Random Dirac operators with time-reversal symmetry

TL;DR

This paper studies quasi-one-dimensional stochastic Dirac operators with time-reversal symmetry, demonstrating they have a single conducting channel and absolutely continuous spectrum, using advanced matrix group analysis and Kotani theory.

Contribution

It adapts existing criteria and Kotani theory to analyze random Dirac operators with time-reversal symmetry, revealing their spectral and conducting properties.

Findings

One conducting channel identified

Absolutely continuous spectrum established

Absence of singular spectrum shown

Abstract

Quasi-one-dimensional stochastic Dirac operators with an odd number of channels, time reversal symmetry but otherwise efficiently coupled randomness are shown to have one conducting channel and absolutely continuous spectrum of multiplicity two. This follows by adapting the criteria of Guivarch-Raugi and Goldsheid-Margulis to the analysis of random products of matrices in the group SO$^*(2L)$, and then a version of Kotani theory for these operators. Absence of singular spectrum can be shown by adapting an argument of Jaksic-Last if the potential contains random Dirac peaks with absolutely continuous distribution.