Universal Fluctuations in Spectra of Disordered Systems at the Anderson Transition

Using the level--spacing distribution and the total probability function of the numbers of levels in a given energy interval we analyze the crossover of the level statistics between the delocalized and the localized regimes. By numerically calculating the electron spectra of systems of up to $32^3$ lattice sites described by the Anderson Hamiltonian it is shown that the distribution $P(s)$ of neighboring spacings is {\em scale- independent} at the metal-insulator transition. For large spacings it has a Poisson-like asymptotic form $P(s)\propto \exp (- A\,s/\Delta )$, where $A \approx 1.9$. At the critical point we obtain a linear relationship between the variance of the number of levels $\langle [\delta n(\varepsilon)]^2 \rangle$ and their average number $\langle n(\varepsilon)\rangle$ within the interval $\varepsilon$. The constant of proportionality is less than unity due to the repulsion of the levels. Both $P(s)$ and $\langle [\delta n(\varepsilon)]^2 \rangle$ are determined by the probability density $Q_{n}(\varepsilon)$ of having exactly $n$ levels in the energy interval $\varepsilon$. The distribution $Q_{n}(\varepsilon)$ at the critical point is found to be size--independent and to obey a Gaussian law near its maximum, where $n\sim \langle n \rangle$.