Statistics of wave functions in mesoscopic systems

We review the results of a recent study of fluctuations of wave functions in confined chaotic systems. The fluctuations can be due to a random potential or be a consequence of a chaotic scattering by the walls. The entire distribution function of the local amplitudes of the wave functions, f₁, and the joint two-point distribution are calculated in various situations. The computation is performed using the supersymmetry technique and employs the studies of a reduced version of the non-linear supersymmetric σ-model developed especially for investigating the properties of a single eigenstate in a discrete spectrum of a chaotic quantum system. For not very large amplitudes, the complete description can be achieved using the zero-dimensional approximation of the σ-model. The distribution function calculated in the limit of various symmetry classes shows the universal behavior known as the Porter-Thomas statistics, and fluctuations at distant points do not correlate. In the crossover regime between the ensembles, the distribution of local amplitudes shows a somewhat more sophisticated behavior: the fluctuations in this case are correlated over distances exceeding the mean free path. For large amplitudes generated by the states the most affected by the localization (we call them prelocalized), the zero-dimensional approximation is no longer valid. Instead, the statistics of their wave functions is determined by nontrivial vacua of the reduced σ-model which is quite similar to the Liouville model known in conformal field theory. In particular, the vacuum state of the reduced σ-model obeys the Liouville equation, which indicates that in two dimensions the prelocalized states have nearly critical properties: we prove their multifractality and power-law statistically averaged envelope |φ(r)|² ∝ r^-2μ at the intermediate range of distances below the localization length with a spectrum of exponents μ<1, as well as obtain a logarithmically-normal tail of the distribution function f₁. We also find an evidence of prelocalized states in quasi-one-dimensional wires with the length shorter than the localization length: their statistically averaged envelope has power-law asymptotics, |φ(x)|² ∝ x^-2, and the tail of the distribution function is similar to that describing localized states in the infinite wires.