Bulk-edge correspondence in entanglement spectra

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 (FQH) 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 (PES), whose counting of eigenvalues for each good quantum number is identical to the counting of bulk quasiholes (up to accidental zero eigenvalues of the reduced density matrix), and the orbital entanglement spectrum (OES), considered by Li and Haldane. By using a set of clustering operators that have their origin in conformal-field-theory (CFT) operator expansions, we show that the counting of the OES 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. Our results can be interpreted as a bulk-edge correspondence in entanglement spectra. Moreover, we show that the counting of the PES and OES is identical even for CFT states that are likely bulk gapless, such as the Gaffnian wave function.