The Bulk-Edge Correspondence for Disordered Chiral Chains

TL;DR

This paper establishes a relationship between bulk and edge topological indices in disordered one-dimensional chiral insulators, introducing a new index formulation via Lyapunov exponents that highlights conditions for topological phase transitions.

Contribution

It proves the bulk-edge correspondence for disordered chiral chains and introduces a novel index formulation based on Lyapunov exponents.

Findings

Bulk and edge indices are equal for nearest neighbor Hamiltonians.

A new index formulation in terms of Lyapunov exponents is provided.

Conditions for topological phase transitions in the mobility gap are identified.

Abstract

We study one-dimensional insulators obeying a chiral symmetry in the single-particle picture. The Fermi level is assumed to lie in a mobility gap. Topological indices are defined for infinite (bulk) or half-infinite (edge) systems, and it is shown that for a given Hamiltonian with nearest neighbor hopping the two indices are equal. We also give a new formulation of the index in terms of the Lyapunov exponents of the zero energy Schr\"odinger equation, which illustrates the conditions for a topological phase transition occurring in the mobility gap regime.