The disordered Su-Schrieffer-Heeger model

TL;DR

This paper derives an analytical expression for the Lyapunov exponent in the disordered SSH model, revealing how disorder affects topological properties and transmission, with strong agreement between theory and numerical simulations.

Contribution

It provides the first analytical formula for the Lyapunov exponent in the disordered SSH model, considering both diagonal and off-diagonal disorder.

Findings

Analytical Lyapunov exponent as a function of energy and disorder

Excellent match between analytical and numerical results

Real space winding number varies with disorder types

Abstract

Quantum topology categorizes physical systems in integer invariants, which are robust to some deformations and certain types of disorder. A prime example is the Su-Schrieffer-Heeger (SSH) model, which has two distinct topological phases, the trivial phase with no edge states and the non-trivial phase with zero-energy edge states. The energy dispersion of the SSH model is dominated by a gap around zero energy, which suppresses the transmission. This exponential suppression of the transmission with system length is determined by the Lyapounov exponent. Here we find an analytical expression of the Lyapounov as a function of energy in the presence of both diagonal and off-diagonal disorder. We obtain this result by finding a recurrence relation for the local density, which can be averaged over different disorder configurations. There is excellent agreement between our analytical expression and the numerical results over a wide range of disorder strengths and disorder types. The real space winding number is evaluated as a function of off-diagonal and on-site disorder for possible applications of quantum topology.