Edge modes in modulated metamaterials based on the three-gap theorem

TL;DR

This paper introduces a novel approach for designing structured metamaterials using the three-gap theorem, unifying concepts of edge states, spectral flow, and localization, with potential applications in photonics and material science.

Contribution

It develops a new theoretical framework based on the three-gap theorem to analyze and predict localized edge modes in modulated metamaterials, extending existing models like SSH.

Findings

Established conditions for localized edge states in finite systems.

Characterized spectral flow and localization patterns of edge modes.

Demonstrated rich edge mode dynamics as parameters vary.

Abstract

We present a new paradigm for generating complex structured materials based on the three-gap theorem that unifies and generalises several key concepts in the study of localised edge states. Our model has both the discretised coupling strengths of the SSH model and a modulation parameter that can be used to characterise the spectral flow of edge modes and produce images reminiscent of the Hofstadter butterfly. By defining a localisation factor associated to each eigenmode, we are able to establish conditions for the existence of localised edge states in finite systems. This allows us to compare their eigenfrequencies with the spectra of the corresponding infinitely periodic problem and characterise the rich pattern of localised edge modes appearing and disappearing (in the sense of becoming delocalised) as the parameters of our three-gap algorithm are varied.