Topological phase transitions in the 1D multichannel Dirac equation with random mass and a random matrix model

TL;DR

This paper links a multichannel disordered 1D Dirac model with a random matrix model to analyze topological phase transitions, deriving exact formulas for the density of states and identifying critical points via a change in a quantum number.

Contribution

It introduces a novel connection between a multichannel Dirac equation with random mass and a deformed Laguerre ensemble, enabling exact analysis of topological phase transitions.

Findings

Exact determinantal formulas for the density of states.

Identification of phase transition points via the Witten index.

Low energy behavior characterized by a power-law density of states.

Abstract

We establish the connection between a multichannel disordered model --the 1D Dirac equation with $N\times N$ matricial random mass-- and a random matrix model corresponding to a deformation of the Laguerre ensemble. This allows us to derive exact determinantal representations for the density of states and identify its low energy ($\varepsilon\to0$) behaviour $\rho(\varepsilon)\sim|\varepsilon|^{\alpha-1}$. The vanishing of the exponent $\alpha$ for $N$ specific values of the averaged mass over disorder ratio corresponds to $N$ phase transitions of topological nature characterised by the change of a quantum number (Witten index) which is deduced straightforwardly in the matrix model.