General Relationship Between the Entanglement Spectrum and the Edge State Spectrum of Topological Quantum States

TL;DR

This paper provides a demonstration of the observation made by Li and Haldane about the relationship between the entanglement spectrum and the spectrum of a physical edge state, using the tools of boundary conformal field theory.

Abstract

We consider (2+1)-dimensional topological quantum states which possess edge states described by a chiral (1+1)-dimensional Conformal Field Theory (CFT), such as e.g. a general quantum Hall state. We demonstrate that for such states the reduced density matrix of a finite spatial region of the gapped topological state is a thermal density matrix of the chiral edge state CFT which would appear at the spatial boundary of that region. We obtain this result by applying a physical instantaneous cut to the gapped system, and by viewing the cutting process as a sudden "quantum quench" into a CFT, using the tools of boundary conformal field theory. We thus provide a demonstration of the observation made by Li and Haldane about the relationship between the entanglement spectrum and the spectrum of a physical edge state.

Figures (2)

• Figure 1: (a) A topological state on a cylinder with a bipartition into two regions $A$ and $B$. (b) The deformed system (see text) with the coupling between $A$ and $B$ regions weighted by a factor $\lambda\in[0,1]$. The system can be understood as two cylinders $A$ and $B$, with edge states propagating along the boundary between $A$ and $B$, coupled by an inter-edge coupling. (c) For small enough $\lambda$, the coupling between the gapped bulk states can be neglected, and the problem can be reduced to an inter-edge coupling problem described by a $(1+1)$-dimensional conformal field theory with a relevant coupling $\lambda H_{\rm int}$.
• Figure 2: Illustration of the RG flow of model (\ref{['FQHedge']}) and (\ref{['FQHint']}) with two parameters $\lambda$ and $R$. Any two points A and B in the gapped phase can be connected by a continuous path (red dash line). By this continuous deformation one can show that the entanglement spectrum of system A is qualitatively the same as system B. The entanglement Hamiltonian of system B can be obtained using the approach in the main text since $\lambda$ is relevant.