Distribution of level curvatures for the Anderson model at the localization-delocalization transition

TL;DR

This paper investigates the distribution of level curvatures in a disordered lattice model, revealing distinct behaviors in metallic, insulating, and critical regimes, and introduces a novel distribution at the transition point.

Contribution

It introduces a new distribution function for level curvatures at the Anderson transition, connecting non-analytical behavior to multifractality of wave functions.

Findings

In metals, $P(k)$ matches random-matrix theory predictions.

In insulators, $P(k)$ is logarithmically-normal.

At the transition, $P(k)$ follows a novel distribution with non-analytical features.

Abstract

We compute the distribution function of single-level curvatures, $P(k)$, for a tight binding model with site disorder, on a cubic lattice. In metals $P(k)$ is very close to the predictions of the random-matrix theory (RMT). In insulators $P(k)$ has a logarithmically-normal form. At the Anderson localization-delocalization transition $P(k)$ fits very well the proposed novel distribution $P(k)\propto (1+k^{\mu})^{3/\mu}$ with $\mu \approx 1.58$, which approaches the RMT result for large $k$ and is non-analytical at small $k$. We ascribe such a non-analiticity to the spatial multifractality of the critical wave functions.