Two-eigenfunction correlation in a multifractal metal and insulator

Abstract

We consider the correlation of two single-particle probability densities ${\ensuremath{\mid}{\ensuremath{\Psi}}_{E}(\mathbf{r})\ensuremath{\mid}}^{2}$ at coinciding points $\mathbf{r}$ as a function of the energy separation $\ensuremath{\omega}=\ensuremath{\mid}E\ensuremath{-}{E}^{\ensuremath{'}}\ensuremath{\mid}$ for disordered tight-binding lattice models (the Anderson models) and certain random-matrix ensembles. We focus on the parameter range close to but not exactly at the Anderson localization transition. We show that even away from the critical point, the eigenfunction statistics exhibit the remnant of multifractality characteristic of the critical states. By a combination of numerical results on the Anderson model and analytical and numerical results for the relevant random-matrix theories, we identified the Gaussian random-matrix ensembles that describe the multifractal features both in the metal and in the insulator phases. This analysis reveals other features of the structure of eigenfunctions: (i) eigenfunction mutual avoidance at large energy separations, (ii) competition between repulsion of centers of localization and enhanced overlap by tails at small energy separations, and (iii) possible existence of a different metallic phase in higher dimensions.