Topological edge states in the Su-Schrieffer-Heeger model

The Su-Schrieffer-Heeger (SSH) model on a two-dimensional square lattice exhibits a topological phase transition which is related to the Zak phase determined by bulk band topology. The strong modulation of electron hopping causes nontrivial charge polarization even in the presence of inversion symmetry. The energy band structures and topological edge states have been calculated numerically in previous studies. Here, however, the full energy spectrum and explicit form of wave functions for two-dimensional bulk and one-dimensional ribbon geometries of the SSH model are analytically derived using the wave mechanics approach. Explicit analytic representations of wave functions provide the information of parity for each subband, localization length, and critical point of the topological phase transition in the SSH ribbon. IThe dimensional crossover of the topological transition point for the SSH model from one to two dimensions is also shown.