Entanglement Entropy and Entanglement Spectrum of the Kitaev Model

In this Letter, we obtain an exact formula for the entanglement entropy of the ground state and all excited states of the Kitaev model. Remarkably, the entanglement entropy can be expressed in a simple separable form $S=S_G+S_F$, with $S_F$ the entanglement entropy of a free Majorana fermion system and $S_G$ that of a $Z_2$ gauge field. The $Z_2$ gauge field part contributes to the universal “topological entanglement entropy” of the ground state while the fermion part is responsible for the nonlocal entanglement carried by the $Z_2$ vortices (visons) in the non-Abelian phase. Our result also enables the calculation of the entire entanglement spectrum and the more general Renyi entropy of the Kitaev model. Based on our results we propose a new quantity to characterize topologically ordered states—the capacity of entanglement, which can distinguish the states with and without topologically protected gapless entanglement spectrum.

Figure 1

The schematic honeycomb lattice is bipartitioned into two parts $A$ and $B$. The partition boundary (dashed line) cuts the links $\overline{a_n{b}_n}$, $n=1,\dots ,2L$. New $Z_2$ gauge variables (see the supplementary material [29]) ${\widehat{w}}_{A,n}$ and ${\widehat{w}}_{B,n}$ are introduced on the new (dotted) links $\overline{a_{2n-1}{a}_{2n}}$ and $\overline{b_{2n-1}{b}_{2n}}$, $n=1,\dots ,L$, respectively.

Figure 2

(a) Schematic picture of a torus and a cylinder, each split to two regions $A$ and $B$. The cylinder is equivalent to a sphere with two quasiparticles. (b) The entanglement spectrum ${\lambda }_n(k_y)$ vs $k_y$ for non-Abelian (red solid lines) and Abelian (blue dotted lines) states on the torus. Here and below, we take the parameters $J_x=J_y=J_z=1$ and next-nearest neighbor coupling $J^{\prime }=0.2$ for the non-Abelian state and $J_x=J_z=1$, $J_y=2.5$, $J^{\prime }=0.2$ for the Abelian state. (c) The entanglement spectrum for the non-Abelian state on cylinder. The blue circle marks an additional state with $\lambda =1/2$ at $k_y=0$. (d) The entropy $S(k_y)$ vs $k_y$ for non-Abelian (red solid line) and Abelian (blue dotted line) states on the torus and for non-Abelian state on cylinder (black dashed line with circles).

Figure 3

(a) Renyi entropy $S_{\alpha }$ and (b) capacity of entanglement $C_E$ defined by Eq. (8) of non-Abelian (red solid line) and Abelian (blue dashed line) states. The black dotted line is a linear fitting. The parameters are the same as those in Fig. 2.