Universal Fluctuations in Spectra of Disordered Systems at the Anderson Transition

TL;DR

This study investigates the spectral fluctuations at the Anderson transition, revealing scale-invariant level-spacing distributions and a Gaussian form of level count probabilities at criticality, using numerical simulations of disordered systems.

Contribution

It demonstrates that the level-spacing distribution is scale-independent at the transition and characterizes the probability distribution of level counts, providing new insights into spectral statistics at the Anderson transition.

Findings

Level-spacing distribution is scale-invariant at the transition.

Distribution of level counts is Gaussian near its maximum.

Asymptotic form of P(s) is Poisson-like with A ≈ 1.9.

Abstract

Using the level--spacing distribution and the total probability function of the numbers of levels in a given energy interval we analyze the crossover of the level statistics between the delocalized and the localized regimes. By numerically calculating the electron spectra of systems of up to $32^3$ lattice sites described by the Anderson Hamiltonian it is shown that the distribution $P(s)$ of neighboring spacings is {\em scale- independent} at the metal-insulator transition. For large spacings it has a Poisson-like asymptotic form $P(s)\propto \exp (- A\,s/\Delta )$, where $A \approx 1.9$. At the critical point we obtain a linear relationship between the variance of the number of levels $\langle [\delta n(\varepsilon)]^2 \rangle $ and their average number $\langle n(\varepsilon)\rangle $ within the interval $\varepsilon$. The constant of proportionality is less than unity due to the repulsion of the levels. Both $P(s)$ and $\langle [\delta n(\varepsilon)]^2 \rangle $ are determined by the probability density $Q_{n}(\varepsilon)$ of having exactly $n$ levels in the energy interval $\varepsilon $. The distribution $Q_{n}(\varepsilon)$ at the critical point is found to be size--independent and to obey a Gaussian law near its maximum, where $n\sim \langle n \rangle $.