Transition from localized to extended eigenstates in the ensemble of power-law random banded matrices

Abstract

We study statistical properties of the ensemble of large NϫN random matrices whose entries H i j decrease in a power-law fashion H i j ϳ͉iϪ j͉ Ϫ␣ . Mapping the problem onto a nonlinear model with nonlocal interaction, we find a transition from localized to extended states at ␣ϭ1. At this critical value of ␣ the system exhibits multifractality and spectral statistics intermediate between the Wigner-Dyson and Poisson statistics. These features are reminiscent of those typical of the mobility edge of disordered conductors. We find a continuous set of critical theories at ␣ϭ1, parametrized by the value of the coupling constant of the model. At ␣Ͼ1 all states are expected to be localized with integrable power-law tails. At the same time, for 1Ͻ␣Ͻ3/2 the wave packet spreading at a short time scale is superdiffusive: ͉͗r͉͘ϳt 1/(2␣Ϫ1) , which leads to a modification of the Altshuler-Shklovskii behavior of the spectral correlation function. At 1/2Ͻ␣Ͻ1 the statistical properties of eigenstates are similar to those in a metallic sample in dϭ(␣Ϫ1/2) Ϫ1 dimensions. Finally, the region ␣Ͻ1/2 is equivalent to the corresponding Gaussian ensemble of random matrices (␣ϭ0). The theoretical predictions are compared with results of numerical simulations.

Restoring Long-Range Order in One-Dimensional Superconductivity by Power-Law Hopping

“…The strength of the on-site interaction between different spins is |U|, with U < 0. To ensure that the effective model in the following stays well-defined in the L → ∞ limit, we limit our discussion to α > 1/2, in which case it has been known [13] that noninteracting 1D weakly disordered systems with hopping proportional to |l − m| −α mimics the properties of a short-range hopping system with d eff (α) = 2/(2α − 1) spatial dimensions.…”

Long range disorder and Anderson transition in systems with chiral symmetry

“…Taking into account higher order terms in the gradient expansion (as in Ref. [23]), we find the following expression for A (2) 0 ,…”

“…At the bulk the spectral correlations are not affected by the block structure and coincide with the nonchiral version of Eq. (3) which has been intensively studied in recent years [23,31]. In this section we study the localization properties at the origin by mapping the chRBM (3) onto a supersymmetric nonlinear σ model [16].…”

“…[23]. By mapping the problem onto a nonlinear σ model with a nonlocal interaction, it was shown that for a 1/r band decay the eigenstates are multifractal and the spectral correlations resemble the ones at the Anderson transition.…”

“…Typical features include: scale invariant spectrum [20], level repulsion, and sub-Poisson number variance [21]. Different generalized random matrix model (gRMM) have been successfully employed to describe critical statistics [22,23].…”