Universality of Parametric Spectral Correlations: Local versus Extended Perturbing Potentials

TL;DR

This paper investigates how different types of external potential perturbations affect spectral correlations in disordered conductors, revealing universal behaviors linked to the perturbation's range and profile, and connecting these to measurable physical properties.

Contribution

The study identifies two universal regimes of parametric spectral statistics based on perturbation characteristics and links translational invariance in Hamiltonian space to physical observables.

Findings

Localized perturbations relate to scattering matrix eigenphases.

Universal spectral correlation regimes depend on perturbation range.

Results extend random matrix theory applicability to disordered systems.

Abstract

We explore the influence of an arbitrary external potential perturbation V on the spectral properties of a weakly disordered conductor. In the framework of a statistical field theory of a nonlinear sigma-model type we find, depending on the range and the profile of the external perturbation, two qualitatively different universal regimes of parametric spectral statistics (i.e. cross-correlations between the spectra of Hamiltonians H and H+V). We identify the translational invariance of the correlations in the space of Hamiltonians as the key indicator of universality, and find the connection between the coordinate system in this space which makes the translational invariance manifest, and the physically measurable properties of the system. In particular, in the case of localized perturbations, the latter turn out to be the eigenphases of the scattering matrix for scattering off the perturbing potential V. They also have a purely statistical interpretation in terms of the moments of the level velocity distribution. Finally, on the basis of this analysis, a set of results obtained recently by the authors using random matrix theory methods is shown to be applicable to a much wider class of disordered and chaotic structures.