Vol. 1, No. 3, 2019
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Characterization of edge states in perturbed honeycomb structures

Alexis Drouot
Vol. 1 (2019), No. 3, 385–445
DOI: 10.2140/paa.2019.1.385
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.

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Keywords
edge states, graphene, Dirac points, Schrödinger operators
Mathematical Subject Classification 2010
Primary: 35P15
Secondary: 35P25, 35Q40, 35Q41
Milestones
Received: 21 December 2018
Revised: 26 February 2019
Accepted: 4 April 2019
Published: 17 July 2019
Authors
Alexis Drouot
Department of Mathematics
Columbia University
New York, NY
United States