Energy-level correlation function and ac conductivity of a finite disordered system

The space- and energy-difference- (ω) dependent Wigner-type correlation function for the energy levels is studied for the Anderson disordered tight-binding model, including some of the effects of localization. For small metallic particles it is confirmed that these correlations, in the absence of spin and magnetic effects, are similar to those of the Gaussian orthogonal ensemble. This is in agreement with the conjecture made by Gor’kov and Eliashberg and the analytical results obtained by Efetov. The results are used to obtain the effective conductivity ${\sigma }_{\mathrm{eff}(\mathrm{\omega }}$), relevant for the absorption of low-frequency electromagnetic radiation, in all regimes, including the effects of localization and screening. It is found that in the orthogonal case, ${\sigma }_{\mathrm{eff}(\mathrm{\omega }}$) does not change at low ω when level correlations become important. The conditions to observe changes in the absorption in this regime due to changes in the ensemble symmetry are formulated. In all cases except for the microscopic (critical) one, ${\sigma }_{\mathrm{eff}\left(\mathrm{\omega }\right)\mathrm{\propto }{\omega }^2}$, and in the latter region, ${\sigma }_{\mathrm{eff}\left(\mathrm{\omega }\right)\mathrm{\propto }{\omega }^{5/3}}$. For systems much larger than the localization length, the level correlations decay in space with a length ξ ln(${\Delta }_{\xi }$/ω), as ω→0 (${\Delta }_{\xi }$ is the average spacing between levels within a localization volume). The connection with Mott’s calculation of the ac conductivity in the insulating phase is made. It is shown that these ideas explain the ‘‘level attraction’’ found in one dimension by Gor’kov, Dorokhov, and Prigara. These considerations are generalized to the evaluation of the space-dependent level correlation function in an arbitrary dimension.