Dimensionality dependence of the wave function statistics at the Anderson transition

The statistics of critical wave functions at the Anderson transition in three and four dimensions are studied numerically. The distribution of the inverse participation ratios (IPR) $P_q$ is shown to acquire a scale-invariant form in the limit of large system size. Multifractality spectra governing the scaling of the ensemble-averaged IPRs are determined. Conjectures concerning the IPR statistics and the multifractality at the Anderson transition in a high spatial dimensionality are formulated.Comment: 4 pages, 4 figure

Distribution of local density of states in disordered metallic samples: logarithmically normal asymptotics

Asymptotical behavior of the distribution function of local density of states (LDOS) in disordered metallic samples is studied with making use of the supersymmetric $\sigma$--model approach, in combination with the saddle--point method. The LDOS distribution is found to have the logarithmically normal asymptotics for quasi--1D and 2D sample geometry. In the case of a quasi--1D sample, the result is confirmed by the exact solution. In 2D case a perfect agreement with an earlier renormalization group calculation is found. In 3D the found asymptotics is of somewhat different type: P(\rho)\sim \exp(-\mbox{const}\,|\ln^3\rho|).Comment: REVTEX, 14 pages, no figure

Fluctuation of the Correlation Dimension and the Inverse Participation Number at the Anderson Transition

The distribution of the correlation dimension in a power law band random matrix model having critical, i.e. multifractal, eigenstates is numerically investigated. It is shown that their probability distribution function has a fixed point as the system size is varied exactly at a value obtained from the scaling properties of the typical value of the inverse participation number. Therefore the state-to-state fluctuation of the correlation dimension is tightly linked to the scaling properties of the joint probability distribution of the eigenstates.Comment: 4 pages, 5 figure

Localization and fluctuations of local spectral density on tree-like structures with large connectivity: Application to the quasiparticle line shape in quantum dots

We study fluctuations of the local density of states (LDOS) on a tree-like lattice with large branching number $m$. The average form of the local spectral function (at given value of the random potential in the observation point) shows a crossover from the Lorentzian to semicircular form at $\alpha\sim 1/m$, where $\alpha= (V/W)^2$, $V$ is the typical value of the hopping matrix element, and $W$ is the width of the distribution of random site energies. For $\alpha>1/m^2$ the LDOS fluctuations (with respect to this average form) are weak. In the opposite case, $\alpha<1/m^2$, the fluctuations get strong and the average LDOS ceases to be representative, which is related to the existence of the Anderson transition at $\alpha_c\sim 1/(m^2\log^2m)$. On the localized side of the transition the spectrum is discrete, and LDOS is given by a set of $\delta$-like peaks. The effective number of components in this regime is given by $1/P$, with $P$ being the inverse participation ratio. It is shown that $P$ has in the transition point a limiting value $P_c$ close to unity, $1-P_c\sim 1/\log m$, so that the system undergoes a transition directly from the deeply localized to extended phase. On the side of delocalized states, the peaks in LDOS get broadened, with a width $\sim\exp\{-{const}\log m[(\alpha-\alpha_c)/\alpha_c]^{-1/2}\}$ being exponentially small near the transition point. We discuss application of our results to the problem of the quasiparticle line shape in a finite Fermi system, as suggested recently by Altshuler, Gefen, Kamenev, and Levitov.Comment: 12 pages, 1 figure. Misprints in eqs.(21) and (28) corrected, section VII added. Accepted for publication in Phys. Rev.

Quasiclassical magnetotransport in a random array of antidots

We study theoretically the magnetoresistance $\rho_{xx}(B)$ of a two-dimensional electron gas scattered by a random ensemble of impenetrable discs in the presence of a long-range correlated random potential. We believe that this model describes a high-mobility semiconductor heterostructure with a random array of antidots. We show that the interplay of scattering by the two types of disorder generates new behavior of $\rho_{xx}(B)$ which is absent for only one kind of disorder. We demonstrate that even a weak long-range disorder becomes important with increasing $B$. In particular, although $\rho_{xx}(B)$ vanishes in the limit of large $B$ when only one type of disorder is present, we show that it keeps growing with increasing $B$ in the antidot array in the presence of smooth disorder. The reversal of the behavior of $\rho_{xx}(B)$ is due to a mutual destruction of the quasiclassical localization induced by a strong magnetic field: specifically, the adiabatic localization in the long-range Gaussian disorder is washed out by the scattering on hard discs, whereas the adiabatic drift and related percolation of cyclotron orbits destroys the localization in the dilute system of hard discs. For intermediate magnetic fields in a dilute antidot array, we show the existence of a strong negative magnetoresistance, which leads to a nonmonotonic dependence of $\rho_{xx}(B)$.Comment: 21 pages, 13 figure

Current correlations and quantum localization in 2D disordered systems with broken time-reversal invariance

We study long-range correlations of equilibrium current densities in a two-dimensional mesoscopic system with the time reversal invariance broken by a random or homogeneous magnetic field. Our result is universal, i.e. it does not depend on the type (random potential or random magnetic field) or correlation length of disorder. This contradicts recent sigma-model calculations of Taras-Semchuk and Efetov (TS&E) for the current correlation function, as well as for the renormalization of the conductivity. We show explicitly that the new term in the sigma-model derived by TS&E and claimed to lead to delocalization does not exist. The error in the derivation of TS&E is traced to an incorrect ultraviolet regularization procedure violating current conservation and gauge invariance.Comment: 8 pages, 3 figure

Frequency Dependence of Magnetopolarizability of Mesoscopic Grains

We calculate average magnetopolarizability of an isolated metallic sample at frequency $\omega$ comparable to the mean level spacing $\Delta$. The frequency dependence of the magnetopolarizability is described by a universal function of $\omega/\Delta$.Comment: 3 pages, 1 figur

Fractional microwave-induced resistance oscillations

We develop a systematic theory of microwave-induced oscillations in magnetoresistivity of a 2D electron gas in the vicinity of fractional harmonics of the cyclotron resonance, observed in recent experiments. We show that in the limit of well-separated Landau levels the effect is dominated by the multiphoton inelastic mechanism. At moderate magnetic field, two single-photon mechanisms become important. One of them is due to resonant series of multiple single-photon transitions, while the other originates from microwave-induced sidebands in the density of states of disorder-broadened Landau levels.Comment: 3 pages, 2 figures; Proceedings of EP2DS17 to be published in Physica E; less technical version of arXiv:0707.099

Critical generalized inverse participation ratio distributions

The system size dependence of the fluctuations in generalized inverse participation ratios (IPR's) $I_{\alpha}(q)$ at criticality is investigated numerically. The variances of the IPR logarithms are found to be scale-invariant at the macroscopic limit. The finite size corrections to the variances decay algebraically with nontrivial exponents, which depend on the Hamiltonian symmetry and the dimensionality. The large-$q$ dependence of the asymptotic values of the variances behaves as $q^2$ according to theoretical estimates. These results ensure the self-averaging of the corresponding generalized dimensions.Comment: RevTex4, 5 pages, 4 .eps figures, to be published in Phys. Rev.