Critical properties in long-range hopping Hamiltonians

TL;DR

This study numerically investigates critical properties of disordered long-range hopping models in two and three dimensions, revealing how correlation dimensions and level spacing distributions depend on coupling strength and dimensionality.

Contribution

It provides new numerical insights into the behavior of correlation dimensions and level spacing distributions in long-range disordered models at criticality, highlighting their dependence on coupling regimes.

Findings

Correlation dimension $d_2$ scales with $b^d$ in strong coupling and inversely in weak coupling.

Level spacing distribution $P_c(s)$ follows an exponential form with a critical exponent depending on coupling.

Coefficients in scaling laws depend only on the system's dimensionality.

Abstract

Some properties of $d$-dimensional disordered models with long-range random hopping amplitudes are investigated numerically at criticality. We concentrate on the correlation dimension $d_2$ (for $d=2$) and the nearest level spacing distribution $P_c(s)$ (for $d=3$) in 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 (i) the extrapolated values of $d_2$ are of the form $d_2=c_db^d$ in the strong coupling limit and $d_2=d-a_d/b^d$ in the case of weak coupling, and (ii) $P_ (s)$ has the asymptotic form $P_c(s)\sim\exp (-A_ds^{\alpha})$ for $s\gg $, with the critical exponent $\alpha=2-a_d/b^d$ for $b^d \gg 1$ and $\alpha=1+c_d b^d$ for $b^d \ll 1$. In these cases the numerical coefficients $A_d$, $a_d$ and $c_d$ depend only on the dimensionality.