Long range disorder and Anderson transition in systems with chiral symmetry

TL;DR

This paper investigates the spectral properties of a chiral random banded matrix with power-law decaying elements, revealing an Anderson transition at a specific decay exponent, with implications for quantum chromodynamics.

Contribution

It provides explicit analytical results for spectral correlations in a chiral random matrix model with long-range hopping, identifying an Anderson transition at a critical decay exponent.

Findings

Analytical expressions for density of states and correlation functions.

Evidence of an Anderson transition at a specific power-law decay exponent.

Potential applications to quantum chromodynamics.

Abstract

We study the spectral properties of a chiral random banded matrix (chRBM) with elements decaying as a power-law ${{\cal H}_{ij}}\sim |i-j|^{-\alpha}$. This model is equivalent to a chiral 1D Anderson Hamiltonian with long range power-law hopping. In the weak disorder limit we obtain explicit nonperturbative analytical results for the density of states and the two-level correlation function by mapping the chRBM onto a nonlinear sigma model. We also put forward, by exploiting the relation between the chRBM at $\alpha=1$ and a generalized chiral random matrix model, an exact expression for the above correlation functions. We give compelling analytical and numerical evidence that for this value the chRBM reproduces all the features of an Anderson transition. Finally we discuss possible applications of our results to QCD.