Characterization of edge states in perturbed honeycomb structures

Abstract

This paper is a mathematical analysis of conduction effects at interfaces between insulators. Motivated by work of Haldane and Raghu (2008), we continue the study of a linear PDE initiated by Fefferman, Lee-Thorp, and Weinstein (2016). This PDE is induced by a continuous honeycomb Schrödinger operator with a line defect.

This operator exhibits remarkable connections between topology and spectral theory. It has essential spectral gaps about the Dirac point energies of the honeycomb background. In a perturbative regime, Fefferman, Lee-Thorp, and Weinstein constructed edge states: time-harmonic waves propagating along the interface, localized transversely. At leading order, these edge states are adiabatic modulations of the Dirac-point Bloch modes. Their envelopes solve a Dirac equation that emerges from a multiscale procedure.

We develop a scattering-oriented approach that derives all possible edge states, at arbitrary precision. The key component is a resolvent estimate connecting the Schrödinger operator to the emerging Dirac equation. We discuss topological implications via the computation of the spectral flow, or edge index.