STATISTICS OF PRELOCALIZED STATES IN DISORDERED CONDUCTORS

The distribution function of local amplitudes, t = \psi(r(0))\(2), of single-particle states in disordered conductors is calculated on the b asis of a reduced version of the supersymmetric sigma model solved usi ng the saddle-point method. Although the distribution of relatively sm all amplitudes can be approximated by the universal Porter-Thomas form ulas known from the random-matrix theory, the asymptotical statistics of large t's is strongly modified by localization effects. In particul ar, we find a multifractal behavior of eigenstates in two-dimensional (2D) conductors which follows from the noninteger power-law scaling fo r the inverse participation numbers (IPN's) with the size of the syste m, Vt(n) proportional to L(-(n-1)d(n)), where d*(n) = 2 - beta(-1)n/( 4 pi nu D) is a function of the index n and disorder. The result is va lid for all fundamental symmetry classes (unitary, beta(u) = 1; orthog onal, beta(o) = 1/2; symplectic, beta(s) = 2). The multifractality is due to the existence of prelocalized states which are characterized by a power-law form of statistically averaged envelopes of wave function s at the tails, \psi(t)(r)\(2) proportional to r(-2 mu), mu = mu(t) < 1. The prelocalized states in short quasi-one-dimensional (1D) wires h ave the tails \psi(x)\(2) proportional to x(-2), too, although their I PN's indicate no fractal behavior. The distribution function of the la rgest-amplitude fluctuations of wave functions in 2D and 3D conductors has logarithmically normal asymptotics.