Level number variance and spectral compressibility in a critical two-dimensional random matrix model

TL;DR

This paper investigates the spectral properties of a 2D random matrix model with power-law decay, identifying a transition from critical to metallic behavior through analytical and numerical methods.

Contribution

It analytically characterizes the level number variance and spectral compressibility across different decay parameters, revealing a phase transition.

Findings

Level number variance is linear at small decay parameters.

Spectral compressibility transitions from between 0 and 1 to zero.

Critical decay parameter marks the phase transition point.

Abstract

We study level number variance in a two-dimensional random matrix model characterized by a power-law decay of the matrix elements. The amplitude of the decay is controlled by the parameter b. We find analytically that at small values of b the level number variance behaves linearly, with the compressibility chi between 0 and 1, which is typical for critical systems. For large values of b, we derive that chi=0, as one would normally expect in the metallic phase. Using numerical simulations we determine the critical value of b at which the transition between these two phases occurs.