Eigenfunction entropy and spectral compressibility for critical random matrix ensembles

Based on numerical and perturbation series arguments we conjecture that for certain critical random matrix models the information dimension of eigenfunctions D_1 and the spectral compressibility chi are related by the simple equation chi+D_1/d=1, where d is the system dimensionality.Comment: 4 pages, 3 figure

Spectral correlations in the crossover between GUE and Poisson regularity: on the identification of scales

Motivated by questions of present interest in nuclear and condensed matter physics we consider the superposition of a diagonal matrix with independent random entries and a GUE. The relative strength of the two contributions is determined by a parameter $\lambda$ suitably defined on the unfolded scale. Using results for the spectral two-point correlator of this model obtained in the framework of the supersymmetry method we focus attention on two different regimes. For $\lambda$ << 1 the correlations are given by Dawson's integral while for $\lambda$ >> 1 we derive a novel analytical formula for the two-point function. In both cases the energy scales, in units of the mean level spacing, at which deviations from pure GUE behavior become noticable can be identified. We also derive an exact expansion of the local level density for finite level number.Comment: 15 pages, Revtex, no figures, to be published in special issue of J. Math. Phys. (1997

Infrared catastrophe in two-quasiparticle collision integral

Relaxation of a non-equilibrium state in a disordered metal with a spin-dependent electron energy distribution is considered. The collision integral due to the electron-electron interaction is computed within the approximation of a two-quasiparticle scattering. We show that the spin-flip scattering processes with a small energy transfer may lead to the divergence of the collision integral for a quasi one-dimensional wire. This divergence is present only for a spin-dependent electron energy distribution which corresponds to the total electron spin magnetization M=0 and only for non-zero interaction in the triplet channel. In this case a non-perturbative treatment of the electron-electron interaction is needed to provide an effective infrared cut-off.Comment: 6 pages, 3 figure