Bulk-Edge Correspondence in the Entanglement Spectra

TL;DR

This paper analytically confirms the bulk-edge correspondence in the entanglement spectra of fractional quantum Hall states by establishing a one-to-one mapping between particle and orbital entanglement spectra, linking bulk quasiholes to edge modes.

Contribution

It provides a rigorous microscopic proof of the bulk-edge correspondence in entanglement spectra for a class of FQH states using conformal field theory techniques.

Findings

The orbital entanglement spectrum counting matches the quasihole counting.

The bulk-edge correspondence holds even for gapless CFT states like the Gaffnian.

A mapping between particle and orbital entanglement spectra is established.

Abstract

Li and Haldane conjectured and numerically substantiated that the entanglement spectrum of the reduced density matrix of ground-states of time-reversal breaking topological phases (fractional quantum Hall states) contains information about the counting of their edge modes when the ground-state is cut in two spatially distinct regions and one of the regions is traced out. We analytically substantiate this conjecture for a series of FQH states defined as unique zero modes of pseudopotential Hamiltonians by finding a one to one map between the thermodynamic limit counting of two different entanglement spectra: the particle entanglement spectrum, whose counting of eigenvalues for each good quantum number is identical (up to accidental degeneracies) to the counting of bulk quasiholes, and the orbital entanglement spectrum (the Li-Haldane spectrum). As the particle entanglement spectrum is related to bulk quasihole physics and the orbital entanglement spectrum is related to edge physics, our map can be thought of as a mathematically sound microscopic description of bulk-edge correspondence in entanglement spectra. By using a set of clustering operators which have their origin in conformal field theory (CFT) operator expansions, we show that the counting of the orbital entanglement spectrum eigenvalues in the thermodynamic limit must be identical to the counting of quasiholes in the bulk. The latter equals the counting of edge modes at a hard-wall boundary placed on the sample. Moreover, we show this to be true even for CFT states which are likely bulk gapless, such as the Gaffnian wavefunction.