Correlations of Eigenfunctions in Disordered Systems

TL;DR

This paper derives formulas for eigenfunction correlations in disordered systems using supermatrix sigma-model, revealing how correlations depend on conductance and distance, especially in the weak localization regime.

Contribution

It introduces a general theoretical framework for eigenfunction correlations in disordered systems, connecting them to the supermatrix sigma-model and conductance.

Findings

Correlations of the same eigenfunction scale as g^{-1} at large distances.

Correlations between different eigenfunctions transition from g^{-1} to g^{-2} with increasing distance.

The results apply specifically to the weak localization regime.

Abstract

Correlations of eigenfunctions, $\langle|\psi_k(r_1)|^2|\psi_l(r_2)|^2\rangle$, in a disordered system are investigated. We derive general formulae expressing these correlation functions in terms of the supermatrix sigma-model. In particular case of weak localization regime we find that the correlations of the same eigenfunction are proportional to $g^{-1}$ for large distances, while the correlations of two different eigenfunctions cross over from $g^{-1}$ behavior for $r_1=r_2$ to $g^{-2}$ one for $\vert r_1 - r_2 \vert \gg l$, with $g$ and $l$ being the dimensionless conductance and the mean free path, respectively.