Local symmetry theory of resonator structures for the real-space control of edge states in binary aperiodic chains

TL;DR

This paper introduces a real-space, symmetry-based framework to understand and control edge-localized states in aperiodic binary chains, linking local symmetries to spectral properties and enabling targeted manipulation of edge states.

Contribution

It presents a novel local symmetry theory that predicts and explains edge states in aperiodic chains, extending understanding beyond traditional spectral analysis.

Findings

Eigenstates localize on reflection-symmetric substructures

Local symmetries predict edge state occurrence

Energy localization correlates with structural features

Abstract

We propose a real-space approach explaining and controlling the occurrence of edge-localized gap states between the spectral quasibands of binary tight binding chains with deterministic aperiodic long-range order. The framework is applied to the Fibonacci, Thue-Morse and Rudin-Shapiro chains, representing different structural classes. Our approach is based on an analysis of the eigenstates at weak inter-site coupling, where they are shown to generically localize on locally reflection-symmetric substructures which we call local resonators. A perturbation theoretical treatment demonstrates the local symmetries of the eigenstates. Depending on the degree of spatial complexity of the chain, the proposed local resonator picture can be used to predict the occurrence of gap-edge states even for stronger couplings. Moreover, we connect the localization behavior of a given eigenstate to its energy, thus providing a quantitative connection between the real-space structure of the chain and its eigenvalue spectrum. This allows for a deeper understanding, based on local symmetries, of how the energy spectra of binary chains are formed. The insights gained allow for a systematic analysis of aperiodic binary chains and offers a pathway to control structurally induced edge states.