Correlations of eigenfunctions in disordered systems

Correlations of eigenfunctions, 〈|${\mathrm{\psi }}_{\mathrm{k}}$(${\mathrm{r}}_1$)${\mathrm{|}}^2$|${\mathrm{\psi }}_{\mathrm{l}}$(${\mathrm{r}}_2$)${\mathrm{|}}^2$〉, in a disordered system are investigated. We derive general formulas expressing these correlation functions in terms of the supermatrix σ model. In the particular case of the weak localization regime we find that the correlations of the same eigenfunction are proportional to ${\mathrm{g}}^{\mathrm{-}1}$ for large distances, while the correlations of two different eigenfunctions cross over from ${\mathrm{g}}^{\mathrm{-}1}$ behavior for ${\mathrm{r}}_1$=${\mathrm{r}}_2$ to ${\mathrm{g}}^{\mathrm{-}2}$ behavior for |${\mathrm{r}}_1$-${\mathrm{r}}_2$|≫:l, with g and l being the dimensionless conductance and the mean free path, respectively.