Random Matrix Theory of the Energy-Level Statistics of Disordered Systems at the Anderson Transition

TL;DR

This paper extends Dyson's mean field theory to weakly confined random matrix ensembles, revealing their eigenvalue statistics match those of the Anderson transition in disordered systems, supported by Monte Carlo simulations.

Contribution

It generalizes Dyson's plasma model to weak confinement potentials and demonstrates the eigenvalue statistics align with the Anderson transition, supported by systematic simulations.

Findings

Eigenvalue statistics deviate from Wigner-Dyson for weak confinement.

Level spacing distribution matches Anderson model at criticality.

Number variance grows linearly with slope ~0.32 at critical point.

Abstract

We consider a family of random matrix ensembles (RME) invariant under similarity transformations and described by the probability density $P({\bf H})= \exp[-{\rm Tr}V({\bf H})]$. Dyson's mean field theory (MFT) of the corresponding plasma model of eigenvalues is generalized to the case of weak confining potential, $V(\epsilon)\sim {A\over 2}\ln ^2(\epsilon)$. The eigenvalue statistics derived from MFT are shown to deviate substantially from the classical Wigner-Dyson statistics when $A<1$. By performing systematic Monte Carlo simulations on the plasma model, we compute all the relevant statistical properties of the RME with weak confinement. For $A_c\approx 0.4$ the distribution function of the energy-level spacings (LSDF) of this RME coincides in a large energy window with the LSDF of the three dimensional Anderson model at the metal-insulator transition. For the same $A_c$, the variance of the number of levels, $\langle n^2\rangle -\langle n\rangle^2$, in an interval containing $\langle n\rangle$ levels on average, grows linearly with $\langle n\rangle$, and its slope is equal to $0.32 \pm 0.02$, which is consistent with the value found for the Anderson model at the critical point.