Universal Fluctuations in Spectra of Disordered Systems at the Anderson Transition

Abstract

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 323 lattice sites described by the Anderson Hamiltonian it is shown that the distribution P(s) of neighboring spacings is scale-independent at the metal–insulator transition. For large spacings it has a Poisson-like asymptotic form P(s)∝exp (- As/Δ), where A ≈1.9. At the critical point we obtain a linear relationship between the variance of the number of levels <[δn(ε)]2 > and their average number <n(ε)> within the interval ε. The constant of proportionality is less than unity due to the repulsion of the levels. Both P(s) and <[δn(ε)]2 > are determined by the probability density Qn(ε) of having exactly n levels in the energy interval ε. The distribution Qn(ε) at the critical point is found to be size-independent and to obey a Gaussian law near its maximum, where n∼<n >.