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.

Exact Multifractality for Disordered N-Flavour Dirac Fermions in Two Dimensions

We present a nonperturbative calculation of all multifractal scaling exponents at strong disorder for critical wavefunctions of Dirac fermions interacting with a non-Abelian random vector potential in two dimensions. The results, valid for an arbitrary number of fermionic flavours, are obtained by deriving from Conformal Field Theory an effective Gaussian model for the wavefunction amplitudes and mapping to the thermodynamics of a single particle in a random potential. Our spectrum confirms that the wavefunctions remain delocalized in the presence of strong disorder.Comment: 4 pages, no figue

Termination of Multifractal Behaviour for Critical Disordered Dirac Fermions

We consider Dirac fermions interacting with a disordered non-Abelian vector potential. The exact solution is obtained through a special type of conformal field theory including logarithmic correlators, without resorting to the replica or supersymmetry approaches. It is shown that the proper treatment of the conformal theory leads to a different multifractal scaling behaviour than initially expected. Moreover, the previous replica solution is found to be incorrect at the level of higher correlation functions.Comment: 4 pages, no figure

New Class of Random Matrix Ensembles with Multifractal Eigenvectors

Three recently suggested random matrix ensembles (RME) are linked together by an exact mapping and plausible conjections. Since it is known that in one of these ensembles the eigenvector statistics is multifractal, we argue that all three ensembles belong to a new class of critical RME with multifractal eigenfunction statistics and a universal critical spectral statitics. The generic form of the two-level correlation function for weak and extremely strong multifractality is suggested. Applications to the spectral statistics at the Anderson transition and for certain systems on the border of chaos and integrability is discussed.Comment: 4 pages RevTeX, resubmitte

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.

Quasi-localized states in disordered metals and non-analyticity of the level curvature distribution function

It is shown that the quasi-localized states in weakly disordered systems can lead to the non-analytical distribution of level curvatures. In 2D systems the distribution function P(K) has a branching point at K=0. In quasi-1D systems the non-analyticity at K=0 is very weak, and in 3D metals it is absent at all. Such a behavior confirms the conjecture that the branching at K=0 is due to the multi-fractality of wave functions and thus is a generic feature of all critical eigenstates. The relationsip between the branching power and the multi-fractality exponent $\eta(2)$ is derived.Comment: 4 pages, LATE

Level Curvature Distribution and the Structure of Eigenfunctions in Disordered Systems

The level curvature distribution function is studied both analytically and numerically for the case of T-breaking perturbations over the orthogonal ensemble. The leading correction to the shape of the curvature distribution beyond the random matrix theory is calculated using the nonlinear supersymmetric sigma-model and compared to numerical simulations on the Anderson model. It is predicted analytically and confirmed numerically that the sign of the correction is different for T-breaking perturbations caused by a constant vector-potential equivalent to a phase twist in the boundary conditions, and those caused by a random magnetic field. In the former case it is shown using a nonperturbative approach that quasi-localized states in weakly disordered systems can cause the curvature distribution to be nonanalytic. In $2d$ systems the distribution function $P(K)$ has a branching point at K=0 that is related to the multifractality of the wave functions and thus should be a generic feature of all critical eigenstates. A relationship between the branching power and the multifractality exponent $d_{2}$ is suggested. Evidence of the branch-cut singularity is found in numerical simulations in $2d$ systems and at the Anderson transition point in $3d$ systems.Comment: 34 pages (RevTeX), 8 figures (postscript

Spectral Compressibility at the Metal-Insulator Transition of the Quantum Hall Effect

The spectral properties of a disordered electronic system at the metal-insulator transition point are investigated numerically. A recently derived relation between the anomalous diffusion exponent $\eta$ and the spectral compressibility $\chi$ at the mobility edge, $\chi=\eta/2d$, is confirmed for the integer quantum Hall delocalization transition. Our calculations are performed within the framework of an unitary network-model and represent a new method to investigate spectral properties of disordered systems.Comment: 5 pages, RevTeX, 3 figures, Postscript, strongly revised version to be published in PR

Multifractal analysis of the electronic states in the Fibonacci superlattice under weak electric fields

Influence of the weak electric field on the electronic structure of the Fibonacci superlattice is considered. The electric field produces a nonlinear dynamics of the energy spectrum of the aperiodic superlattice. Mechanism of the nonlinearity is explained in terms of energy levels anticrossings. The multifractal formalism is applied to investigate the effect of weak electric field on the statistical properties of electronic eigenfunctions. It is shown that the applied electric field does not remove the multifractal character of the electronic eigenfunctions, and that the singularity spectrum remains non-parabolic, however with a modified shape. Changes of the distances between energy levels of neighbouring eigenstates lead to the changes of the inverse participation ratio of the corresponding eigenfunctions in the weak electric field. It is demonstrated, that the local minima of the inverse participation ratio in the vicinity of the anticrossings correspond to discontinuity of the first derivative of the difference between marginal values of the singularity strength. Analysis of the generalized dimension as a function of the electric field shows that the electric field correlates spatial fluctuations of the neighbouring electronic eigenfunction amplitudes in the vicinity of anticrossings, and the nonlinear character of the scaling exponent confirms multifractality of the corresponding electronic eigenfunctions.Comment: 10 pages, 9 figure

Interactions, Localization, and the Integer Quantum Hall Effect

We report on numerical studies of the influence of Coulomb interactions on localization of electronic wavefunctions in a strong magnetic field. Interactions are treated in the Hartree-Fock approximation. Localization properties are studied both by evaluating participation ratios of Hartree-Fock eigenfunctions and by studying the boundary-condition dependence of Hartree-Fock eigenvalues. We find that localization properties are independent of interactions. Typical energy level spacings near the Fermi level and the sensitivity of those energy levels to boundary condition show similar large enhancements so that the Thouless numbers of the Hartree-Fock eigenvalues are similar to those of non-interacting electrons.Comment: 10 pages, latex (revtex 3.0), 3 figures are avaiable from S.R. Eric Yang (e-mail [email protected]