Entanglement entropy and entanglement spectrum of the Kitaev model (1001.1165v2)

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.

• The paper presents an exact solution for entanglement entropy in the Kitaev model, showing it decomposes into contributions from a free Majorana fermion system and a Z2 gauge field.
• This decomposition reveals the topological entanglement entropy and non-local entanglement properties that distinguish Abelian from non-Abelian phases through their entanglement spectra.
• The study introduces "capacity of entanglement" to quantify states based on their entanglement spectrum and highlights potential for leveraging non-Abelian anyon signatures in quantum computing.

Analyzing Entanglement and Topological Order in the Kitaev Model

The paper by Hong Yao and Xiao-Liang Qi delivers an exact solution for the entanglement entropy within the Kitaev model. The Kitaev model, pivotal for understanding topological phases and quantum computation, supports both Abelian and non-Abelian anyons. This paper advances the theoretical analysis of such models by presenting an analytical expression for entanglement entropy across all quantum states—ground and excited—of the system. The key finding is the decomposition of entanglement entropy into two distinct contributions: $S_F$ from a free Majorana fermion system, and $S_G$ from a $Z_2$ gauge field. This separation highlights important topological distinctions between Abelian and non-Abelian phases.

The contribution from the $Z_2$ gauge field, $S_G$, accounts for the topological entanglement entropy (TEE), a universal constant reflecting the topological nature of a quantum state independently of its Hamiltonian specifics. It is shown that $S_G=(L-1)\log 2$, with $L$ indicating the interface's length between subsystem $A$ and $B$. The derived TEE, $S_{\text{topo}} = -\log 2$, aligns with theoretical expectations and corroborates earlier predictions about topological invariants in both phases of the Kitaev model.

Furthermore, the allocation $S = S_G + S_F$ enables a profound insight: while $S_G$ reflects the topological properties tied to $Z_2$ gauge fields, $S_F$ reveals non-local entanglement characteristics that differentiate Abelian from non-Abelian phases. Non-Abelian phases, notable for their potential in topological quantum computing, exhibit a gapless entanglement spectrum in contrast to the gapped spectrum of Abelian phases. This is vividly demonstrated through numerical calculations of the entanglement spectrum and majorana correlation functions.

The authors introduce the term "capacity of entanglement" to denote a novel quantitative measure that differentiates states based on their entanglement spectrum. This measure draws an analogy to heat capacity, emphasizing changes in the entanglement spectrum's structure across different topological phases. Crucially, this concept yields higher values in the presence of gapless spectra in the non-Abelian phase—analogous to critical behaviors in thermodynamics.

From a theoretical perspective, these findings deepen the understanding of entanglement in quantum many-body systems and enhance the classification of topological orders. Practically, this research underpins ongoing developments in quantum computing where entanglement properties are pivotal. Moving forward, translating these exact theoretical results to experimental techniques remains a significant challenge but is essential for leveraging the topological nature of quantum states in real applications. Exploiting the distinct entanglement signatures of non-Abelian anyons could provide ways to harness these states for practical quantum computation. Extensions of this analysis to other strongly correlated systems or higher-dimensional models could open exciting avenues in topological matter studies.