Level Curvature Distribution and the Structure of Eigenfunctions in Disordered Systems

TL;DR

This paper analyzes the distribution of level curvature in disordered quantum systems, revealing how it relates to eigenfunction structure, multifractality, and the effects of different T-breaking perturbations, with both analytical and numerical insights.

Contribution

It provides a detailed analytical and numerical study of level curvature distribution in disordered systems, highlighting the impact of T-breaking perturbations and the connection to multifractality at criticality.

Findings

The correction to the curvature distribution differs for constant vector-potential and random magnetic field perturbations.

Quasi-localized states can cause nonanalytic behavior in the curvature distribution.

The distribution exhibits a branching point at K=0 related to multifractality, confirmed numerically.

Abstract

The level curvature distribution function is studied both analytically and numerically for the case of T-breaking perturbations over the orthogonal ensemble. The leading correction to the shape of the curvature distribution beyond the random matrix theory is calculated using the nonlinear supersymmetric sigma-model and compared to numerical simulations on the Anderson model. It is predicted analytically and confirmed numerically that the sign of the correction is different for T-breaking perturbations caused by a constant vector-potential equivalent to a phase twist in the boundary conditions, and those caused by a random magnetic field. In the former case it is shown using a nonperturbative approach that quasi-localized states in weakly disordered systems can cause the curvature distribution to be nonanalytic. In $2d$ systems the distribution function $P(K)$ has a branching point at K=0 that is related to the multifractality of the wave functions and thus should be a generic feature of all critical eigenstates. A relationship between the branching power and the multifractality exponent $d_{2}$ is suggested. Evidence of the branch-cut singularity is found in numerical simulations in $2d$ systems and at the Anderson transition point in $3d$ systems.