Lyapunov spectra for all symmetry classes of quasi-one-dimensional disordered systems of non-interacting Fermions

TL;DR

This paper introduces a random phase property for classical group products, enabling explicit calculation of Lyapunov spectra in disordered Fermion systems across all symmetry classes, revealing boundary conductance features.

Contribution

It proposes a universal random phase property for classical groups and applies it to compute Lyapunov spectra for all symmetry classes of disordered Fermion systems.

Findings

Lyapunov spectra can be explicitly calculated in a perturbative regime.

Boundary states in topological classes have zero Lyapunov exponents.

Boundary spectra are almost surely absolutely continuous, indicating gapless modes.

Abstract

A random phase property is proposed for products of random matrices drawn from any one of the classical groups associated with the ten Cartan symmetry classes of non-interacting disordered Fermion systems. It allows to calculate the Lyapunov spectrum explicitly in a perturbative regime. These results apply to quasi-one-dimensional random Dirac operators which can be constructed as representatives for each of the ten symmetry classes. For those symmetry classes that correspond to two-dimensional topological insulators or superconductors, the random Dirac operators describing the one-dimensional boundaries have vanishing Lyapunov exponents and almost surely an absolutely continuous spectrum, reflecting the gapless and conducting nature of the boundary degrees of freedom.