Statistics of pre-localized states in disordered conductors

The distribution function of local amplitudes of single-particle states in disordered conductors is calculated on the basis of the supersymmetric $\sigma$-model approach using a saddle-point solution of its reduced version. Although the distribution of relatively small amplitudes can be approximated by the universal Porter-Thomas formulae known from the random matrix theory, the statistics of large amplitudes is strongly modified by localization effects. In particular, we find a multifractal behavior of eigenstates in 2D conductors which follows from the non-integer power-law scaling for the inverse participation numbers (IPN) with the size of the system. This result is valid for all fundamental symmetry classes (unitary, orthogonal and symplectic). The multifractality is due to the existence of pre-localized states which are characterized by power-law envelopes of wave functions, $|\psi_t(r)|^2\propto r^{-2\mu}$, $\mu <1$. The pre-localized states in short quasi-1D wires have the power-law tails $|\psi (x)|^2\propto x^{-2}$, too, although their IPN's indicate no fractal behavior. The distribution function of the largest-amplitude fluctuations of wave functions in 2D and 3D conductors has logarithmically-normal asymptotics.