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

We consider ($2+1$)-dimensional topological quantum states which possess edge states described by a chiral ($1+1$)-dimensional conformal field theory, 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 conformal field theory 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 conformal field theory, 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.

Figure 1

(a) A topological state on a cylinder with a bipartition into two regions $A$ and $B$. (b) The deformed system (see the 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 interedge coupling. (c) For small enough $\lambda$, the coupling between the gapped bulk states can be neglected, and the problem can be reduced to an interedge coupling problem described by a ($1+1$)-dimensional conformal field theory with a (RG-)relevant coupling $\lambda H_{\mathrm{int}}$.