Statistics of spectra of disordered systems near the metal-insulator transition

Abstract

We study the nearest-level-spacing distribution function P(s) in a disordered system near the metal-insulator transition. We claim that in the limit of an infinite system there are only three possible functions P(s): Wigner surmise ${\mathit{P}}_{\mathit{W}}$(s) in a metal, Poisson law ${\mathit{P}}_{\mathit{P}}$(s) in an insulator, and a third one ${\mathit{P}}_{\mathit{T}}$(s), exactly at the transition. The function ${\mathit{P}}_{\mathit{T}}$ is an interesting hybrid of ${\mathit{P}}_{\mathit{W}}$(s) and ${\mathit{P}}_{\mathit{P}}$(s), it has the small-s behavior of the former and the large-s behavior of the latter one. A scaling theory of critical behavior of P(s) in finite samples is proposed and verified numerically.