Universal Topological Data for Gapped Quantum Liquids in Three Dimensions and Fusion Algebra for Non-Abelian String Excitations
UUniversal Topological Data for Gapped Quantum Liquids in Three Dimensionsand Fusion Algebra for Non-Abelian String Excitations
Heidar Moradi and Xiao-Gang Wen
1, 2 Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5 Canada Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Dated: July 6, 2018)Recently we conjectured that a certain set of universal topological quantities characterize topo-logical order in any dimension. Those quantities can be extracted from the universal overlap ofthe ground state wave functions. For systems with gapped boundaries, these quantities are rep-resentations of the mapping class group
MCG ( M ) of the space manifold M on which the systemslives. We will here consider simple examples in three dimensions and give physical interpretationof these quantities, related to fusion algebra and statistics of particle and string excitations. Inparticular, we will consider dimensional reduction from 3+1D to 2+1D, and show how the induced2+1D topological data contains information on the fusion and the braiding of non-Abelian stringexcitations in 3D. These universal quantities generalize the well-known modular S and T matricesto any dimension. I. INTRODUCTION
For more than two decades exotic quantum states have attracted a lot attention from the condensed mat-ter community. In particular gapped systems with non-trivial topological order, which is a reflection of long-range entanglement of the ground state, have beenstudied intensely in 2 + 1 dimensions. Recently, peoplestarted to work on a general theory of topological orderin higher than 2 + 1 dimensions. In a recent work Ref. 19, we conjectured that for agapped system on a d -dimensional manifold M of volume V with the set of degenerate ground states {| ψ α (cid:105)} Nα =1 on M , we have the following overlaps (cid:104) ψ α | ˆ O A | ψ β (cid:105) = e − αV + o (1 /V ) M Aα,β , (1)where ˆ O A are transformations on the wave functions in-duced by the automorphisms A : M → M , α is a non-universal constant and M A is a universal matrix up toan overall U (1) phase. Here M A form a projective rep-resentation of the automorphism group AMG ( M ), whichis robust against any local perturbations that do notclose the bulk gap. In Ref. 19 we conjectured thatsuch projective representations for different space mani-fold topologies fully characterize topological orders withfinite ground state degeneracy in any dimension. Fur-thermore, we conjectured that projective representationsof the mapping class groups
MCG ( M ) = π [ AMG ( M )] clas-sify topological order with gapped boundaries. Thesequantities can be used as order parameters for topologicalorder and detect transitions between different phases. In this paper we will study these universal quantitiesfurther in 3-dimensions for one of the most simple man-ifolds, the 3-torus M = T . The mapping class group ofthe 3-torus is MCG ( T ) = SL (3 , Z ). This group is gener-ated by two elements of the form ˆ˜ S = , ˆ˜ T = . (2) These matrices act on the unit vectors by ˆ˜ S : ( ˆ x , ˆ y , ˆ z ) (cid:55)→ ( ˆ z , ˆ x , ˆ y ) and similarly ˆ˜ T : ( ˆ x , ˆ y , ˆ z ) (cid:55)→ ( ˆ x + ˆ y , ˆ y , ˆ z ). Thus˜ S corresponds to a rotation, while ˜ T is shear transforma-tion in the xy -plane.In this paper, we will study the SL (3 , Z ) represen-tations generated by a very simple class of Z N mod-els in detail and then consider models for any finitegroup G , which are 3-dimensional versions of Kitaevsquantum double models . One can also generalize intotwisted versions of these based on the group cohomol-ogy H ( G, U (1)) by direct generalization of Ref. 26 into3+1D.
We will consider dimensional reduction of a 3D topo-logical order C D to 2D by making one direction of the3D space into a small circle. In this limit, the 3D topo-logically ordered states C D can be viewed as several 2Dtopological orders C Di , i = 1 , , · · · which happen to havedegenerate ground state energy. We denote such a dimen-sional reduction process as C D = (cid:77) i C Di . (3)We can compute such a dimensional reduction using therepresentation of SL (3 , Z ) that we have calculated.We consider SL (2 , Z ) ⊂ SL (3 , Z ) subgroup and thereduction of the SL (3 , Z ) representation R D to the SL (2 , Z ) representations R Di : R D = (cid:77) i R Di . (4)We will refer to this as branching rules for the SL (2 , Z )subgroup. The SL (3 , Z ) representation R D describesthe 3D topological order C D and the SL (2 , Z ) represen-tations R Di describe the 2D topological orders C Di . Thedecomposition (4) gives us the dimensional reduction (3).Let us use C G to denote the topological order describedby the gauge theory with the finite gauge group G . Using a r X i v : . [ c ond - m a t . s t r- e l ] A p r the above result, we find that C DG = | G | (cid:77) n =1 C DG (5)for Abelian G where | G | is the number of the group ele-ments. For non-Abelian group G C DG = (cid:77) C C DG C (6)where (cid:76) C sums over all different conjugacy classes C of G , and G C is a subgroup of G which commutes with anelement in C . The results for G = Z N were mentionedin our previous paper. We also found that the reduction of SL (3 , Z ) repre-sentation, eqn. (4), encodes all the information aboutthe three-string statistics discussed in Ref. 20 and 21 forAbelian groups. For non-Abelian groups, we will have a“non-Abelian” string braiding statistics and a non-trivialstring fusion algebra. We also have a “non-Abelian”three-string braiding statistics and a non-trivial three-string fusion algebra. Within the dimension reductionpicture, the 3D strings reduces to particles in 2D, andthe (non-Abelian) statistics of the particles encode the(non-Abelian) statistics of the strings. II. Z N MODEL IN 3-DIMENSIONS
In this section we will define and study the excitationsof a Z N model in detail and compute the 3-torus uni-versal matrices, eq. (1).Consider a simple cubic lattice with a local Hilbertspace on each link isomorphic to the group algebra of Z N , H i ≈ C [ Z N ] ≈ C N ≈ span C {| σ (cid:105)| σ ∈ Z N } . Give thelinks on the lattice an orientation as in figure 1 and letthere be a natural isomorphism H i ∼ → H i (cid:63) for link i andits reversed orientation i (cid:63) as | σ i (cid:105) (cid:55)→ | σ i (cid:63) (cid:105) = | − σ i (cid:105) . Letthis basis be orthonormal. Define two local operators Z i | σ i (cid:105) = ω σ i | σ i (cid:105) , X i | σ i (cid:105) = | σ i − (cid:105) , where ω = e πiN . These operators have the importantcommutation relation X i Z i = ωZ i X i . Note that theseoperators are unitary and satisfy X Ni = Z Ni = 1. Foreach lattice site s and plaquette p define A s = (cid:89) i ∈ s + Z i (cid:89) j ∈ s − Z † j , B p = (cid:89) i ∈ ∂p + X † i (cid:89) j ∈ ∂p − X j . Here s + is the set of links pointing into s , while s − isthe set of links pointing away from s . B p creates a stringaround plaquette p with orientation given by the normaldirection using the right hand thumb rule. Then ∂p ± arethe set of links surrounding plaquette p with the sameor opposite orientation as the lattice. One can directlycheck that all these operators commute for all sites andplaquettes. s xy z (a) pp'p'' xy z (b) FIG. 1. (a) Lattice site of 3D cubic lattice. A s act on spinsconnected to site s . (b) 2D plaquettes. B p acts on the fourspins surrounding p . Choose a righthanded ( x, y, z ) frame,and let all links be oriented wrt. to these directions. Thisassociates a natural orientation to 2 D plaquettes on the duallattice. We can now define the Z N model by the Hamiltonian H D, Z N = − J e (cid:88) s (cid:0) A s + A † s (cid:1) − J m (cid:88) p (cid:0) B p + B † p (cid:1) , where we will assume J e , J m ≥ A s + A † s ) = { πN q ) } N − , and the similar for B p + B † p , the ground state is the state satisfying A s | GS (cid:105) = | GS (cid:105) , B p | GS (cid:105) = | GS (cid:105) , (7)for all s and p . We can easily construct hermitian pro-jectors to the state with eigenvalue 1 for all vertices andplaquettes ρ s = 1 N N − (cid:88) k =0 A ks , ρ p = 1 N N − (cid:88) k =0 B kp . The ground state is thus | GS (cid:105) = (cid:81) s ρ s (cid:81) p ρ p | ψ (cid:105) , for anyreference state | ψ (cid:105) such that | GS (cid:105) is non-zero. For thechoice | ψ (cid:105) = | . . . (cid:105) ≡ | (cid:105) , the ρ s is trivial and theground state is thus | GS (cid:105) = (cid:89) p (cid:32) N N − (cid:88) k =0 B kp (cid:33) | (cid:105) = N (cid:88) Z N string nets | loops (cid:105) . The first condition in equation (7) requires that theground state consists of Z N string-nets, while the sec-ond requires that these appear with equal superpositions.Note that if we had used eigenstates of X i instead, wewould find that the ground state is a membrane conden-sate on the dual lattice.
1. String and Membrane Operators
Now let l ab denote a curve on the lattice from site a to b , with the orientation that it points from a to b .And let Σ C denote an oriented surface on the dual latticewith ∂ Σ C = C . Using these, define string and membraneoperators W [ l ab ] = (cid:89) i ∈ l − ab X i (cid:89) j ∈ l + ab X † j , Γ[Σ C ] = (cid:89) i ∈ Σ −C Z † i (cid:89) j ∈ Σ + C Z j . Again l ± ab and Σ ±C are defined wrt. the orientation ofthe lattice. Note that B p = W [ ∂p ], where ∂p denotes aclosed loop around plaquette p with right hand thumbrule orientation wrt. the normal direction. Similarly, A s = Γ[star( s )], where star( s ) is the closed surface on thedual lattice surrounding site s with inward orientation.It is clear that the following operators commute (cid:104) W [ l ab ] , B p (cid:105) = 0 , ∀ p, and (cid:104) Γ[Σ C ] , A s (cid:105) = 0 , ∀ s. Furthermore it is easy to show that (cid:104) W [ l ab ] , A s (cid:105) = 0 , s (cid:54) = a, b, (cid:104) Γ[Σ C ] , B p (cid:105) = 0 , p (cid:54)∈ C , while A a W [ l ab ] = ω − W [ l ab ] A a , A b W [ l ab ] = ω W [ l ab ] A b , and B p Γ[Σ C ] = ω ± Γ[Σ C ] B p , p ∈ C , where ± depends on orientation of Σ C .
2. Ground States on 3-Torus
The ground state degeneracy depends on the topol-ogy of the manifold on which the theory is defined, takefor example the 3-torus T . Let l x , l y and l z be non-contractible loops along the three cycles on the lattice,with the orientation of the lattice. Similarly, let Σ x ,Σ y and Σ z be non-contractible surfaces along the three-directions, with the orientation of the dual lattice. Wecan define the operators W i ≡ W [ l i ] = (cid:89) j ∈ l i X † j , Γ i ≡ Γ[Σ i ] = (cid:89) j ∈ Σ i Z i , i = x, y, z. These operators have the commutation relations W i Γ i = ω − Γ i W i , i = x, y, z. (8)We can thus find three commuting (independent) non-contractible operators to get N fold ground state degen-eracy. For example | α, β, γ (cid:105) = ( W x ) α ( W y ) β ( W z ) γ | GS (cid:105) ,where α, β, γ = 0 , . . . , N −
1. This basis correspondto eigenstates of the surface operators Γ i | α , α , α (cid:105) = ω α i | α , α , α (cid:105) . Note that on the torus we get the extraset of constraints (cid:81) s A s = 1, (cid:81) p B p = 1. Let G be thegroup generated by B p for all p , modulo B p B p (cid:48) = B p (cid:48) B p , B Np = 1 and (cid:81) p B p = 1. Furthermore define the groups G αβγ ≡ ( W x ) α ( W y ) β ( W z ) γ G , then we can write theground states as | α, β, γ (cid:105) = 1 (cid:112) | G αβγ | (cid:88) g ∈ G αβγ | g (cid:105) , where | g (cid:105) ≡ g | (cid:105) . In 2D, the quasiparticle basis corresponds to the basisin which there is well-defined magnetic and electric fluxalong one cycle of the torus. We can try to do the samein three-dimensions. Γ x , W y , W z all commute with eachother and we can consider the basis which diagonalizesall of them. This basis is given by | ψ abc (cid:105) = 1 N (cid:88) βγ ω − βb − γc | a, β, γ (cid:105) , (9)where a, b, c = 0 , . . . , N −
1. These are clearly eigenstatesof Γ x , and furthermore we have that W y | ψ abc (cid:105) = ω b | ψ abc (cid:105) and W z | ψ abc (cid:105) = ω c | ψ abc (cid:105) . This basis is a 3D version ofminimum entropy states (MES).
3. Excitations
Now lets go back to, say, this theory on S and lookat elementary excitations of our model. An excitationcorrespond to a state in which the conditions (7) are vi-olated in a small region. Using the string operators, wecan create a pair of particles by |− q e , q e (cid:105) = W [ l ab ] q e | GS (cid:105) with the electric charges A a |− q e , q e (cid:105) = ω − q e |− q e , q e (cid:105) , A b |− q e , q e (cid:105) = ω q e |− q e , q e (cid:105) . This excitation has an energy cost of ∆ E particles =2 J e [1 − cos( πN q e )]. Furthermore we have oriented stringexcitations by using the membrane operators |C , q m (cid:105) =Γ[Σ C ] q m | GS (cid:105) , with the magnetic flux B p |C , q m (cid:105) = ω ± q m |C , q m (cid:105) , p ∈ C , where the ± depend on the orientation of C . Thisexcitation comes with the energy penalty ∆ E string =Lenght( C ) J m [1 − cos( πN q m )].One can easily show that all the particles have trivialself and mutual statistics, and the same with the strings.Mutual statistics between particles and strings can benon-trivial however, taking a charge q e particle througha flux q m string gives the anyonic phase ω ± q e q m , wherethe ± depend on the orientations. See figure 2. III. REPRESENTATIONS OF
MCG ( T ) = SL (3 , Z ) Let us now go back to T and consider the universalquantities as defined in (1). In the | α, β, γ (cid:105) basis, therepresentation of the SL (3 , Z ) generators (2) is given by˜ S αβγ,α (cid:48) β (cid:48) γ (cid:48) = δ α,β (cid:48) δ β,γ (cid:48) δ γ,α (cid:48) , (10)and ˜ T αβγ,α (cid:48) β (cid:48) γ (cid:48) = δ α,α (cid:48) δ β,α (cid:48) + β (cid:48) δ γ,γ (cid:48) . (11)In the 3D quasiparticle basis (9) these are given by˜ S abc, ¯ a ¯ b ¯ c = 1 N δ b, ¯ c e πiN (¯ ac − a ¯ b ) , ˜ T abc, ¯ a ¯ b ¯ c = δ a, ¯ a δ b, ¯ b δ c, ¯ c e πiN ab . FIG. 2. String and particle excitations. The red curve is theboundary of a membrane on the dual lattice and correspondto a string excitation. The blue links are the ones affected bythe membrane operator and the green plaquettes are the oneson which B p can measure the presence of the string excita-tion. The green line correspond to a string operator on thelattice, in which the end point are particles. Mutual statisticsbetween strings and particles can be calculated by creating aparticle-antiparticle pair from the vacuum, moving one parti-cle around the string excitation and annihilating the particles. For example in the simplest case N = 2, which is the 3DToric code, we have˜ T = − − , and ˜ S = 12 − − − − − − − −
10 0 1 − −
10 0 1 − − .
4. Interpretation of ˜ T These matrix elements in this particular ground statebasis, actually contain some physical information aboutstatistics of excitations. In order to see this, we can asso-ciate a collection of excitations to each ground state onthe 3-torus.
FIG. 3. The result of cutting open the 3-torus along the x -axis, can be represented by a hollow solid cylinder wherethe inner and outer surfaces are identified, but there are twoboundaries along x . In the above, the compactified directionis y and the radial direction is z , while the open direction is x . We can see the N possible excitations on the boundarieswhich give rise to 3-torus ground states uppon gluing. Thefour first states correspond to | (cid:105) , | e a (cid:105) , | m y,c (cid:105) and | m z,b (cid:105) . First cut the 3-torus along the x -axis such thatit now has two boundaries. We can measure thepresence of excitations on the boundary using theoperators Γ x , W y and W z . First take the statewith no particle, | (cid:105) = N (cid:80) βγ | β, γ (cid:105) , in which alloperators have eigenvalue 1. Here | β, γ (cid:105) are stateswith β and γ non-contractible electric loops alongthe y and z axis, respectively. Now add excita-tions on the boundary using open string and mem-brane operators (see fig. 3) | e a (cid:105) = ( W [ l ]) a | (cid:105) , | m y,c (cid:105) = (Γ[Σ C y ]) c | (cid:105) , | m z,b (cid:105) = (Γ[Σ C z ]) b | (cid:105) , | e a m y,c (cid:105) =( W [ l ]) a (Γ[Σ C y ]) c | (cid:105) , | e a m z,b (cid:105) = ( W [ l ]) a (Γ[Σ C z ]) b | (cid:105) , | m y,c m z,b (cid:105) = (Γ[Σ C y ]) c (Γ[Σ C z ]) b | (cid:105) and | e a m y,c m z,b (cid:105) =( W [ l ]) a (Γ[Σ C y ]) c (Γ[Σ C z ]) b | (cid:105) , where a, b, c =1 , . . . , N −
1. Or more compactly, | e a m y,c m z,b (cid:105) ,where a, b, c = 0 , . . . , N −
1. Here l is a curve fromone edge to the other, Σ C y is a membrane between edgeswrapping along the y -cycle and Σ C z is a membranebetween edges wrapping along z -cycle. All these havethe same orientation as the (dual) lattice. These stateshave well-defined electric and magnetic flux wrt. Γ x , W y and W z . Here m y and m z correspond to the stringson the boundaries, wrapping around the y and z cycles,respectively.If we now glue the two boundaries together, we see thatfor each of these excitations we have a 3-torus groundstate | (cid:105) = | ψ (cid:105) , | e a m ,c (cid:105) = | ψ a c (cid:105) , | e a (cid:105) = | ψ a (cid:105) , | e a m ,b (cid:105) = | ψ ab (cid:105) , | m ,c (cid:105) = | ψ c (cid:105) , | m ,c m ,b (cid:105) = | ψ bc (cid:105) , | m ,b (cid:105) = | ψ b (cid:105) , | e a m ,c m ,b (cid:105) = | ψ abc (cid:105) . FIG. 4. The Dehn twist ˜ T is along the x − y plane, thus itis natural to think of T as a solid hollow 2-torus where theinner and outer boundaries are identified, here the thickeneddirection is z . In this picture, we can think of ˜ T just as ausual Dehn twist of a 2-torus. We can add other string excitations on the boundary,however they will not give rise to new 3-torus groundstates after gluing. We thus see a generalization of thesituation in 2D, where there is a direct relation betweennumber of excitation types and GSD on the torus.Now lets to back to the open boundaries, and con-sider making a 2 π twist of one of the boundaries, whichwill give some kind of 3D analogue of topological spin .It can be seen that most states will be invariant undersuch an operation by appropriately deforming and recon-necting the string and membrane operators. For exam-ple | e a (cid:105) → | e a (cid:105) , which implies that the particles e a arebosons. However we pick up a factor of ω ab for | e a m ,b (cid:105) and | e a m ,c m ,b (cid:105) , since the string corresponding to parti-cle e a has to cross the membrane corresponding to m ,b .Physically this is a consequence of mutual statistics ofthe particle and string excitation. We can consider theseas 3D analogue of topological spin.Now notice that this operation precisely correspondsto the ˜ T Dehn twist on the 3-torus by gluing theboundaries (see fig.4). Thus ˜ T , as calculated from theground state, should contain information about statis-tics of excitations. Writing ˜ T abc, ¯ a ¯ b ¯ c = δ a, ¯ a δ b, ¯ b δ c, ¯ c e πiN ab ≡ δ a, ¯ a δ b, ¯ b δ c, ¯ c ˜ T abc , we get the following 3D topological spins ˜ T = ˜ T = 1 , ˜ T e a = ˜ T a = 1 , ˜ T m ,c = ˜ T c = 1 , ˜ T m ,b = ˜ T b = 1 , ˜ T e a m ,c = ˜ T a c = 1 , ˜ T e a m ,b = ˜ T ab = e πiN ab , ˜ T m ,c m ,b = ˜ T bc = 1 , ˜ T e a m ,c m ,b = ˜ T abc = e πiN ab . This exactly match the properties of the excitations.Thus the universal quantity ˜ T calculated from the groundstate alone, contain direct physical information aboutstatistics of excitations in the system. Note that ele-ments like ˜ T m ,c m ,b can be non-trivial in theories withnon-trivial string-string statistics. D → D Dimensional Reduction
We can actually relate these universal quantities to thewell-known S and T matrices in two dimensions. Con-sider now the SL (2 , Z ) subgroup of SL (3 , Z ) generatedbyˆ T yx ≡ and ˆ S yx ≡ − (12)One can directly compute the representation of this sub-group for the above Z N model, which is given by S yxabc, ¯ a ¯ b ¯ c = 1 N δ c, ¯ c e − πiN ( a ¯ b +¯ ab ) , T yxabc, ¯ a ¯ b ¯ c = δ a, ¯ a δ b, ¯ b δ c, ¯ c e πiN ab . Note that S D Z N = (cid:76) Nn =1 S D Z N and T D Z N = (cid:76) Nn =1 T D Z N . Inparticular, for the toric code N = 2 we have S yx = 12 − − − − − − − − − − − − , and T yx = − − . These N blocks are distinguished by eigenvalues of W z .Consider the 2D limit of the three-dimensional Z N modelwhere the x and y directions are taken to be verylarge compared to the z direction. In this limit a non-contractible loop along the z -cycle becomes very smalland the following perturbation is essentially local H = H D, Z N − J z (cid:0) W z + W † z (cid:1) , (13)where W z creates a loop along z . Since this perturba-tion commutes with the original Hamiltonian, besidesthe conditions (7) the ground state must also satisfy W z | GS (cid:105) = | GS (cid:105) . Thus the N -fold degeneracy is not sta-ble in the 2D limit and the N remaining ground statesare now | D, a, b (cid:105) ≡ | ψ ab (cid:105) . The gap to the state | ψ abc (cid:105) is ∆ E c = J c [1 − cos( πN c )].It is easy to see that S yx and T yx on this set of groundstates exactly correspond the two dimensional Z N modu-lar matrices and can be used to construct the correspond-ing UMTC. Thus the 3D Z N model and our universalquantities exactly reduce to the 2D versions in this limit.Furthermore, the 3D quasiparticle basis also directly re-duce to the 2D quasiparticle basis. IV. QUANTUM DOUBLE MODELS INTHREE-DIMENSIONS
In this section we will construct exactly soluble mod-els in three-dimensions for any finite group G . These arenothing but a natural generalization of Kitaev’s quan-tum double models to three-dimensions and are closelyrelated to discrete gauge theories with gauge group G .These models will have the above Z N models as a specialcase, but formulated in a slightly different way.Consider a simple cubic lattice with the orientationused above. Let there be a Hilbert space H l ≈ C [ G ] oneach link l , where G is a finite group, and let there bean isomorphism H l ∼ → H l (cid:63) for the link l and its reverseorientation l (cid:63) as | g l (cid:105) (cid:55)→ | g l (cid:63) (cid:105) = | g − l (cid:105) . Furthermore letthe natural basis of the group algebra be orthonormal.The following local operators will be useful L g + | z (cid:105) = | gz (cid:105) , T h + | z (cid:105) = δ h,z | z (cid:105) ,L g − | z (cid:105) = | zg − (cid:105) , T h − | z (cid:105) = δ h − ,z | z (cid:105) . To each two dimensional plaquette p , associate a orienta-tion wrt. to the lattice orientation using the right-handrule. For such a plaquette, define the following operator B h ( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z D z L z U z R p (cid:43) = δ z U z − R z − D z L ,h (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z D z L z U z R p (cid:43) , and similar for other orientations of plaquettes. Notethat the order of the product is important for non-Abelian groups. To each lattice site s , define the operator A g ( s ) = (cid:89) l − L g − ( l − ) (cid:89) l + L g + ( l + ) , where l − are the set of links pointing into s while l + arethe links pointing away from s . In particular we havethat A g ( s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z z x x y y s (cid:43) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z g − gz x g − gx gy y g − s (cid:43) . From these we have two important operators A ( s ) = 1 | G | (cid:88) g ∈ G A g ( s ) , and B ( p ) ≡ B ( p ), where 1 ∈ G is the identity element.One can show that both these operators are hermitianprojectors. Furthermore one can check that they all com-mute together (cid:2) A ( s ) , B ( p ) (cid:3) = 0 , ∀ s, p, (cid:2) B ( p ) , B ( p (cid:48) ) (cid:3) = 0 , ∀ p, p (cid:48) , (cid:2) A ( s ) , A ( s (cid:48) ) (cid:3) = 0 , ∀ s, s (cid:48) . We can now define the Hamiltonian of the three-dimensional quantum double model as H = − J e (cid:88) s A ( s ) − J m (cid:88) p B ( p ) . (14)Since the Hamiltonian is just a sum of commuting pro-jectors, the ground states of the system must satisfy A ( s ) | GS (cid:105) = B ( p ) | GS (cid:105) = | GS (cid:105) , for all s and p . The ground state can be con-structed using the following hermitian projector ρ GS = (cid:81) s A ( s ) (cid:81) p B ( p ). If we take as reference state | (cid:105) = | l l . . . (cid:105) , we can write | GS (cid:105) = ρ GS | (cid:105) = (cid:89) s A ( s ) | (cid:105) . A. Ground states on T The easiest way to construct the ground states on thethree-torus is to consider the minimal torus, which is justa single cube where the boundaries are identified. Theminimal torus has one site s aab bcc s and three plaquettes p , p , p ba b ap ca c ap cc b ap One can readily show that the subspace H B =1 satisfying B ( p ) | GS (cid:105) ! = | GS (cid:105) for p = p , p , p , is spanned by thevectors | a, b, c (cid:105) such that ab = ba , bc = cb and ac = ca .The last condition is A ( s ) | GS (cid:105) = | GS (cid:105) where on the basisvectors A ( s ) | a, b, c (cid:105) = 1 | G | (cid:88) g ∈ G | gag − , gbg − , gcg − (cid:105) . In the case of Abelian groups G , this condition is clearlytrivial and then we have GSD = | G | . In general we canfind the ground state degeneracy by taking the trace ofthe projector A ( s ) in H B =1 . This is given by GSD = (cid:88) { a,b,c } (cid:104) a, b, c | A ( s ) | a, b, c (cid:105) = 1 | G | (cid:88) g ∈ G (cid:88) { a,b,c } δ ag,ga δ bg,gb δ cg,gc , where { a, b, c } is triplets of commuting group elements.One can actually easily check that the following vectorsspan the ground state subspace | ψ [ a,b,c ] (cid:105) = 1 | G | (cid:88) g ∈ G | gag − , gbg − , gcg − (cid:105) , (15)where [ a, b, c ] = { (˜ a, ˜ b, ˜ c ) ∈ G × G × G | (˜ a, ˜ b, ˜ c ) =( gag − , gbg − , gcg − ) , g ∈ G } is the three-element con-jugacy class and a, b, c are representatives of the class. B. D ˜ S and ˜ T matrices and the SL (2 , Z ) subgroup We can now readily compute the overlaps (1) for theabove model for any group G . We find the followingrepresentations of MCG ( T ) = SL (3 , Z )˜ S [ a,b,c ] , [¯ a, ¯ b, ¯ c ] = (cid:104) ψ [ a,b,c ] | ˜ S | ψ [¯ a, ¯ b, ¯ c ] (cid:105) = δ [ a,b,c ] , [¯ b, ¯ c, ¯ a ] and ˜ T [ a,b,c ] , [¯ a, ¯ b, ¯ c ] = (cid:104) ψ [ a,b,c ] | ˜ T | ψ [¯ a, ¯ b, ¯ c ] (cid:105) = δ [ a,b,c ] , [¯ a, ¯ a ¯ b, ¯ c ] , since ˜ S | ψ [ a,b,c ] (cid:105) = | ψ [ b,c,a ] (cid:105) and ˜ T | ψ [ a,b,c ] (cid:105) = | ψ [ a,ab,c ] (cid:105) .Once again we can consider the subgroup SL (2 , Z ) ⊂ SL (3 , Z ) generated by (12). The representation of thissubgroup can be directly computed and is given by S yx [ a,b,c ] , [¯ a, ¯ b, ¯ c ] = (cid:104) ψ [ a,b,c ] | S yx | ψ [¯ a, ¯ b, ¯ c ] (cid:105) = δ [ a,b,c ] , [¯ b, ¯ a − , ¯ c ] and T yx [ a,b,c ] , [¯ a, ¯ b, ¯ c ] = (cid:104) ψ [ a,b,c ] | T yx | ψ [¯ a, ¯ b, ¯ c ] (cid:105) = δ [ a,b,c ] , [¯ a, ¯ a ¯ b, ¯ c ] . Note that since c is not independent of a and b , in generalwe don’t have the decomposition S DG = (cid:76) | G | n =1 S DG and T DG = (cid:76) | G | n =1 T DG , unless the group is Abelian. C. Branching Rules and Dimensional Reduction
With the above formulas, we can directly compute the˜ S and ˜ T generators for any group G. In the limit whereone direction of the 3-torus is taken to be very small, wecan view the 3D topological order as several 2D topolog-ical orders.The branching rules (3) for the dimensional reductioncan be directly computed by studying how a representa-tion of SL (3 , Z ) decomposes into representations of thesubgroup SL (2 , Z ) ⊂ SL (3 , Z ). For example, for someof the simplest non-Abelian finite groups we find thebranching rules C DS = C DS ⊕ C D Z ⊕ C D Z , C DD = 2 C DD ⊕ C DD ⊕ C D Z , C DD = C DD ⊕ C D Z ⊕ C D Z , C DS = C DS ⊕ C DD ⊕ C DD ⊕ C D Z ⊕ C D Z . In general we find the following branching in the di-mensional reduction C DG = (cid:76) C C DG C , where (cid:76) C sumsover all different conjugacy classes C of G , and G C isthe centralizer subgroup of G for some representative g C ∈ C . Similar to the G = Z N case above (13), thedegeneracy between the different sectors can be liftedby a perturbation creating Wilson loops along the smallnon-contractible cycle of T , which is essentially a localperturbation in the 2D limit.We like to remark that the above branching result fordimensional reduction can be understood from a “gaugesymmetry breaking” point of view. In the dimensionalreduction, we can choose to insert gauge flux throughthe small compactified circle. The different choices ofthe gauge flux is given by the conjugacy classes C of G .Such gauge flux break the “gauge symmetry” from G to G C . So, such a compactification leads to a 2D gauge the-ory with gauge group G C and reduces the 3D topologicalorder C DG to a 2D topological order C DG C . The differ-ent choices of gauge flux lead to different degenerate 2Dtopological ordered states, each described by C DG C for acertain G C . This gives us the result eqn. (6). It is quiteinteresting to see that the branching (4) of the represen-tation of the mapping class group SL (3 , Z ) → SL (2 , Z )is closely related to the “gauge symmetry breaking” inour examples.In order to gain a better understanding of the informa-tion contained in these branching rules, we will considera simple example. V. EXAMPLE: G = S A. Two-Dimensional D ( S ) Let us consider the simplest non-Abelian group G = S . Let us first recall the 2D quantum double models.The excitations of these models are given by irreduciblerepresentations of the Drinfeld Quantum Double D ( G ).The states can be labelled by | C, ρ (cid:105) , where C denote aconjugacy class of G while ρ is a representation of thecentralizer subgroup G C ≡ Z ( a ) = { g ∈ G | ag = ga } ofsome element in a ∈ C (note that Z ( a ) ≈ Z ( gag − )).The symmetric group G = S consists of the elements { () , (23) , (12) , (123) , (132) , (13) } , where ( . . . ) is the stan-dard notation for cycles (cyclic permutations). There arethree conjugacy classes A = { () } , B = { (12) , (13) , (23) } and C = { (123) , (132) } , with the corresponding central-izer subgroups G A = S , G B = Z , G C = Z . The num-ber of irreducible representations for each group is equalto the number of conjugacy classes, 3 for G A and G C while 2 for G B . For simplicity we will label the particlescorresponding to the three different conjugacy classes by( , A , A ), ( B, B ) and ( C, C , C ). Here the particleswithout a superscript, B and C , are pure fluxes (trivialrepresentation), A and A are pure charges (trivial con-jugacy class), while B , C and C are charge-flux com-posites. The fusion rules for the two-dimensional D ( S ) ⊗ A A B B C C C A A B B C C C A A A B B C C C A A A ⊕ A ⊕ A B ⊕ B B ⊕ B C ⊕ C C ⊕ C C ⊕ C B B B B ⊕ B ⊕ A ⊕ C ⊕ C ⊕ C A ⊕ A ⊕ C ⊕ C ⊕ C B ⊕ B B ⊕ B B ⊕ B B B B B ⊕ B A ⊕ A ⊕ C ⊕ C ⊕ C ⊕ A ⊕ C ⊕ C ⊕ C B ⊕ B B ⊕ B B ⊕ B C C C C ⊕ C B ⊕ B B ⊕ B ⊕ A ⊕ C C ⊕ A C ⊕ A C C C C ⊕ C B ⊕ B B ⊕ B C ⊕ A ⊕ A ⊕ C C ⊕ A C C C C ⊕ C B ⊕ B B ⊕ B C ⊕ A C ⊕ A ⊕ A ⊕ C TABLE I. Fusion rules of two-dimensional D ( S ) model. Here B and C correspond to pure flux excitations, A and A purecharge excitations, the vacuum sector while B , C and C are charge-flux composites. model is given in table I. B. Three-Dimensional G = S Model
In three dimensions, the S model has two point-liketopological excitations, which are pure charge excitationsthat can be labelled by A D and A D . Here A is the one-dimensional irreducible representation of S and A thetwo-dimensional irreducible representation of S . Underthe dimensional reduction to 2D, they become the 2Dcharge particles labelled by A and A . The S modelalso has two string-like topological excitations, labelledby the non-trivial conjugacy classes B D and C D . Un-der the dimensional reduction to 2D, they become the2D particles with pure fluxes described by B and C .(For details, see the discussion below.) We can alsoadd a 3D charged particle to a 3D string and obtain aso called mixed string-charge excitation. Those mixedstring-charge excitations are labelled by B D , C D , and C D , and, under the dimensional reduction, become the2D particles B , C , and C .We like to remark that, since a 3D string carries gaugeflux described by a conjugacy class B or C , the S “gaugesymmetry” is broken down to G B = Z on the B D string, and down to G C = Z on the C D string.Under the symmetry breaking S → Z , the two irre-ducible representations A and A of S reduce to theirreducible representations 1 and e of Z : A → e and A → ⊕ e . Thus fusing the S charge A D to a B D string give us the mixed string-charge excitation B D .But fusing the S charge A D to a B D string gives usa composite mixed string-charge excitation B D ⊕ B D .(The physical meaning of the composite topological ex-citations B D ⊕ B D is explained in Ref. 32.) So fusingthe two non-trivial S charges to a B D string only giveus one mixed string-charge excitation B D .Under the symmetry breaking S → Z , the two irre-ducible representations A and A of S reduce to theirreducible representations 1, e and e of Z : A → A → e ⊕ e . Thus fusing the S charge A to a C D string still gives us the string excitation C D . But fusingthe S charge A D to a C D string gives us a compos-ite mixed string-charge excitation C D ⊕ C D . So fusingthe two non-trivial S charges to a C string give us two ab c FIG. 5. Three string configuration, where two loops of type b and c are threaded by a string of type a . mixed string-charge excitations C D and C D . We seethat the fusion between point S charges and the stringsis consistent with fusion of the corresponding 2D parti-cles.Now, we would like to understand the fusion andbraiding properties of the 3D strings B D and C D .To do that, let us consider the dimension reduction C DS = C DS ⊕ C D Z ⊕ C D Z . Let us choose the gauge fluxthrough the small compactified circle to be B . In thiscase C DS → C D Z . C D Z is a Z topological order in 2Dand contains four particle-like topological excitations , e, m, f , where is the trivial excitations. e is the Z charge and m the Z vortex, which are both bosons. f isthe bound state of e and m which is a fermion. The triv-ial 2D excitation comes from the trivial 3D excitation D , and the Z charge e comes from the 3D charge ex-citation A . The 3D string excitations B and B , wrap-ping around the small compactified circle, give rise to twoparticle-like excitations in 2D – the Z vortex m and thefermion f . In the dimensional reduction, the gauge flux B through the small compactified circle forbids the 3Dstring excitations C D , C D , and C D to wrap around thesmall compactified circle. So there is no 2D excitationsthat correspond to the 3D string excitations C D , C D ,and C D . Because of the symmetry breaking S → Z caused by the gauge flux B , the 3D particle A D reducesto ⊕ e in 2D.The above results have a 3D understanding. Let usconsider the situation where two loops, b and c , arethreaded by string a (see Fig. 5). If the a -string is thetype- B D string, then the b and c -strings must also bethe type- B D string. So the type B D string in the cen-ter forbids the 3D strings C D , C D , and C D to looparound it. This is just like the gauge flux B throughthe small compactified circle forbids the 3D string exci-tations C D , C D , and C D to wrap around the smallcompactified circle. So the type- B D string in the cen-ter corresponds to the gauge flux B through the smallcompactified circle.The fusion and braiding of the 2D particle e is verysimple: it is an boson with fusion e ⊗ e = . This is con-sistent with the fact that the corresponding 3D particle A D is a boson with fusion A D ⊗ A D = D . The fu-sion and braiding of the 2D particle m is also very simple,since it is also an boson m ⊗ m = . This suggests thatthe 3D type- B D string excitations has a simple fusionand braiding property, provided that those 3D string ex-citations are threaded by a type- B D string going throughtheir center (see Fig. 5). For example, from the 2Dfusion rule m ⊗ m = , we find that the fusion of twotype- B D loops give rise to a trivial string B D ⊗ B D = D . (16)As suggested by the 2D braiding of two m particles,when a type- B D string going around another type- B D string, the induced phase is zero ( i.e. the mutual braiding“statistics” is trivial).Similarly, we can choose the gauge flux through thesmall compactified circle to be C . In this case C DS →C D Z , and C D Z is a Z topological order in 2D which has9 particle types: , e , e , m , m , e i m j | i,j =1 , . In thiscase, the gauge flux C through the small compactifiedcircle forbids the 3D string excitations B D and B D towrap around the small compactified circle. So there isno 2D excitations that correspond to the 3D string ex-citations B D and B D . The 3D string excitation C D wrapping around the small compactified circle gives riseto a composite Z vortex m ⊕ m in 2D. (This is be-cause there are two non-trivial group elements in S thatcommute with a group element in the conjugacy class C ).Also, from the S → Z symmetry breaking: A → A → e ⊕ e , we see that the 3D A D charge reduces totype- particle in 2D, and the 3D A D charge reduce toa composite particle e ⊕ e in 2D.The fusion of the composite 2D particle c = m ⊕ m is given by c ⊗ c = 2 ⊕ c. (17)This leads to the corresponding fusion rule for the 3Dtype- C D loops C D ⊗ C D = 2 D ⊕ C D or D ⊕ A D ⊕ C D , (18) provided that those 3D loops are threaded by a type- C D string going through their center (see Fig. 5). (The am-biguity arises because the 3D charge A D reduces to in2D.)Now, let us choose the gauge flux through the smallcompactified circle to be trivial. In this case C DS → C DS ,which has 8 particle types: , A , A , B , B , C , C , C . The 3D string excitation B D and C D wrappingaround the small compactified circle gives rise to the 2D a A B C Symmetry Breaking S → S S → Z S → Z D → A D → A e A D → A ⊕ e e ⊕ e B D → B m - B D → B em - C D → C - m ⊕ m C D → C - e m ⊕ e m C D → C - e m ⊕ e m TABLE II. The situation of figure 5, where strings arewrapped around another string of type a = A, B, C . De-pending on a , fusion algebra and braiding statistics of eachstring will be related to a particle of some 2D topological or-der, as computed from the branching rules (6). See the textfor more details. excitation B and C . The fusion of the 2D particle C isgiven by C ⊗ C = ⊕ A ⊕ C. (19)This leads to the corresponding fusion rule for the 3Dtype- C D loops C D ⊗ C D = D ⊕ A D ⊕ C D , (20) provided that those 3D loops are not threaded by any non-trivial string . The above fusion rule implies that whenwe fusion two C D loops, we obtain three accidentallydegenerate states: the first one is a non-topological exci-tation, the second one is a S charge A D , and the thirdone is a S string C D .Similarly, the fusion of the 2D particle B is given by B ⊗ B = ⊕ A ⊕ C ⊕ C ⊕ C . (21)This leads to the corresponding fusion rule for the 3Dtype- B D loops B D ⊗ B D = D ⊕ A D ⊕ C D ⊕ C D ⊕ C D . (22)This way, we can obtain the fusion algebra between allthe 3D excitations A D , A D , B D , B D , C D , C D , C D .On the other hand, since the above 3D string loops arenot threaded by any non-trivial string, we can shrink asingle loop into a point. So we should be able to com-pute the fusion of 3D loops by shrinking them into apoints. Mathematically we will define shrinking opera-tion S , which describes the shrinking process of loops.Let E denote the set of 3D particle and string exci-tations. We would like to make sure that the shrinkingoperation is consistent with the fusion rules, ie S ( a ⊗ b ) = S ( a ) ⊗ S ( b ) for a, b ∈ E . One can indeed check that thisis the case for the following shrinking operations S ( C D ) = D ⊕ A D , S ( C D ) = A D , S ( C D ) = A D , S ( B D ) = D ⊕ A D , S ( B D ) = A D ⊕ A D . So indeed, we can compute the fusion of 3D loops byshrinking them into points. In particular, we find that0the topological degeneracy for N type- C D loops is 2 N / B D loops is 2.The topological degeneracy for N type- B D loops is oforder 3 N in large N limit.The above example suggests the following. Given atopological order in 3D, C D , one may want to considerthe situation illustrated in figure 5 where two loops b and c are threaded with a string a , and ask about the three-string braiding statistics. One way to compute this isto put the system on a 3-torus and compute the quan-tities (1), which give rise to a SL (3 , Z ) representation.Then by finding the branching rules of this representationwrt. to the subgroup SL (2 , Z ) ⊂ SL (3 Z ), one finds howthe systems decomposes in the 2D limit C D = (cid:76) i C Di ,where there will be a sector i for each string type. Thethree-string statistics with string a in the middle, will berelated to the 2D topological order C Da . To summarize: • The representation branching rule (4) for SL (3 , Z ) → SL (2 , Z ) leads to the dimensionreduction branching rule (3). • The number of the SL (2 , Z ) representations (or thenumber of induced 2D topological orders) is equalto the number of 3D string types in the 3D topo-logical order C D . • The SL (2 , Z ) representations also contains infor-mation about two-string/three-string fusion, asdescribed by eqns. (16,18,20,22). The two-string/three-string braiding can be obtained di-rectly from the correspond 2D braiding of the cor-responding particles. VI. SOME GENERAL CONSIDERATIONS
To calculate the braiding statistics of strings and par-ticles, we first need to know the topological degeneracy D in the presence of strings and particles before theybraid. This is because the unitary matrix that describethe braiding is D by D matrix. To compute the topo-logical degeneracy D , we need to know the topologicaltypes of strings and the particles since the topologicaldegeneracy D depends on those types.We have seen that, from the branching rules of SL (3 , Z ) representation under SL (3 , Z ) → SL (2 , Z ) (seeeqn. (4)) we can obtain the number of the string types.How to obtain the number of the particle types?To compute the number of the particle types, we startwith a 3D sphere S , and then remove two small ballsfrom it. The remaining 3D sphere will have two S sur-faces. This two surfaces may surround a particle andanti-particle. So the number of the particle types canbe obtained by calculating the ground state degeneracy.But there is one problem with this approach, the twosurfaces may carry gapless boundary excitations or someirrelevant symmetry breaking states. To fix this problem, we note that the 3D space S × I also have have two S surfaces, where I is the 1D seg-ment: I = [0 , S × I onto the3D sphere S with two balls removed, along the two 2Dspheres S . The resulting space is S × S . This way, weshow that the topological degeneracy on S × S is equalto the number of the particle types.For the gauge theory of finite gauge group G , the topo-logically degenerate ground states on S × S are la-belled by the group elements g ∈ G (which describe themonodromy along the non-contractible loop in S × S ),but not in an one-to-one fashion. Two elements g and g (cid:48) = h − gh label the same ground state since g and g (cid:48) are related by a gauge transformation. So the topolog-ical degeneracy on S × S is equal to the number ofconjugacy classes of G . The number of conjugacy classesis equal to the number of irreducible representations of G , which is also the number of the particle types, a wellknown result for gauge theory.Once we know the types of particles and strings, thesimple fusion and braiding of those excitations can beobtained from the dimensional reduction as described inthis paper. VII. CONCLUSION
In a recent work Ref. 19, we proposed that for a gapped d -dimensional theory on a manifold M , the overlaps (1)give rise to a representation of MCG ( M ) and that these arerobust against any local perturbation that do not closethe energy gap. In this paper we studied a simple class of Z N models on M = T and computed the correspondingrepresentations of MCG ( T ) = SL (3 , Z ). We argued that,similar to in 2D, the ˜ T generator contains informationabout particle and string excitations above the groundstate, although computed from the ground states. Inan independent work Ref. 21, the authors studied thematrices (1) using some Abelian models on T . Theyargued that the generator ˜ S contains information aboutbraiding processes involving three loops.Furthermore we studied a dimensional reduction pro-cess in which the 3D topological order can be viewedas several 2D topological orders C D = (cid:76) i C Di . Thisdecomposition can be computed from branching rulesof a SL (3 , Z ) representation into representations of a SL (2 , Z ) ⊂ SL (3 , Z ) subgroup. Interestingly, this re-duction encodes all the information about three-stringstatistics discussed in Ref. 20 for Abelian groups. Thisapproach, however, also provide information about fusionand braiding statistics of non-Abelian string excitationsin 3D.We also discussed how to obtain information aboutparticles by putting the theory on S × S . All this lendssupport for our conjecture , that the overlaps (1) fordifferent manifold topologies M , completely characterizetopological order with finite ground state degeneracy inany dimension.1This research is supported by NSF Grant No. DMR-1005541, NSFC 11074140, and NSFC 11274192. It is alsosupported by the John Templeton Foundation. Research at Perimeter Institute is supported by the Governmentof Canada through Industry Canada and by the Provinceof Ontario through the Ministry of Research. K. von Klitzing, G. Dorda, and M. Pepper, Phys. Rev.Lett. , 494 (1980). D. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys. Rev.Lett. , 1559 (1982). R. B. Laughlin, Phys. Rev. Lett. , 1395 (1983). V. Kalmeyer and R. B. Laughlin, Phys. Rev. Lett. , 2095(1987). X.-G. Wen, F. Wilczek, and A. Zee, Phys. Rev. B ,11413 (1989). N. Read and S. Sachdev, Phys. Rev. Lett. , 1773 (1991). X.-G. Wen, Phys. Rev. B , 2664 (1991). R. Moessner and S. L. Sondhi, Phys. Rev. Lett. , 1881(2001). G. Moore and N. Read, Nucl. Phys. B , 362 (1991). X.-G. Wen, Phys. Rev. Lett. , 802 (1991). R. Willett, J. P. Eisenstein, H. L. Str¨ormer, D. C. Tsui,A. C. Gossard, and J. H. English, Phys. Rev. Lett. ,1776 (1987). I. P. Radu, J. B. Miller, C. M. Marcus, M. A. Kastner,L. N. Pfeiffer, and K. W. West, Science , 899 (2008). X.-G. Wen, Phys. Rev. B , 7387 (1989). X.-G. Wen and Q. Niu, Phys. Rev. B , 9377 (1990). X.-G. Wen, Int. J. Mod. Phys. B , 239 (1990). X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B ,155138 (2010), arXiv:1004.3835. M. Levin and X.-G. Wen, Phys. Rev. B , 045110 (2005),cond-mat/0404617. K. Walker and Z. Wang, (2011), arXiv:1104.2632. H. Moradi and X.-G. Wen, (2014), arXiv:1401.0518 [cond-mat.str-el]. C. Wang and M. Levin, (2014), arXiv:1403.7437. S. Jiang, A. Mesaros, and Y. Ran, (2014),arXiv:1404.1062. E. Keski-Vakkuri and X.-G. Wen, Int. J. Mod. Phys. B ,4227 (1993). H. He, H. Moradi, and X.-G. Wen, (2014),arXiv:1401.5557. S. M. Trott, Canadian Mathematical Bulletin , 245(1962). A. Y. Kitaev, Ann. Phys. (N.Y.) , 2 (2003). Y. Hu, Y. Wan, and Y.-S. Wu, Phys. Rev. B , 125114(2013), arXiv:1211.3695. J. Wang and X.-G. Wen, to appear (2014). Two-dimensional version of this model has previously beenstudied in for example Ref. 33. Y. Zhang, T. Grover, A. Turner, M. Oshikawa, andA. Vishwanath, Phys. Rev. B , 235151 (2012),arXiv:1111.2342. A. Y. Kitaev, Annals Phys. , 2 (2003), arXiv:quant-ph/9707021 [quant-ph]. The model can easily be defined on arbitrary triangula-tions, but for simplicity we will consider the cubic lattice. T. Lan and X.-G. Wen, (2013), arXiv:1311.1784. M. D. Schulz, S. Dusuel, R. Or´us, J. Vidal, and K. P.Schmidt, New Journal of Physics14