contrctible loop γ 0 implies tht Φ( g )(γ) only depends on the homotopy clss of γ. Hence Φ( g ) is well defined mp from π 1 (Σ, v 0 ) to G. It is cler tht it is lso group morphism, hence Φ( g ) Hom (π 1 (Σ, v 0 ), G). Now we show tht mp Φ is onto nd in fct G N -to-1. Given ny τ Hom(π 1 (Σ, v 0 ), G), we construct preimge g of τ s follows. The vlue of g on edges of the mximl spnning tree T is rbitrry, nd the vlue on other edges is to be determined. For ny edge α not in T, let 0 α nd 1 α be the two end vertices of α. By construction, there is unique pth γ i in T connecting v 0 to i α, i = 0, 1. Let γ = γ 0 α γ 1 be the closed pth, where γ 1 mens the pth γ 1 with reversed direction. Nmely, γ reches 0 α long γ 0 from v 0, trvels through the edge α, nd then goes bck to v 0 long γ 1. Define g α to be the unique group element such tht g γ0 g α g γ1 = τ(γ). It cn be checked tht such defined g is in S, nd Φ( g ) = τ. Since we hve G N choices when defining g, the mp Φ is G N -to-1 On the other hnd, for ech given g, if we only llow to pply guge trnsformtions on g t vertices other thn v 0, there re in totl G N such trnsformtions. (One needs to check ll the possible G N trnsformtions re indeed different with ech other.) If two g, g re relted by guge trnsformtions t vertices other thn v 0, then Φ( g ) = Φ( g ). We conclude tht the preimge of τ contins precisely those g s which re relted by guge trnsformtions t vertices other thn v 0. If we perform guge trnsformtion A h (v 0 ) t v 0 to g, then it is esy to see tht Φ(A h (v 0 ) g ) = hφ( g ) h. Thus we hve one-to-one correspondence between guge clsses in S nd orbits in Hom(π 1 (Σ), G). 22

23 s 1 s 0 = t Figure 14: An exmple of ribbon t connecting the site s 0 to s 1. Since the fundmentl group of the sphere is trivil, the ground stte degenercy on the sphere is 1. On the torus, the fundmentl group is Z Z. A morphism from Z Z to G is given by pir of group elements (g 1, g 2 ) such tht g 1 g 2 = g 2 g 1. The ction of G on the pir (g 1, g 2 ) is given by conjugtion: (g 1, g 2 ) (gg 1 ḡ, gg 2 ḡ). Hence the ground stte degenercy on the torus is given by the number of commuting pirs in G G, up to conjugtion by G; this is equl to the number of irreps of DG. 4.3 Excittions of the Model (Summry) We give summry of excittions, fusion, nd briding, delying the technicl detils until next section. From now on, ssume the lttice is on sphere nd denote by E the unique ground stte. In generl, n excittion occupies site. However, s in the cse of toric code, single site excittion does not exist. (But such excittion cn exist on surfces with nontrivil topology.) The tool to study excittions is ribbon opertors which re nlogous to string opertors in toric code. Just s string connects two vertices, ribbon connects two sites. See Figure 14. A ribbon t cn be thought of s thin strip with pir of prllel strings, one in the lttice nd the other in the dul lttice. The ribbon t is ssumed to be directed; it strts from the site s 0 := 0 t nd ends t the site s 1 := 1 t. (More precise definition of ribbons will be given in next section.) As nottion, site s i lwys mens the pir (v i, p i ). Associted with ech ribbon t re set of opertors {F (h,g) (t) : h, g G}. Ech F (h,g) (t) cts non-trivilly only on edges tht re contined in or crossed by t. We give severl properties of these opertors (without proof) nd use them to study excittions. (We hve not shown wht the opertors F (h,g) (t) ctully re. But hopefully this will not ffect understnding of the sttements below.) 1. Let F(t) = spn C {F (h,g) (t) : h, g G}. (37) Elements in F(t) re clled ribbon opertors. The dimension of F(t) is G 2. Hence {F (h,g) (t) : h, g G} is bsis of F(t); it is clled the group elements bsis. A differ- 23

24 ent bsis, which is Fourier trnsformtion of the group elements bsis, is given by {F (C,χ;u,u ) (t) : (C, χ) Irr(DG), u nd u ech enumertes bsis of V (C,χ).} (38) This is clled the representtion bsis. 2. F (h,g) (t) commutes with ll A(v) nd B(p) where v v 0, v 1 nd p p 0, p 1. The spce of 2-prticle excittions t s 0 nd s 1 is given by nd it hs two orthonorml bses, { h, g := F (h,g) (t) E : h, g G}, L(s 0, s 1 ) = {F E : F F(t)}, (39) { C, χ; u, u := F (C,χ;u,u ) E : (C, χ) Irr(DG), u nd u ech enumertes bsis of V (C,χ) }, (40) which re respectively gin clled the group elements bsis nd the representtion bsis. 3. The locl opertors D(s 0 ) nd D(s 1 ) preserve the spce L(s 0, s 1 ). They re the commutnt of ech other in the spce of opertors on L(s 0, s 1 ), hence vlidte the notion of being clled locl opertors. Indeed, ny locl opertor tht cts on few qudits ner the site s 0 must commute with the ction of D(s 1 ). Although the ction of the locl opertor does not necessrily preserve the subspce L(s 0, s 1 ), when projecting the ction to L(s 0, s 1 ), it will coincide with one of the opertors in D(s 0 ). The ction of D(s i ) on the representtion bsis hs prticulrly nice form. Explicitly, the ction of D(s 1 ) on C, χ; u, u trnsforms the u prt ccording to the irrep (C, χ) while the ction of D(s 0 ) trnsforms the u prt ccording to the the dul irrep (C, χ). For instnce, B h (s 1 )A g (s 1 ) C, χ; u, u = ũ Γ (C,χ) ũ u ( D(h,g) ) C, χ; u, ũ. (41) Thus the representtion bsis gives decomposition of L(s 0, s 1 ): L(s 0, s 1 ) = V(C,χ) V (C,χ), (42) (C,χ) Irr(DG) nd under this decomposition D(s 0 ) (resp. D(s 1 )) cts on the first (resp. second) fctor. When fixing (C, χ), we cn think of the u index s living on site s 0 nd the u index s living on site s 1. Moreover, the locl opertors only chnge the indices u nd u, but not the irrep (C, χ). Therefore, we cll (C, χ) qusi-prticle type or n nyon type, nd the types of qusi-prticles re in one-to-one correspondence with the irreps of DG (strictly speking, with isomorphism clsses of irreps). 4. The stte F (h,g) (t) E is unchnged s we deform the ribbon t. 5. To obtin the spce of 3-prticle excittions t sites s 0, s 1, s 2, we simply connect one of the sites, sy s 0, to ech of the other two by ribbon. See Figure 15 (Left). Then L(s 0, s 1, s 2 ) = spn{ h 1, g 1 ; h 2, g 2 := F (h 2,g 2 ) (t 2 )F (h 1,g 1 ) (t 1 ) E : h i, g i G}. (43) 24

25 s 1 s 2 s 1 s 2 s n = = = = = t 1 t 2 t 1 t 2 t n s 0 s 0 Figure 15: Ribbons connecting more thn two sites We lso hve the representtion bsis: { C 1, χ 1 ; u 1, u 1;C 2, χ 2 ; u 2, u 2 }. (44) One cn think of the u 1 index s living on s 1, u 2 on s 2, u 1 nd u 2 on s 0. The locl opertors D(s i ), i = 1, 2 trnsform u i ccording to (C i, χ i ), while D(s 0 ) trnsforms u 1 u 2 ccording to (C 1, χ 1 ) (C 2, χ 2 ). (Note tht the ltter sttement is not obvious.) By generl representtion theory, (C 1, χ 1 ) (C 2, χ 2 ) decomposes into direct sums of irreps: (C 1, χ 1 ) (C 2, χ 2 ) = (C 3, χ 3 ) (C 4, χ 4 ) (45) Ech irrep in the decomposition is clled totl chrge/type of (C 1, χ 1 ) nd (C 2, χ 2 ). In generl, given ny two types α, Irr(DG), we cn decompose α = N γ αγ, (46) γ Irr(DG) where N γ α = 0, 1 nd these {N γ α } re clled fusion rules. 6. More generlly, consider the spce L(s 0, s 1,, s n ). We connect s i to s 0 with ribbon t i, i = 1,, n. See Figure 15(Right). Assume ech ribbon t i is ssocited with the type α i := (C i, χ i ). For i = 1,, n, D(s i ) cts on u i ccording to α i nd D(s 0 ) cts on u 1,, u n ccording to the product α 1 α n. Let Inv(α 1 α n ) be the subspce of α 1 α n corresponding to the trivil representtions of D(s 0 ). Then the spce α 1 α n Inv(α 1 α n ) corresponds to the subspce without excittion t the site s 0. In nother word, this is the spce of n qusi-prticles of types α 1,, α n whose totl type is trivil. Note tht the locl opertors D(s i ) hve no ccess to the spce Inv(α 1 α n ); this is the logicl spce where we cn encode informtion. For n > 3, the spce Inv(α 1 α n ) could hve dimension greter thn one, but in generl it lcks tensor product structure. 5 Ribbon Opertors In this section, we study ribbon opertors nd excittions. An elementry nd detiled discussion of ribbon opertors cn be found in [6]. Let L be lttice on which the Hmiltonin is defined, nd let L be its dul lttice. Recll tht ech edge α in L is rbitrrily ssigned direction (orienttion). We now orient 25

26 s 1 s 0 Figure 16: A ribbon consisting of sequence of tringles. 0 τ = 1 τ = τ τ 0 τ 1 τ Figure 17: (Left) type-i (or dul) tringle; (Right) type-ii (or direct) tringle the dul edge α so tht α crosses α from right to left. (This only mkes sense if the surfce is oriented, which is lwys our ssumption.) Roughly speking, ribbon is thin strip in the combined lttice L L tht connects two sites. See Figure 16, where, nd throughout the section, we will drw n edge in L s solid blck line, n edge in L s dshed blck line, nd site s solid red line. More precisely, ribbon is built up from two types of tringles, clled type-i tringles (or dul tringles) nd type-ii tringles (or direct tringles). See Figure 17. A dul tringle consists of two sites nd dul edge, while direct tringle consists of two sites nd direct edge (tht is, n edge in L). For dul/direct tringle τ, by choosing one site s 0 τ nd the other s 1 τ, we sy τ is directed nd it strts t 0 τ nd ends t 1 τ. We often use n rrow to indicte the direction. See Figure 17. A (directed) ribbon t is defined to be sequence (τ 1,, τ n ) of tringles (type I or II) such tht 1 τ i = 0 τ i+1, nd tht the edge contined in ny τ i (not including the sites) is not the sme s nd lso not dul to the edge in ny other τ j. The second condition bove mens the ribbon does not repet or cross itself. Denote by 0 t = 0 τ 1 nd 1 t = 1 τ n. We sy t is directed ribbon, strting t 0 t nd ending t 1 t. Similrly, we lso use n rrow to indicte the direction. Note tht it is possible tht 0 t = 1 t. Such ribbon is clled closed ribbon, (though we will not tlk bout it here). For ech pir of group elements (h, g) G G nd directed ribbon t, we wish to define n opertor F (h,g) (t). We first define it for the cse of tringle. See Figure 18 for grphicl illustrtion. Nmely, if t is tringle, then F (h,g) (t) cts non-trivilly only on the edge contined in it. If t is type-i tringle nd x is bsis stte in the Hilbert spce of the edge α t contined in t, the ction F (h,g) (t) sends x to δ g,e hx if the direction of t coincides with tht of α t, nd to δ g,e xĥ otherwise. If t is type-ii tringle nd gin x is bsis stte, then the ction sends x to δ g,x x if the direction of t coincides with tht of its edge, nd to δḡ,x x otherwise. Note tht for type-ii tringle, the ction is projector 26

27 = x δ g,e = hx x = x δ g,x = = x δ g,e = x h x = x δḡ,x = Figure 18: (Left block) The ction of F (h,g) (τ) for type-i tringle τ; (Right block) The ction of F (h,g) (τ) for type-ii tringle τ. For ech type, there re two cses determined by whether the direction of the tringle coincides with the direction of the edge in it. In ll cses, x G is group element such tht x represents bsis stte in the corresponding Hilbert spce. nd it does not depend on h. We now define the opertor F (h,g) (t) for generl ribbon t inductively. If t consists of more thn one tringle, then split t = t 1 t 2 s disjoint union of two sub ribbons, nd define F (h,g) (t) := k G F (h,k) (t 1 )F ( khk, kg) (t 2 ). (47) With Eqution 47, we cn define F (h,g) (t) inductively for ribbons of rbitrry length. However, one must check tht F (h,g) (t) does not depend on the wy the ribbon is split. To prove this, it suffices to show the following. Let t = t 1 t 2 t 3. To compute F (h,g) (t), one cn first split t into t 1 t 2 nd t 3, nd then split the former into t 1 nd t 2, or one cn insted first split t into t 1 nd t 2 t 3, nd then split the ltter into t 2 nd t 3. One needs to check the two wys of splitting give the sme nswer. This is stright forwrd: F (h,g) (t) = k G F (h,k) (t 1 t 2 )F ( khk, kg) (t 3 ) = F (h,l) (t 1 )F ( lhl, lk) (t 2 )F ( khk, kg) (t 3 ), k,l G F (h,g) (t) = l G F (h,l) (t 1 )F ( lhl, lg) (t 2 t 3 ) = m,l G F (h,l) (t 1 )F ( lhl,m) (t 2 )F ( m lhlm, m lg) (t 3 ) lm k = k,l G F (h,l) (t 1 )F ( lhl, lk) (t 2 )F ( khk, kg) (t 3 ), (48) 27

28 where in the lst equlity, we pplied chnge of vrible lm k. Remrk 5.1. Although it is direct to verify tht F (h,g) (t) is well defined, it is still not cler how the prticulr combintion on the right hnd side of Eqution 47 is chosen. In fct, we cn rewrite Eqution 47 s F (h,g) (t) = h 1,g 1,h 2,g 2 Ω (h,g) (h 1,g 1 ),(h 2,g 2 ) F (h 1,g 1 ) (t 1 )F (h 2,g 2 ) (t 2 ), (49) for some tensor Ω (h,g) (h 1,g 1 ),(h 2,g 2 ). For compctness, we use bold letters, b, c,, to represent pir of group elements. Then we hve F c (t) =,b Ω c bf (t 1 )F b (t 2 ). (50) Tht F (t) does not depend on the wy the ribbon is split is equivlent to the condition, Ω m bω d mc = Ω d nω n bc (51) n m If we tke G 2 dimensionl Hilbert spce with bsis given by {f : G G} nd define multipliction by f f b := c Ω c bf c, (52) then Eqution 51 is equivlent to the property tht the multipliction is ssocitive, which mkes the Hilbert spce n ssocitive lgebr. In fct, one cn check directly tht the Ω, tensor here is exctly the multipliction tensor in the quntum double DG: D D b = c Ω c bd c. (53) By the inductive formul, the opertor F (h,g) (t) cts non-trivilly only on edges contined in or crossed by the ribbon. For typicl ribbon, the explicit formul is given in Figure 19. Let F(t) = spn C {F (h,g) (t) : h, g G}. (54) Any opertor in F(t) is clled ribbon opertor. We stte some properties of ribbon opertors. Although they ll cn be proved formlly, it is esier to use the explicit expression in Figure 19 to verify the properties. Let t be ribbon with i t = s i = (v i, p i ), i = 0, 1. It is stright forwrd to show F (h 1,g 1 ) (t)f (h 2,g 2 ) (t) = δ g1,g 2 F (h 1h 2,g 2 ) (t). (55) Hence F(t) is n lgebr. Also, F (h,g) (t) commutes with ll the A(v) nd B(p) for which v v i, p p i, i = 0, 1. In fct, F (h,g) (t) even commutes with ll A g (v) for ny g G. Recll tht t ech site s 28

29 v 0 x 1 x 2 x 3 v 1 s 1 s 0 p 0 y 1 y 2 y 3 y 4 p 1 F (h,g) (t) v 0 x 1 x 2 x 3 v 1 s 1 δ g,x1x 2x 3 s 0 p 0 hy 1 x 1 hx 1 y 2 p 1 x 2 x 1 hx 1 x 2 y 3 x 3 x 2 x 1 hx 1 x 2 x 3 y 4 Figure 19: Explicit formul for the opertor F (h,g) (t), where t strts t s 0 = (v 0, p 0 ) nd ends t s 1 = (v 1, p 1 ). Ech x i, y j is group element representing bsis stte of the totl Hilbert spce. The ction is zero unless g = x 1 x 2 x 3. we hve the lgebr of opertors D(s) which gives the ction of DG on the totl Hilbert spce. The ribbon opertor in generl does not commute with the opertors in D(s i ). Their commuttion reltions re given s follows. A g (s 0 )F (h,g) (t) = F (g hḡ,g g) (t)a g (s 0 ) B h (s 0 )F (h,g) (t) = F (h,g) (t)b h h(s 0 ) (56) A g (s 1 )F (h,g) (t) = F (h,gḡ ) (t)a g (s 1 ) B h (s 1 )F (h,g) (t) = F (h,g) (t)bḡ hgh (s 1 ) (57) Denote by E the unique ground stte (ssuming the surfce is sphere). Let 0 be the bsis stte in the totl Hilbert spce which hs vlue e t ech edge. Hence 0 is fixed by ll B(p) s. Then up to normliztion, E is given by E = v A(v) 0. (58) In prticulr, ll the bsis sttes contined in the expnsion of E hve positive coefficients. Moreover, for ny bsis stte in the expnsion, the product of the group elements long ny closed loop is the identity e. Assume t is not closed nd define h, g := F (h,g) (t) E. (59) By the formul in Figure 19, F (h,g) (t) projects out ll bsis sttes in E for which the product of the group elements long ny pth connecting v 0 to v 1 is not equl to g, nd it modifies ll remining terms by expressions involving h. Then the bsis sttes in the expnsion of h, g 29