Density of quasiparticle states for a two-dimensional disordered system: Metallic, insulating, and critical behavior in the class D thermal quantum Hall effect

We investigate numerically the quasiparticle density of states $\varrho(E)$ for a two-dimensional, disordered superconductor in which both time-reversal and spin-rotation symmetry are broken. As a generic single-particle description of this class of systems (symmetry class D), we use the Cho-Fisher version of the network model. This has three phases: a thermal insulator, a thermal metal, and a quantized thermal Hall conductor. In the thermal metal we find a logarithmic divergence in $\varrho(E)$ as $E\to 0$, as predicted from sigma model calculations. Finite size effects lead to superimposed oscillations, as expected from random matrix theory. In the thermal insulator and quantized thermal Hall conductor, we find that $\varrho(E)$ is finite at E=0. At the plateau transition between these phases, $\varrho(E)$ decreases towards zero as $|E|$ is reduced, in line with the result $\varrho(E) \sim |E|\ln(1/|E|)$ derived from calculations for Dirac fermions with random mass.Comment: 8 pages, 8 figures, published versio

Quantum Hall Transition in the Classical Limit

We study the quantum Hall transition using the density-density correlation function. We show that in the limit h->0 the electron density moves along the percolating trajectories, undergoing normal diffusion. The localization exponent coincides with its percolation value \nu=4/3. The framework provides a natural way to study the renormalization group flow from percolation to quantum Hall transition. We also confirm numerically that the critical conductivity of a classical limit of quantum Hall transition is \sigma_{xx} = \sqrt{3}/4.Comment: 8 pages, 4 figures; substantial changes include the critical conductivity calculatio

On the distribution of transmission eigenvalues in disordered wires

We solve the Dorokhov-Mello-Pereyra-Kumar equation which describes the evolution of an ensamble of disordered wires of increasing length in the three cases $\beta=1,2,4$. The solution is obtained by mapping the problem in that of a suitable Calogero-Sutherland model. In the $\beta=2$ case our solution is in complete agreement with that recently found by Beenakker and Rejaei.Comment: 4 pages, Revtex, few comments added at the end of the pape

Universal eigenvector statistics in a quantum scattering ensemble

We calculate eigenvector statistics in an ensemble of non-Hermitian matrices describing open quantum systems [F. Haake et al., Z. Phys. B 88, 359 (1992)] in the limit of large matrix size. We show that ensemble-averaged eigenvector correlations corresponding to eigenvalues in the center of the support of the density of states in the complex plane are described by an expression recently derived for Ginibre's ensemble of random non-Hermitian matrices.Comment: 4 pages, 5 figure

Effect of a magnetic flux on the critical behavior of a system with long range hopping

We study the effect of a magnetic flux in a 1D disordered wire with long range hopping. It is shown that this model is at the metal-insulator transition (MIT) for all disorder values and the spectral correlations are given by critical statistics. In the weak disorder regime a smooth transition between orthogonal and unitary symmetry is observed as the flux strength increases. By contrast, in the strong disorder regime the spectral correlations are almost flux independent. It is also conjectured that the two level correlation function for arbitrary flux is given by the dynamical density-density correlations of the Calogero-Sutherland (CS) model at finite temperature. Finally we describe the classical dynamics of the model and its relevance to quantum chaos.Comment: 5 pages, 4 figure

Universal Spectral Correlations at the Mobility Edge

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 $\langle N\rangle^\gamma$ where $\gamma=1-(\nu d)^{-1}$ and $\nu$ is the correlation length exponent. The inequality $\gamma<1$, 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.Comment: REVTeX, 12pages, +1 postscript figure (included

THE ANOMALOUS DIFFUSION IN HIGH MAGNETIC FIELD AND THE QUASIPARTICLE DENSITY OF STATES

We consider a disordered two-dimensional electronic system in the limit of high magnetic field at the metal-insulator transition. Density of states close to the Fermi level acquires a divergent correction to the lowest order in electron-electron interaction and shows a new power-law dependence on the energy, with the power given by the anomalous diffusion exponent $\eta$. This should be observable in the tunneling experiment with double-well GaAs heterostructure of the mobility $\propto 10^{4}V/s$ at temperatures of $\propto 10 mK$ and voltages of $\propto 1 \mu V$.Comment: 12 pages, LATEX, one figure available at request, accepted for publication in Phys. Rev.

Multifractality and critical fluctuations at the Anderson transition

Critical fluctuations of wave functions and energy levels at the Anderson transition are studied for the family of the critical power-law random banded matrix ensembles. It is shown that the distribution functions of the inverse participation ratios (IPR) $P_q$ are scale-invariant at the critical point, with a power-law asymptotic tail. The IPR distribution, the multifractal spectrum and the level statistics are calculated analytically in the limits of weak and strong couplings, as well as numerically in the full range of couplings.Comment: 14 pages, 13 eps figure

Quantum and classical localisation, the spin quantum Hall effect and generalisations

We consider network models for localisation problems belonging to symmetry class C. This symmetry class arises in a description of the dynamics of quasiparticles for disordered spin-singlet superconductors which have a Bogoliubov - de Gennes Hamiltonian that is invariant under spin rotations but not under time-reversal. Our models include but also generalise the one studied previously in the context of the spin quantum Hall effect. For these systems we express the disorder-averaged conductance and density of states in terms of sums over certain classical random walks, which are self-avoiding and have attractive interactions. A transition between localised and extended phases of the quantum system maps in this way to a similar transition for the classical walks. In the case of the spin quantum Hall effect, the classical walks are the hulls of percolation clusters, and our approach provides an alternative derivation of a mapping first established by Gruzberg, Read and Ludwig, Phys. Rev. Lett. 82, 4254 (1999).Comment: 11 pages, 5 figure

Disordered Electrons in a Strong Magnetic Field: Transfer Matrix Approaches to the Statistics of the Local Density of States

We present two novel approaches to establish the local density of states as an order parameter field for the Anderson transition problem. We first demonstrate for 2D quantum Hall systems the validity of conformal scaling relations which are characteristic of order parameter fields. Second we show the equivalence between the critical statistics of eigenvectors of the Hamiltonian and of the transfer matrix, respectively. Based on this equivalence we obtain the order parameter exponent $\alpha_0\approx 3.4$ for 3D quantum Hall systems.Comment: 4 pages, 3 Postscript figures, corrected scale in Fig.