Level-statistics in Disordered Systems: A single parametric scaling and Connection to Brownian Ensembles

TL;DR

This paper demonstrates that level statistics in disordered systems undergoing metal-insulator transitions can be modeled by single parametric Brownian ensembles, providing analytical support for single parameter scaling.

Contribution

It establishes an analogy between level statistics in disordered systems and Brownian ensembles, enabling analytical insights into level correlations at criticality.

Findings

Level statistics follow single parametric Brownian ensemble behavior.

Analytical evidence supports single parameter scaling in disordered systems.

Provides a method to compute level correlations at the critical point.

Abstract

We find that the statistics of levels undergoing metal-insulator transition in systems with multi-parametric Gaussian disorders and non-interacting electrons behaves in a way similar to that of the single parametric Brownian ensembles \cite{dy}. The latter appear during a Poisson $\to$ Wigner-Dyson transition, driven by a random perturbation. The analogy provides the analytical evidence for the single parameter scaling of the level-correlations in disordered systems as well as a tool to obtain them at the critical point for a wide range of disorders.