Critical level spacing distribution in long-range hopping Hamiltonians

TL;DR

This study numerically analyzes the critical level spacing distribution in disordered long-range hopping Hamiltonians in 1D and 2D, revealing how the distribution's tail behavior varies with coupling strength.

Contribution

It provides a detailed numerical investigation of the critical level spacing distribution in long-range disordered models, identifying the asymptotic form and critical exponents in different coupling regimes.

Findings

The distribution follows an exponential form with a power-law exponent depending on coupling.

Critical exponents vary linearly with the inverse coupling parameter in both regimes.

Results enhance understanding of spectral statistics at the metal-insulator transition.

Abstract

The nearest level spacing distribution $P_c(s)$ of $d$-dimensional disordered models ($d=1$ and 2) with long-range random hopping amplitudes is investigated numerically at criticality. We focus on both the weak ($b^d \gg 1$) and the strong ($b^d \ll 1$) coupling regime, where the parameter $b^{-d}$ plays the role of the coupling constant of the model. It is found that $P_c(s)$ has the asymptotic form $P_c(s)\sim\exp [-A_ds^{\alpha}]$ for $s\gg 1$, with the critical exponent $\alpha=2-a_d/b^d$ in the weak coupling limit and $\alpha=1+c_d b^d$ in the case of strong coupling.