Statistics of energy levels and eigenfunctions in disordered systems

The article reviews recent developments in the theory of fluctuations and c orrelations of energy levels and eigenfunction amplitudes in diffusive meso scopic samples. Various spatial geometries are considered, with emphasis on low-dimensional (quasi-1D and 2D) systems. Calculations are based on the s upermatrix sigma-model approach. The method reproduces, in so-called zero-m ode approximation, the universal random matrix theory (RMT) results for the energy-level and eigenfunction fluctuations. Going beyond this approximati on allows us to study system-specific deviations from universality, which a re determined by the diffusive classical dynamics in the system. These devi ations are especially strong in the far "tails" of the distribution functio n of the eigenfunction amplitudes (as well as of some related quantities, s uch as local density of states, relaxation time, etc.). These asymptotic "t ails" are governed by anomalously localized states which are formed in rare realizations of the random potential. The deviations of the level and eige nfunction statistics from their RMT form strengthen with increasing disorde r and become especially pronounced at the Anderson metal-insulator transiti on. In this regime, the wave functions are multifractal, while the level st atistics 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 stud ied within the cr-model approach. For a ballistic system with rough surface an appropriately modified ("ballistic") sigma-model is used. Finally, the interplay of the fluctuations and the electron-electron interaction in smal l samples is discussed, with application to the Coulomb blockade spectra. ( C) 2000 Elsevier Science B.V. All rights reserved.