Edge states in square lattice media and their deformations

TL;DR

This paper investigates edge states in 2D square lattice media with domain walls, analyzing how magnetic perturbations create topologically protected edge modes and deriving effective models for their bifurcations.

Contribution

It introduces a detailed analysis of edge states in square lattice media with magnetic perturbations, deriving effective edge Hamiltonians for bifurcation analysis.

Findings

Edge states traverse the band gap, confirming bulk-edge correspondence.

Effective Hamiltonians describe bifurcation of edge states near degeneracies.

Numerical simulations validate analytical predictions.

Abstract

Edge states are time-harmonic solutions of conservative wave systems which are plane wave-like parallel to and localized transverse to an interface between two bulk media. We study a class of 2D edge Hamiltonians modeling a medium which slowly interpolates between periodic bulk media via a domain wall across a "rational" line defect. We consider the cases of (1) periodic bulk media having the symmetries of a square lattice, and (2) linear deformations of such media. Our bulk Hamiltonians break time-reversal symmetry due to perturbation by a magnetic term, which opens a band gap about the band structure degeneracies of the unperturbed bulk Hamiltonian. In case (1), these are quadratic band degeneracies; in case (2), they are pairs of conical degeneracies. We demonstrate that this band gap is traversed by two distinct edge state curves, consistent with the bulk-edge correspondence principle of topological physics. Blow-ups of these curves near the bulk band degeneracies are described by effective (homogenized) edge Hamiltonians derived via multiple-scale analysis which control the bifurcation of edge states. In case (1), the bifurcation is governed by a matrix Schr\"{o}dinger operator; in case (2), it is governed by a pair of Dirac operators. We present analytical results and numerical simulations for both the full 2D edge Hamiltonian spectral problem and the spectra of effective edge Hamiltonians.