Fluctuations of the Energy Levels Number at the Mobility Edge

TL;DR

This paper analyzes the fluctuations in the number of energy levels near the mobility edge in disordered systems, revealing a linear variance behavior linked to anomalous contributions near the Anderson transition.

Contribution

It provides a detailed analysis of level correlation functions and uncovers a linear variance behavior near the mobility edge, highlighting the role of ballistic contributions.

Findings

Variance of level number is linear in energy near the transition.

Anomalous contributions affect the sum rule for correlation functions.

Behavior observed in both Anderson transition and 2D systems.

Abstract

We present a detailed analysis of the behavior of two--level correlation function $R(s)$ in the disordered sample. We show that in the vicinity of the Anderson transition as well as in $2d$, the variance of the number of levels in an energy band of a width $E$ has a linear behavior as a function of $E$. This is related with an ``anomalous'' contribution to the sum rule for $R(s)$, which comes from the ``ballistic'' region of $s$.