Transfer Matrix Approach for Topological Edge States

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.