arXiv subjects
Publications and source records attributed to I. V. Lerner.
The dependence of the transition temperature $T_g$ in terms of the concentration of magnetic impurities $c$ in spin glasses is explained on the basis of a screened RKKY interaction. The two observed power laws, $T_g ~ c$ at low $c$ and $T_g ~ c^{2/3}$ for intermediate $c$, are described in a unified approach.
The statistics of energy levels for a disordered conductor are considered in the critical energy window near the mobility edge. It is shown that, if critical wave functions are multifractal, the one-dimensional gas of levels on the energy axis is ``compressible'', in the sense that the variance of the level number in an interval is $< (δN)^{2} > = χ $ for $ >> 1$. The compressibility, $χ=η/2d$, is given ``exactly'' in terms of the multifractal exponent $η=d-D_2$ at the mobility edge in a $d$-dimensional system.
For a disordered system near the Anderson transition we show that the nearest-level-spacing distribution has the asymptotics $P(s)\propto \exp(-A s^{2-γ})$ for $s\gg \av{s}\equiv 1$ which is universal and intermediate between the Gaussian asymptotics in a metal and the Poisson in an insulator. (Here the critical exponent $0<γ<1$ and the numerical coefficient $A$ depend only on the dimensionality $d>2$). It is obtained by mapping the energy level distribution to the Gibbs distribution for a classical one-dimensional gas with a pairwise interaction. The interaction, consistent with the universal asymptotics of the two-level correlation function found previously, is proved to be the power-law repulsion with the exponent $-γ$.
It is shown that the distribution functions of the diffusion coefficient are very similar in the standard model of quantum diffusion in a disordered metal and in a model of classical diffusion in a disordered medium: in both cases the distribution functions have lognormal tails, their part increasing with the increase of the disorder. The similarity is based on a similar behaviour of the high-gradient operators determining the high-order cumulants. The one-loop renormalization-group corrections make the anomalous dimension of the operator that governs the $s$-th cumulant proportional to $s(s-1)$ thus overtaking for large $s$ the negative normal dimension. As behaviour of the ensemble-averaged diffusion coefficient is quite different in these models, it suggests that a possible universality in the distribution functions is independent of the behaviour of average quantities.
We demonstrate the level statistics in the vicinity of the Anderson transition in $d>2$ dimensions to be universal and drastically different from both Wigner-Dyson in the metallic regime and Poisson in the insulator regime. The variance of the number of levels $N$ in a given energy interval with $\langle N\rangle\gg1$ is proved to behave as $\langle N\rangle^γ$ where $γ=1-(νd)^{-1}$ and $ν$ is the correlation length exponent. The inequality $γ<1$, shown to be required by an exact sum rule, results from nontrivial cancellations (due to the causality and scaling requirements) in calculating the two-level correlation function.