Absolutely Continuous Edge Spectrum of Hall Insulators on the Lattice

TL;DR

This paper proves that two-dimensional lattice quantum Hall systems with nonzero Hall conductance have edge modes with spectra that are absolutely continuous and fill the entire bulk gap, supporting ballistic transport.

Contribution

It establishes that gapped lattice quantum Hall systems with Hall conductance support edge modes with absolutely continuous spectra filling the bulk gap.

Findings

Edge modes support ballistic transport.

Edge spectrum fills the entire bulk gap.

Edge spectrum is absolutely continuous.

Abstract

The presence of chiral modes on the edges of quantum Hall samples is essential to our understanding of the quantum Hall effect. In particular, these edge modes should support ballistic transport and therefore, in a single particle picture, be supported in the absolutely continuous spectrum of the single-particle Hamiltonian. We show in this note that if a free fermion system on the two-dimensional lattice is gapped in the bulk, and has a nonvanishing Hall conductance, then the same system put on a half-space geometry supports edge modes whose spectrum fills the entire bulk gap and is absolutely continuous.