Statistics of prelocalized states in disordered conductors

The distribution function of local amplitudes, t=‖ψ(${\mathbf{r}}_0$)${\mathrm{\Vert }}^2$, 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, ${\mathit{Vt}}_{\mathit{n}}$∝${\mathit{L}}^{\mathrm{-}\left(\mathit{n}\mathrm{-}1\right)\mathit{d}\mathrm{*}}$(n), where ${\mathit{d}}^{\mathrm{*}}$(n)=2-${\mathrm{\beta }}^{\mathrm{-}1}$n/(4${\mathrm{\pi }}^2$νD) is a function of the index n and disorder. The result is valid for all fundamental symmetry classes (unitary, ${\mathrm{\beta }}_{\mathit{u}}$=1; orthogonal, ${\mathrm{\beta }}_0$=1/2; symplectic, ${\mathrm{\beta }}_{\mathit{s}}$=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, ‖${\mathrm{\psi }}_{\mathit{t}}$(r)${\mathrm{\Vert }}^2$∝${\mathit{r}}^{\mathrm{-}2\mathrm{\mu }}$, μ=μ(t)<1. The prelocalized states in short quasi-1D wires have the tails ‖ψ(x)${\mathrm{\Vert }}^2$∝${\mathit{x}}^{\mathrm{-}2}$, 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.