Shape Analysis of the Level Spacing Distribution around the Metal Insulator Transition in the Three Dimensional Anderson Model

TL;DR

This paper introduces a new numerical method for analyzing the level spacing distribution around the metal-insulator transition in the 3D Anderson model, revealing a distinct distribution shape near the transition.

Contribution

It applies shape analysis techniques to eigenvalue distributions, providing new insights into the critical behavior at the metal-insulator transition in the Anderson model.

Findings

The level spacing distribution near the MIT differs from Brody and Izrailev distributions.

The distribution near the MIT fits a specific form with a parameter β≈0.2.

The method accurately estimates the transition point and critical exponent.

Abstract

We present a new method for the numerical treatment of second order phase transitions using the level spacing distribution function $P(s)$. We show that the quantities introduced originally for the shape analysis of eigenvectors can be properly applied for the description of the eigenvalues as well. The position of the metal--insulator transition (MIT) of the three dimensional Anderson model and the critical exponent are evaluated. The shape analysis of $P(s)$ obtained numerically shows that near the MIT $P(s)$ is clearly different from both the Brody distribution and from Izrailev's formula, and the best description is of the form $P(s)=c_1\,s\exp(-c_2\,s^{1+\beta})$, with $\beta\approx 0.2$. This is in good agreement with recent analytical results.