Theory of Anomalous Quantum Hall Effects in Graphene

Recent successes in manufacturing of atomically thin graphite samples (graphene) have stimulated intense experimental and theoretical activity. The key feature of graphene is the massless Dirac type of low-energy electron excitations. This gives rise to a number of unusual physical properties of this system distinguishing it from conventional two-dimensional metals. One of the most remarkable properties of graphene is the anomalous quantum Hall effect. It is extremely sensitive to the structure of the system; in particular, it clearly distinguishes single- and double-layer samples. In spite of the impressive experimental progress, the theory of quantum Hall effect in graphene has not been established. This theory is a subject of the present paper. We demonstrate that the Landau level structure by itself is not sufficient to determine the form of the quantum Hall effect. The Hall quantization is due to Anderson localization which, in graphene, is very peculiar and depends strongly on the character of disorder. It is only a special symmetry of disorder that may give rise to anomalous quantum Hall effects in graphene. We analyze the symmetries of disordered single- and double-layer graphene in magnetic field and identify the conditions for anomalous Hall quantization.Comment: 13 pages (article + supplementary material), 5 figure

Topological delocalization of two-dimensional massless Dirac fermions

The beta function of a two-dimensional massless Dirac Hamiltonian subject to a random scalar potential, which e.g., underlies the theoretical description of graphene, is computed numerically. Although it belongs to, from a symmetry standpoint, the two-dimensional symplectic class, the beta function monotonically increases with decreasing $g$. We also provide an argument based on the spectral flows under twisting boundary conditions, which shows that none of states of the massless Dirac Hamiltonian can be localized.Comment: 4 pages, 2 figure

A Renormalization-Group approach to the Coulomb Gap

The free energy of the Coulomb Gap problem is expanded as a set of Feynman diagrams, using the standard diagrammatic methods of perturbation theory. The gap in the one-particle density of states due to long-ranged interactions corresponds to a renormalization of the two-point vertex function. By collecting the leading order logarithmic corrections we have derived the standard result for the density of states in the critical dimension, d=1. This method, which is shown to be identical to the approach of Thouless, Anderson and Palmer to spin glasses, allows us to derive the strong-disorder behaviour of the density of states. The use of the renormalization group allows this derivation to be extended to all disorders, and the use of an epsilon-expansion allows the method to be extended to d=2 and d=3. We speculate that the renormalization group equations can also be derived diagrammatically, allowing a simple derivation of the crossover behaviour observed in the case of weak disorder.Comment: 16 pages, LaTeX. Diagrams available on request from [email protected]. Changes to figure 4 and second half of section

Hall plateau diagram for the Hofstadter butterfly energy spectrum

We extensively study the localization and the quantum Hall effect in the Hofstadter butterfly, which emerges in a two-dimensional electron system with a weak two-dimensional periodic potential. We numerically calculate the Hall conductivity and the localization length for finite systems with the disorder in general magnetic fields, and estimate the energies of the extended levels in an infinite system. We obtain the Hall plateau diagram on the whole region of the Hofstadter butterfly, and propose a theory for the evolution of the plateau structure with increasing disorder. There we show that a subband with the Hall conductivity $n e^2/h$ has $|n|$ separated bunches of extended levels, at least for an integer $n \leq 2$. We also find that the clusters of the subbands with identical Hall conductivity, which repeatedly appear in the Hofstadter butterfly, have a similar localization property.Comment: 9 pages, 12 figure

Magnetic-Field Dependence of the Localization Length in Anderson Insulators

Using the conventional scaling approach as well as the renormalization group analysis in $d=2+\epsilon$ dimensions, we calculate the localization length $\xi(B)$ in the presence of a magnetic field $B$. For the quasi 1D case the results are consistent with a universal increase of $\xi(B)$ by a numerical factor when the magnetic field is in the range \ell\ll{\ell_{\!{_H}}}\alt\xi(0), $\ell$ is the mean free path, ${\ell_{\!{_H}}}$ is the magnetic length $\sqrt{\hbar c/eB}$. However, for $d\ge 2$ where the magnetic field does cause delocalization there is no universal relation between $\xi(B)$ and $\xi(0)$. The effect of spin-orbit interaction is briefly considered as well.Comment: 4 pages, revtex, no figures; to be published in Europhysics Letter

Classical and quantum regimes of the superfluid turbulence

We argue that turbulence in superfluids is governed by two dimensionless parameters. One of them is the intrinsic parameter q which characterizes the friction forces acting on a vortex moving with respect to the heat bath, with 1/q playing the same role as the Reynolds number Re=UR/\nu in classical hydrodynamics. It marks the transition between the "laminar" and turbulent regimes of vortex dynamics. The developed turbulence described by Kolmogorov cascade occurs when Re >> 1 in classical hydrodynamics, and q << 1 in the superfluid hydrodynamics. Another parameter of the superfluid turbulence is the superfluid Reynolds number Re_s=UR/\kappa, which contains the circulation quantum \kappa characterizing quantized vorticity in superfluids. This parameter may regulate the crossover or transition between two classes of superfluid turbulence: (i) the classical regime of Kolmogorov cascade where vortices are locally polarized and the quantization of vorticity is not important; and (ii) the quantum Vinen turbulence whose properties are determined by the quantization of vorticity. The phase diagram of the dynamical vortex states is suggested.Comment: 12 pages, 1 figure, version accepted in JETP Letter

Weak levitation of 2D delocalized states in a magnetic field.

The deviation of the energy position of a delocalized state from the center of Landau level is studied in the framework of the Chalker-Coddington model. It is demonstrated that introducing a weak Landau level mixing results in a shift of the delocalized state up in energy. The mechanism of a levitation is a neighboring - Landau level - assisted resonant tunneling which ``shunts'' the saddle-points. The magnitude of levitation is shown to be independent of the Landau level number.Comment: Latex file (12 pages) + 3 Postscript figures

Subgap states in dirty superconductors and their effect on dephasing in Josephson qubits

We present a theory of the subgap tails of the density of states in a diffusive superconductor containing magnetic impurities. We show that the subgap tails have two contributions: one arising from mesoscopic gap fluctuations, previously discussed by Lamacraft and Simons, and the other associated to the long-wave fluctuations of the concentration of magnetic impurities. We study the latter both in small superconducting grains and in bulk systems [$d=1,2,3$], and establish the dimensionless parameter that controls which of the two contributions dominates the subgap tails. We observe that these contributions are related to each other by dimensional reduction. We apply the theory to estimate the effects of a weak concentration of magnetic impurities [$\approx 1 {\rm p.p.m}$] on the phase coherence of Josephson qubits. We find that at these typical concentrations, magnetic impurities are relevant for the dephasing in large qubits, designed around a $10 {\rm \mu m}$ scale, where they limit the quality factor to be $Q<10^4-10^5$.Comment: 13 pages, 1 figur

Dynamical properties of the Landau-Ginzburg model with long-range correlated quenched impurities

We investigate the critical dynamics of the time-dependent Landau-Ginzburg model with non conserved n-component order parameter (Model A) in the presence of long-range correlated quenched impurities. We use a special kind of long-range correlations, previously introduced by Weinrib and Halperin. Using a double expansion in \epsilon and \delta we calculate the critical exponent z up to second order on the small parameters. We show that the quenched impurities of this kind affect the critical dynamics already in first order of \epsilon and \delta, leading to a relevant correction for the mean field value of the exponent zComment: 7 pages, REVTEX, to be published in Phys. Rev.