content_image/409832.jpg content_image/409837.jpg
divided by two parallel circles into A and B regions, as shown in Fig. 2 (a). The boundary circle is along the ˆ y direction. On the torus, the free Majorana fermion Hamiltonian can be block-diagonalized in the basis of k$_{y}$ : H = ∑ i,j η$_{i}$η$_{j}$h$_{ij}$ = ∑ x,x ′ ,k$_{y}$ η † $_{x}$( k$_{y}$ ) h$_{xx}$ ′ ( k$_{y}$ ) η$_{x}$ ′ ( k$_{y}$ ) . Thus the system consists of decoupled one-dimensional subsystems of each k$_{y}$ . The EE is given by [31] 

<img src='content_image/409832.jpg'>

where λ$_{n}$ ( k$_{y}$ ) are the eigenvalues of the single-particle correlation function C$_{xx}$ ′ ( k$_{y}$ ) = 〈 η † $_{x}$( k$_{y}$ ) η$_{x}$ ′ ( k$_{y}$ ) 〉 for each k$_{y}$ . λ$_{n}$ plays the role of Fermi-Dirac distribution 1 / ( e βϵ$_{n}$ + 1) in thermal entropy, so that λ$_{n}$ = 0 (1) cor- responds to fully occupied (unoccupied) states, respec- tively. The “entanglement spectrum” (ES) λ$_{n}$ ( k$_{y}$ ) has been computed numerically for both non-Abelian and Abelian phases, as shown in Fig. 2 (b), for the Kitaev model with three-spin terms J ′ [18]. The ES is gapped for the Abelian phase, and gapless for the non-Abelian phase, similar to the edge states in the energy spectrum. Similar observation has been made in topological insu- lators and superconductors [32–35] and in FQH systems [17]. The two gapless branches in the ES come from the two boundaries between A and B . Since λ$_{n}$ ( k$_{y}$ )’s 

<img src='content_image/409837.jpg'>

are smooth functions of k$_{y}$ , we see from Eq. (7) that in the continuum limit S$_{F}$ = ∑ k$_{y}$ S ( k$_{y}$ ) ≃ L ∫ S ( k$_{y}$ ) dk$_{y}$ 2 π satisfies the area law. It is interesting to note that a “gap” always exists between the edge states and other bulk states with λ$_{n}$ ( k$_{y}$ ) close to 0 or 1, which is analo- gous to the “entanglement gap” studied in Ref. [36] for FQH system. 

The situation becomes more interesting when we con- sider a cylinder with periodic boundary condition (PBC) and the partition shown in Fig. 2 (a). As shown in Fig. 2 (c), in the non-Abelian phase the numerical cal- culation gives only one branch of “gapless” states in the entanglement spectrum. Physically, this is because the coupling through the other boundary between A and B is removed by the open boundary condition. However, at k$_{y}$ = 0 there is one isolated additional state with λ = 1 / 2, as shown by the blue circle in Fig. 2 (c), which is due to the non-local entanglement between the two Majorana zero modes at the open boundary. Consequently, the en- tropy S ( k$_{y}$ ) is not a smooth function of k$_{y}$ but has an $_{additional log}$√$_{2 contributed by}$ k$_{y}$ = 0, as shown in Fig. 2 (d). Compared with the torus case, in the ther- modynamic limit we get S$_{F}$ = αL $_{+ log}$√$_{2, which shows}$ explicitly that in the non-Abelian phase a cylinder with PBC is topologically equivalent to a sphere with two non- Abelian quasiparticles (usually named as σ particles), as $_{illustrated in Fig. 2 (a). Each particle carries a log}$√$_{2}$ entropy which is solely contributed by the fermion sector. 

Besides the EE, more information is contained in our result. The fact that Eq. (4) holds for any posi- tive integer n indicates that the many-body entangle- ment spectrum—the eigenvalue spectrum of ρ$_{A}$ is the direct product of the ones of ρ$_{A,G}$ and ρ$_{A,F}$ . From Tr$_{A,G}$ [ ρ n A,G ] = 2 − ( L − 1)( n − $^{1)}$, one can know that ρ$_{A,G}$ has 2 L − 1 nonzero eigenvalues, all of which are degen- erate and have the value of 2 − ( L − $^{1)}$. Consequently all non-vanishing eigenvalues of ρ$_{A}$ are given by those of the Majorana fermion reduced density matrix ρ$_{A,F}$ times 2 − ( L − $^{1)}$. Thus the low “energy” ( i.e. , close to the maxi- mal eigenvalue of ρ$_{A}$ ) feature in the entanglement spectra of ρ$_{A}$ can be entirely characterized by its fermionic part, which is gapped in the Abelian phase and gapless in the non-Abelian phase, as shown in Fig. 2 (b). 

Such a qualitative difference in the entanglement spec- trum can be characterized by the Renyi entropy [37] S$_{α}$ = 1 1 − α log Tr ρ $^{α}$, which reduces to the EE (or von Neu- mann entropy) at α → 1. According to Eq. (4) the Renyi entropy of Kitaev model is given by S$_{α}$ = S$_{Fα}$ + S$_{Gα}$ for any α , with S$_{Gα}$ and S$_{Fα}$ the contribution from Z$_{2}$ gauge fields and fermions, respectively. From Tr$_{A,G}$[ ρ n $_{A,G}$] = 2 − ( L − 1)( n − $^{1)}$, one can see that S$_{Gα}$ = S$_{G}$ = ( L − 1) log 2. Thus the TEE in S$_{Gα}$ is α independent, which is a generic property of the string-net models[38, 39]. The α depen- dence of S$_{Fα}$ in the Abelian and non-Abelian phases has qualitative difference due to their different entan- 

