Level spacings at the metal-insulator transition in the Anderson Hamiltonians and multifractal random matrix ensembles

TL;DR

This paper analyzes spectral statistics at the metal-insulator transition in Anderson Hamiltonians, introducing a generalized sine kernel that interpolates between Wigner-Dyson and Poisson distributions, fitting numerical data across symmetry classes.

Contribution

It provides analytic expressions for level spacing distributions in a new class of random matrix ensembles with multifractal eigenstates, bridging theoretical models and numerical results.

Findings

Universal spectral correlations governed by a generalized sine kernel.

Analytic level spacing distributions that interpolate between Wigner-Dyson and Poisson.

Excellent fit of theoretical results to numerical data for Anderson Hamiltonians.

Abstract

We consider orthogonal, unitary, and symplectic ensembles of random matrices with (1/a)(ln x)^2 potentials, which obey spectral statistics different from the Wigner-Dyson and are argued to have multifractal eigenstates. If the coefficient $a$ is small, spectral correlations in the bulk are universally governed by a translationally invariant, one-parameter generalization of the sine kernel. We provide analytic expressions for the level spacing distribution functions of this kernel, which are hybrids of the Wigner-Dyson and Poisson distributions. By tuning the single parameter, our results can be excellently fitted to the numerical data for three symmetry classes of the three-dimensional Anderson Hamiltonians at the metal-insulator transition, previously measured by several groups using exact diagonalization.