Correlations of Eigenfunctions in Disordered Systems

Abstract

Correlations of eigenfunctions, $⟨|{\psi }_k(r_1){|}^2|{\psi }_l(r_2){|}^2⟩$, 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 $|r_1-r_2|\gg l$, with $g$ and $l$ being the dimensionless conductance and the mean free path, respectively.

Revised and extended version; some errors are corrected. 4 pages

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