Statistics of prelocalized states in disordered conductors

The distribution function of local amplitudes, t=||ψ(r₀)||², of single-particle states in disordered conductors is calculated on the basis of a reduced version of the supersymmetric σ 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 t's 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 (IPN's) with the size of the system, Vt_n~L^-(n-1)d*(n), where d^*(n)=2-β^-1n/(4π²νD) is a function of the index n and disorder. The result is valid for all fundamental symmetry classes (unitary, β_u=1 orthogonal, β₀=1/2 symplectic, β_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 functions at the tails, ||ψ_t(r)||²~r^-2μ, μ=μ(t)<1. The prelocalized states in short quasi-1D wires have the tails ||ψ(x)||²~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.