2 To obtain ρ$_{A}$ , we trace out the spin degree of free- dom in B , which normally can be carried out in terms of fermions and gauge fields. However, the gauge fields on the partition boundary are shared by A and B ; so we regroup those gauge fields on the boundary links to introduce new Z$_{2}$ gauge variables which lives in A and B exclusively, as shown in Fig. 1. (see supplementary ma- terial [29] for details.) The calculation of Tr[ ρ n $_{A}$] requires some careful treatment of the gauge transformation but is a well-defined mathematical procedure. Thus, we will leave the details involved in obtaining ρ$_{A}$ and Tr ρ n A to the supplementary material [29] and present only the fi- nal result here: mute and also commute with the Hamiltonian. Since ˆ u 2 ij = 1, ˆ u$_{ij}$ can be considered as c -numbers with values u$_{ij}$ = ± 1, so that the Kitaev model is equivalent to a free model of η Majorana fermions coupled to static Z$_{2}$ gauge fields [16, 18, 19, 24–27]. The ground state of such a model is given by the direct product of a Z$_{2}$ gauge con- figuration | u 〉 and the corresponding Majorana fermion ground state | φ ( u ) 〉 . Here the configuration u is deter- mined by minimizing the fermion ground state energy. There is a macroscopic ground state degeneracy in the enlarged Hilbert space, because each state | u 〉 ⊗ | φ ( u ) 〉 is degenerate with all the states | u $^{′}$〉 ⊗  | φ ( u $^{′}$) 〉 with u ′ gauge equivalent to u . However, such a degeneracy is removed when the constraint D$_{i}$ = 1 is applied. The physical ground state is the “gauge” average of the degenerate states, implemented by the projection [16]: Tr$_{A}$[ ρ n $_{A}$] = Tr$_{A,G}$ [ ρ n $_{A,G}$] · Tr$_{A,F}$ [ ρ n $_{A,F}$] , (4) | Ψ 〉 = 1 √$_{2}$ N +1 ∑ g D$_{g}$ | u 〉 ⊗ | φ ( u ) 〉 (2) for any positive integer n . Here ρ$_{A,F}$ = Tr$_{B}$[ | φ ( u ) 〉 〈 φ ( u ) | ] and ρ$_{A,G}$ = Tr$_{B}$[ | G ( u ) 〉 〈 G ( u ) | ] are the reduced density matrices for the free Majorana fermion state | φ ( u ) 〉 , and a pure Z$_{2}$ gauge field [9], respectively, and the ground state of the Z$_{2}$ gauge field | G ( u ) 〉 is given by a equal weight superposition of all the 2 N − 1 gauge field con- figurations | ˜ u 〉 that are gauge equivalent to | u 〉 , i.e. , | G ( u ) 〉 = 2 − ( N − 1) / 2 ∑ ˜ u ≃ u | ˜ u 〉 . Physically, such a sim- plification occurs because the effect of the gauge trans- formation D$_{g}$ on the fermion state in region B is canceled out once the trace over gauge field configurations is taken. Combining Eq. (4) and Eq. (3), it is now obvious that the EE S can be separated into gauge field part S$_{G}$ and fermion part S$_{F}$ as follows: S = S$_{G}$ + S$_{F}$ . (5) where N is the total number of lattice sites, g denotes a set of lattice sites, and D$_{g}$ = ∏ i ∈ g D$_{i}$ . We define D = ∏ i $_{∈L}$D$_{i}$ with L the set of all lattice sites. The sum ∑ g is taken over all possible subsets g of L . Note that, in Eq. (2), we implicitly assumed that D | u 〉 ⊗ | ψ ( u ) 〉 = | u 〉 ⊗ | ψ ( u ) 〉 because states with D = − 1 will be annihilated by the projection. Consequently, we have D$_{g}$ = DD$_{¯}$ g for the complement ¯ g = L − g , so that D$_{g}$ | u 〉 ⊗ | φ ( u ) 〉 = D$_{¯}$ g = L− g | u 〉 ⊗ | φ ( u ) 〉 . In other words there are only 2 N − 1 inequivalent gauge transformations, as expected. We define the Kitaev model on a torus and bipar- tite the lattice into subsystems A and B , as shown in Fig. 1. The EE between A and B is defined as S = − Tr$_{A}$ [ ρ$_{A}$ log ρ$_{A}$ ], where ρ$_{A}$ = Tr$_{B}$ ρ = Tr$_{B}$ | Ψ 〉 〈 Ψ | is the reduced density matrix of A . To calculate the EE, we will follow the “ replica trick ” introduced in Ref. [28] Eq. (4) and (5) are among the central results of this work. By explicit calculation [29] one can obtain Tr$_{A,G}$[ ρ n $_{A,G}$] = 2 − ( L − 1)( n − $^{1)}$, so that S$_{G}$ = ( L − 1) log 2. As will be shown below, the fermion part has the form S$_{F}$ = αL + o (1), where α is a positive constant and o (1) represents terms which vanish as L → ∞ . In the thermodynamic limit, the total entanglement entropy is given by S = ( α + log 2) L − log 2 , (6) from which we conclude that the TEE is S$_{topo}$ = − log 2. Our derivation is valid for all phases of Kitaev model, including the Abelian ( Z$_{2}$ gauge theory), non-Abelian (Ising anyon) phases, and also gapless phases. Thus our result directly proves that the TEE for the Abelian and non-Abelian phases are identical, as expected from the total quantum dimensions of their quasiparticles [30]. Despite its trivial contribution to TEE, the fermion sector S$_{F}$ is responsible for all the essential differences between the Abelian and non-Abelian phase of Kitaev model in their quantum entanglement properties. The EE of a free fermion system can be computed by the method introduced in Ref. [31]. To obtain an explicit understanding to the fermion EE we consider a torus a$_{1}$ b$_{1}$ a$_{2}$ b$_{2}$ a$_{2L}$ b$_{2L}$ A B a$_{3}$ b$_{3}$ b$_{2L-}$$_{1}$ a$_{2L-}$$_{1}$ b$_{4}$ a$_{4}$ FIG. 1: The schematic honeycomb lattice is bipartitioned into two parts A and B . The partition boundary (dashed line) cuts the links a n b n , n = 1 , · · · , 2 L . New Z$_{2}$ gauge variables (see text) ˆ w A,n and ˆ w B,n are introduced on the new (dotted) links a$_{2}$ n$_{−}$ $_{1}$a$_{2}$ n and b$_{2}$ n$_{−}$ $_{1}$b$_{2}$ n , n = 1 , · · · , L , respectively. S = − Tr$_{A}$ [ ρ$_{A}$ log ρ$_{A}$ ] = − ∂ ∂n Tr$_{A}$ [ ρ n A ] ∣ n $_{=1}$. (3) The entanglement entropy can be obtained if we can com- pute Tr$_{A}$ [ ρ n $_{A}$] for arbitrary positive integer n and then extrapolate the result to n ∈ R .