Characterization of edge states in perturbed honeycomb structures

Abstract

This paper is a mathematical analysis of conduction effects at interfaces between insulators. Motivated by Haldane-Raghu [HR08, RH08], we continue the study of a linear PDE initiated in Fefferman-Lee- FLW16b]. 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, the authors of [FLW16a] construct 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 envelops 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.

The bulk-edge correspondence for continuous honeycomb lattices

“…We proved an analogous statement in [Dr18b,Lemma 4.3] for different values of the parameters ξ and z. Here we require |ξ − ξ | ≤ ε 0 and z ∈ ∂D µ ± δ (ξ), r δ (ξ) instead of |ξ −ξ | ≤ δ 1/3 and z ∈ D E , θ ·r δ (ξ) for some θ ∈ (0, 1).…”

“…The associated time-harmonic waves propagate in opposite directions. In [Dr18b] we proved that all edge states take this form, and showed that the corresponding spectral flow vanishes. This agrees with the bulk-edge correspondence.…”

“…As stated, Theorem 2 is more general than [Dr18b, Corollary 4]. We will nonetheless derive Theorem 2 following the approach of [Dr18b]. As a preparation, we fix E δ (ξ) depending smoothly on (δ, ξ) ∈ [0, δ ) × R 2 , 2πΛ *periodic in ξ, such that:…”

“…When δ 1, edge states of operators similar to P δ were constructed in [FLW16,LWZ17,Dr18b] under the no-fold condition. This condition requires that the dispersion surfaces ξ → λ 0,n (ξ) and ξ → λ 0,n+1 (ξ) do not fold over E except at {ξ , −ξ } + 2πΛ * -see [FLW16,§1.3].…”

“…We can then apply the techniques of [Dr18b] to compute the spectral flow of P δ −E . This relies on a resolvent expansion of P δ .…”