Critical Level-Spacing Distribution for General Boundary Conditions

TL;DR

This paper investigates the universality of the level-spacing distribution at the Anderson transition, showing that integrating over all boundary conditions slightly modifies the semi-Poisson distribution, and identifies a Gaussian-like distribution in the crossover regime.

Contribution

The study numerically computes the critical level-spacing distribution over all boundary conditions, challenging the semi-Poisson universality and revealing a Gaussian-like distribution in the crossover.

Findings

Semi-Poisson distribution fits well but has small deviations at large S.

Integrating over boundary conditions slightly alters the distribution.

Crossover between ballistic and localized regimes is Gaussian-like.

Abstract

It is believed that the semi-Poisson function $P(S)=4S\exp(-2S)$ describes the normalized distribution of the nearest level-spacings $S$ for critical energy levels at the Anderson metal-insulator transition from quantum chaos to integrability, after an average over four obvious boundary conditions (BC) is taken (Braun {\it et} {\it al} \cite{1}). In order to check whether the semi-Poisson is the correct universal distribution at criticality we numerically compute it by integrating over all possible boundary conditions. We find that although $P(S)$ describes very well the main part of the obtained critical distribution small differences exist particularly in the large $S$ tail. The simpler crossover between the integrable ballistic and localized limits is shown to be universally characterized by a Gaussian-like $P(S)$ distribution instead.