Topological Transitions and Bulk Wavefunctions in the SSH Model

Working in the context of the Su-Schreiffer-Heeger (SSH) model, the effect of topological transitions on the structure and properties of bulk position-space wavefunctions is studied for a particle undergoing a quantum walk in a one-dimensional lattice. In particular, we consider what happens when the wavefunction reaches a boundary at which the Hamiltonian changes suddenly from one topological phase to another. An exact solution is constructed for the wavefunction on both sides of the boundary. Under some conditions, it is found that the probability of transition into the region of the second, topologically distinct Hamiltonian is strongly suppressed. When the boundary is encountered, the wavefunctions tend to be strongly reflected, and by appropriate choice of system parameters leakage into the second region can be made negligible. Therefore, it is possible to arrange a high degree of bulk wavefunction localization within in each region. This `topologically-assisted' suppression of transitions, although not of direct topological origin itself, exists only because of the presence of a change in the topological properties of the Hamiltonian. We give a quantitative examination of the reflection and transmission coefficients of incident waves at the boundary between regions of different winding number.