Spectral correlations from the metal to the mobility edge

Abstract

We have studied numerically the spectral correlations in a metallic phase and at the metal-insulator transition. We have calculated directly the two-point correlation function of the density of states R(s,s′). In the metallic phase, it is well described by the random-matrix theory. We also find numerically the diffusive corrections for the number variance 〈δ${\mathit{n}}^2$(s)〉 predicted by Al’tshuler and Shklovskiĭ. At the transition, at small energy scales, R(s-s′) starts linearly, with a slope larger than in a metal. At large separations ‖s-s′‖≫1, it is found to decrease as a power law R(s,s′)∼-c/‖s-s′${\mathrm{\Vert }}^{2\mathrm{-}\gamma }$ with c∼0.041 and γ∼0.83, in good agreement with recent microscopic predictions. At the transition, we have also calculated the form factor K̃(t), Fourier transform of R(s-s′). At large s, the number variance contains two terms 〈δ${\mathit{n}}^2$(s)〉=B〈n${\mathrm{〉}}^{\gamma }$+2πK̃(0)〈n〉, where K̃(0) is the limit of the form factor for t→0.