Footprint of a topological phase transition on the density of states

Abstract

For a generalized Su–Schrieffer–Heeger model, the energy zero is always critical and hyperbolic in the sense that all reduced transfer matrices commute and have their spectrum off the unit circle. Disorder-driven topological phase transitions in this model are characterized by a vanishing Lyapunov exponent at the critical energy. It is shown that away from such a transition the density of states vanishes at zero energy with an explicitly computable Hölder exponent, while it has a characteristic divergence (Dyson spike) at the transition points. The proof is based on renewal theory for the Prüfer phase dynamics and the optional stopping theorem for martingales of suitably constructed comparison processes.