Statistics of energy levels and eigenfunctions in disordered systems

The article reviews recent developments in the theory of fluctuations and correlations of energy levels and eigenfunction amplitudes in diffusive mesoscopic samples. Various spatial geometries are considered, with emphasis on low-dimensional (quasi-1D and 2D) systems. Calculations are based on the supermatrix σ-model approach. The method reproduces, in so-called zero-mode approximation, the universal random matrix theory (RMT) results for the energy-level and eigenfunction fluctuations. Going beyond this approximation allows us to study system-specific deviations from universality, which are determined by the diffusive classical dynamics in the system. These deviations are especially strong in the far “tails” of the distribution function of the eigenfunction amplitudes (as well as of some related quantities, such as local density of states, relaxation time, etc.). These asymptotic “tails” are governed by anomalously localized states which are formed in rare realizations of the random potential. The deviations of the level and eigenfunction statistics from their RMT form strengthen with increasing disorder and become especially pronounced at the Anderson metal–insulator transition. In this regime, the wave functions are multifractal, while the level statistics acquires a scale-independent form with distinct critical features. Fluctuations of the conductance and of the local intensity of a classical wave radiated by a point-like source in the quasi-1D geometry are also studied within the σ-model approach. For a ballistic system with rough surface an appropriately modified (“ballistic”) σ-model is used. Finally, the interplay of the fluctuations and the electron–electron interaction in small samples is discussed, with application to the Coulomb blockade spectra.