Energy fluctuations, Thouless energy, and conductance in the Anderson model in the ballistic and diffusive regimes

Abstract

We perform a numerical calculation of long-range energy fluctuations of two- and three-dimensional Anderson models in the ballistic and diffusive regimes. In calculating the energy fluctuations, averages are taken over different realizations of disorder, and not over energy windows at different levels. For windows of width $E$ smaller than the critical energy $E_c,$ fluctuations follow the logarithmic behavior characteristic of random matrix theory (RMT), no matter the degree of disorder. For energies higher than $E_c,$ fluctuations are nearly constant and below RMT in the ballistic case, and they are higher than RMT and increase with energy in the diffusive case. The results allow a reasonably accurate estimate of $E_c.$ The expected behavior of the critical energy with the system size and energy is reproduced by our numerical results. An efficient implementation of Kubo’s formula has been used to calculate the conductance of the system. In the diffusive regime the numerical results for the adimensional conductance are in reasonable agreement with the numerical results for $E_c.$ It is also shown that the asymptotic expression derived by Altshuler and Shklovskii for fluctuations in the diffusive regime gives results much smaller than those reported here.