Toward an invariant matrix model for the Anderson Transition

TL;DR

This paper explores invariant matrix models with log-normal weights, revealing a complex energy landscape with multiple equilibrium states and instanton connections, which may shed light on the Anderson transition and symmetry breaking.

Contribution

It introduces a detailed analysis of the energy landscape and instanton effects in invariant matrix models, advancing understanding of their role in the Anderson transition.

Findings

Eigenvalue distributions are intermediate between Wigner-Dyson and Poisson.

The energy landscape has exponentially many equilibrium configurations.

Instantons connect different saddle points, influencing eigenvalue correlations.

Abstract

We consider invariant matrix models with log-normal (asymptotic) weight. It is known that their eigenvalue distribution is intermediate between Wigner-Dyson and Poissonian, which candidates these models for describing a system intermediate between the extended and localized phase. We show that they have a much richer energy landscape than expected, with their partition functions decomposable in a large number of equilibrium configurations, growing exponentially with the matrix rank. Within each of these saddle points, eigenvalues are uncorrelated and confined by a different potential felt by each eigenvalue. The equilibrium positions induced by the potentials differ in different saddles. Instantons connecting the different equilibrium configurations are responsible for the correlations between the eigenvalues. We argue that these instantons can be linked to the SU(2) components in which the rotational symmetry can be decomposed, paving the way to understand the conjectured critical breaking of U(N) symmetry in these invariant models.