Transfer Matrix Approach for Topological Edge States

TL;DR

This paper introduces a transfer matrix method using Iwasawa decomposition to analyze topological edge states in periodic systems, providing simple conditions and visual tools for understanding topological properties and bulk-edge correspondence.

Contribution

It presents a novel transfer matrix approach with Iwasawa decomposition for topological analysis, applicable to generalized SSH models and photonic crystals, offering new insights and visualizations.

Findings

Derived simple conditions for topologically protected edge states.

Applied method to generalized SSH and photonic crystal models.

Provided pictorial proof of Zak phase bulk-edge correspondence.

Abstract

We suggest and develop a novel approach for describing topological properties of a periodic system purely from the transfer matrix associated to a unit cell. Our approach uses the Iwasawa decomposition to parametrise the transfer matrix uniquely in terms of three real numbers. This allows us to obtain simple conditions for the existence of topologically protected edge states and to provide a visual illustration of all possible solutions. In order to demonstrate our method in action, we apply it to study some generalisations of the Su-Schrieffer-Heeger (SSH) model, such as the tetramer SSH4 model and a dimerised one-dimensional photonic crystal. Finally, we also obtained a simple pictorial proof of the Zak phase bulk-edge correspondence for any one dimensional system using this approach.