Exploring topological phases with extended Su-Schrieffer-Heeger models

The Su-Schrieffer-Heeger (SSH) model describes a tight-binding one-dimensional (1D) lattice with alternating nearest-neighbor amplitudes. Despite its mathematically simple and physically intuitive structure, the SSH model is capable of supporting a 1D topological phase that is characterized by the presence of zero energy eigenstates (zero modes) localized at each end of the lattice. For this reason, many studies in the area of topological phases of matter often consider the SSH model as a subject for various extensions that give rise to more sophisticated topological phenomena. The purpose of this article is to review, in sufficient detail, existing approaches to extending the SSH model. This includes extensions by increasing the dimensionality of the lattice, enlarging the size of its unit cell, or adding extra terms that represent various physical effects. For each approach, some extended SSH models studied in relevant existing literature are discussed as case studies. Noteworthy properties of such models, which are of topological origin, are further comprehensively elaborated.