Entanglement entropy and entanglement spectrum of the Kitaev model

TL;DR

This work derives an exact and modular formula for the entanglement entropy of the Kitaev model, expressing it as $S = S_G + S_F$ with a universal gauge-field contribution and a free Majorana-fermion contribution. It shows the gauge part yields the topological entanglement entropy $S_{ ext{topo}} = -\log 2$ and that the fermionic part encodes nonlocal entanglement from visons, with the entanglement spectrum being gapless in the non-Abelian phase and gapped in the Abelian phase; the spectrum factorizes as a product of gauge and fermion sectors, enabling full Renyi entropies $S_\alpha = S_{F\alpha} + S_{G\alpha}$. The paper introduces the capacity of entanglement $C_E(t)$, which distinguishes topologically ordered states with gapless versus gapped entanglement spectra and provides a diagnostic linked to conformal field theory data in the appropriate limit. Overall, the results supply an exact, broad framework for entanglement in Kitaev-like models and offer new tools for characterizing topological order through entanglement spectra and capacity measures.

Abstract

In this paper, 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 non-local 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.

Figures (3)

• Figure 1: The schematic honeycomb lattice is bipartitioned into two parts $A$ and $B$. The partition boundary (dashed line) cuts the links $\overline{a_nb_n}$, $n=1,\cdots,2L$. New $Z_2$ gauge variables (see text) $\hat{w}_{A,n}$ and $\hat{w}_{B,n}$ are introduced on the new (dotted) links $\overline{a_{2n-1}a_{2n}}$ and $\overline{b_{2n-1}b_{2n}}$, $n=1,\cdots,L$, respectively.
• Figure 2: (Color online) (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 quasi-particles. (b) The entanglement spectrum $\lambda_n(k_y)$ versus $k_y$ for non-Abelian (red solid lines) and Abelian (blue dotted lines) state on torus. Here and below, we take the parameters $J_x=J_y=J_z=1$ and next-nearest neighbor coupling $J'=0.2$ for the non-Abelian state, and $J_x=J_z=1,~J_y=2.5,~J'=0.2$ 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 $S(k_y)$ versus $k_y$ 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: (Color online) (a) Renyi entropy $S_\alpha$, and (b) capacity of entanglement $C_E$ defined by Eq. (\ref{['Ecapacity']}), 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.