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# Entanglement entropy and entanglement spectrum of the Kitaev model 

$^{1}$Department of Physics, University of California, Berkeley, CA 94720, USA 

Hong Yao 1, 2 and Xiao-Liang Qi 3, 4 

$^{2}$Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA 

$^{3}$Microsoft Research, Station Q, Elings Hall, University of California, Santa Barbara, CA 93106, USA 

$^{4}$Department of Physics, Stanford University, Stanford, CA 94305, USA 

(Dated: August 19, 2010) 

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. 

Exotic phases such as fractional quantum Hall (FQH) states, which are not in the paradigm of conventional symmetry breaking, were termed as topologically ordered [1] since they have robust ground state degeneracy which is protected against all local perturbations, but sensitive to the topology of the system [2]. A topologically or- dered state has non-local pattern of quantum entangle- ment, which is essential for the proposal of topological quantum computation [3–5]. 

By bipartitioning a system spatially, the entanglement entropy (EE) measures how closely entangled the two subsystems are. For a gapped system, EE is usually pro- portional to the area of the interface between the two subsystems in the thermodynamic limit. However, as discovered [6, 7] by Levin and Wen as well as Kitaev and Preskill, the entanglement entropy of a topologically or- dered state contains a universal constant term, which is uniquely determined by the topological order of the state, named as topological entanglement entropy (TEE). TEE enables a direct characterization of topological ordered states without referring to the Hamiltonian. EE and TEE are properties of a many-body state and are usually hard to compute. EE and/or TEE have been computed ex- actly or numerically for several models such as toric code model[3, 8, 9], FQH states[10–12] and quantum dimer models[13–15]. So far there has been no exact result for the EE of topologically ordered states whose quasiparti- cles obey non-Abelian statistics. 

This paper serves to fill in that gap by providing a simple but exact method to compute the EE for any eigenstate (either ground or excited states) of the Ki- taev model [16], which is one of the most important exact solvable models with non-Abelian anyons. The essence of our method is a rigorous proof that the EE of the Kitaev model is equal to that of two decoupled systems: a sourceless Z$_{2}$ gauge field and a free Majorana fermion system. Although the TEE of the ground state 

comes only from the Z$_{2}$ gauge field, the fermionic part is responsible for all nontrivial entanglement properties of the non-Abelian phase. Besides the EE, our method also enables the computation of the whole entanglement spectrum (ES), i.e. , the eigenvalue spectrum of the re- duced density matrix [17]. We show that the entangle- ment spectrum is gapless or gapped in the non-Abelian and Abelian phase of the Kitaev model, respectively. We propose a new quantity, the capacity of entanglement , which can be used to distinguish different topological states with gapped and gapless entanglement spectrum. 

Kitaev model is a spin-1/2 model originally proposed on the honeycomb lattice [16] with the Hamiltonian 

<img src='content_image/377209.jpg'>

where x, y, z -link stand for the three types of links. It has a non-Abelian phase when the time-reversal symmetry is broken either explicitly by magnetic field [16] or three- spin couplings [18], or spontaneously by decorating the honeycomb lattice [19]. For simplicity, hereafter we will present our exact results of EE and entanglement spec- trum for the Kitaev model on honeycomb lattice, but our approach can be generalized straightforwardly to a broad class of Hamiltonians, including the Kitaev model on any trivalent lattice and Gamma matrix models [20–24]. 

The Kitaev model can be solved by introducing the Majorana representation of the Pauli matrices [16]: σ α i = iγ α i η$_{i}$ ( α = x, y, z ), where γ α i and η$_{i}$ are Majorana fermion operators. γ α i and η$_{i}$ on each lattice site de- fine a 4-dimensional Hilbert space, so that the Majo- rana representation of a spin-1 / 2 is redundant. The physical Hilbert space is defined by a constraint D$_{i}$ = − iσ x i σ y i σ z i = γ x i γ y i γ z i η$_{i}$ = 1. In other words, a state | Ψ 〉 is physical only if D$_{i}$ | Ψ 〉 = | Ψ 〉 for every i . In the Majorana representation we have σ α i σ α j = γ α i γ α j η$_{i}$η$_{j}$ = − i ˆ u$_{ij}$η$_{i}$η$_{j}$ , in which the link operators ˆ u$_{ij}$ = iγ α i γ α j mutually com- 

