Statistics of prelocalized states in disordered conductors

The distribution function of local amplitudes, t=(r0)2, of single-particle states in disordered conductors is calculated on the basis of a reduced version of the supersymmetric model solved using the saddle-point method. Although the distribution of relatively small amplitudes can be approximated by the universal Porter-Thomas formulas known from the random-matrix theory, the asymptotical statistics of large ts is strongly modified by localization effects. In particular, we find a multifractal behavior of eigenstates in two-dimensional (2D) conductors which follows from the noninteger power-law scaling for the inverse participation numbers (IPNs) with the size of the system, VtnL-(n-1)d*(n), where d*(n)=2-β-1n/(4π2) is a function of the index n and disorder. The result is valid for all fundamental symmetry classes (unitary, β=u1; orthogonal, β0=1/2; symplectic, βs=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 functions at the tails, t(r)2r-2, =(t)<1. The prelocalized states in short quasi-1D wires have the tails ψ(x)2x-2, too, although their IPNs indicate no fractal behavior. The distribution function of the largest-amplitude fluctuations of wave functions in 2D and 3D conductors has logarithmically normal asymptotics.