Network Models in Class C on Arbitrary Graphs

We consider network models of quantum localisation in which a particle with a two-component wave function propagates through the nodes and along the edges of an arbitrary directed graph, subject to a random SU(2) rotation on each edge it traverses. The propagation through each node is specified by an arbitrary but fixed S-matrix. Such networks model localisation problems in class C of the classification of Altland and Zirnbauer, and, on suitable graphs, they model the spin quantum Hall transition. We extend the analyses of Gruzberg, Ludwig and Read and of Beamond, Cardy and Chalker to show that, on an arbitrary graph, the mean density of states and the mean conductance may be calculated in terms of observables of a classical history-dependent random walk on the same graph. The transition weights for this process are explicitly related to the elements of the S-matrices. They are correctly normalised but, on graphs with nodes of degree greater than 4, not necessarily non-negative (and therefore interpretable as probabilities) unless a sufficient number of them happen to vanish. Our methods use a supersymmetric path integral formulation of the problem which is completely finite and rigorous.Comment: 17 pages, 3 figure

Critical fixed points in class D superconductors

We study in detail a critical line on the phase diagram of the Cho-Fisher network model separating three different phases: metallic and two distinct localized phases with different quantized thermal Hall conductances. This system describes non-interacting quasiparticles in disordered superconductors that have neither time-reversal nor spin-rotational invariance. We find that in addition to a tricritical fixed point $W_T$ on that critical line there exist an additional repulsive fixed point $W_N$ (where the vortex disorder concentration $W_N<W_T$), which splits RG flow into opposite directions: toward a clean Ising model at W=0 and toward $W_T$.Comment: 3 pages, one figur

Universal critical exponent in class D superconductors

We study a physical system consisting of non-interacting quasiparticles in disordered superconductors that have neither time-reversal nor spin-rotation invariance. This system belongs to class D within the recent classification scheme of random matrix ensembles (RME) and its phase diagram contains three different phases: metallic and two distinct localized phases with different quantized thermal Hall conductances. We find that critical exponents describing different transitions (insulator-to-insulator and insulator-to-metal) are identical within the error of numerical calculations and also find that critical disorder of the insulator-to-metal transition is energy independent.Comment: 3.5 pages 4 figure

Decoherence induced by magnetic impurities in quantum Hall system

Scattering by magnetic impurities is known to destroy coherence of electron motion in metals and semiconductors. We investigate the decoherence introduced in a single act of electron scattering by a magnetic impurity in a quantum Hall system. To this end we introduce a fictitious nonunitary scattering matrix $\mathcal{S}$ for electrons that reproduces the exactly calculated scattering probabilities. The strength of decoherence is identified by the deviation of eigenvalues of the product $\mathcal{S}\mathcal{S}^{\dagger}$ from unity. Using the fictitious scattering matrix, we estimate the width of the metallic region at the quantum Hall effect inter-plateau transition and its dependence on the exchange coupling strength and the degree of polarization of magnetic impurities.Comment: 13 pages, 4 figure

Hyperfine interaction induced critical exponents in the quantum Hall effect

We study localization-delocalization transition in quantum Hall systems with a random field of nuclear spins acting on two-dimensional (2d) electron spins via hyperfine contact (Fermi) interaction. We use Chalker-Coddington network model, which corresponds to the projection onto the lowest Landau level. The inhomogeneous nuclear polarization acts on the electrons as an additional confining potential, and, therefore, introduces additional parameter $p$ (the probability to find a polarized nucleus in the vicinity of a saddle point of random potential) responsible for the change from quantum to classical behavior. In this manner we obtain two critical exponents corresponding to quantum and classical percolation. We also study how the 2d extended state develops into the one-dimensional (1d) critical state.Comment: 9 pages, 3 figure

Quantum Hall effects in layered disordered superconductors

Layered singlet paired superconductors with disorder and broken time reversal symmetry are studied. The phase diagram demonstrates charge-spin separation in transport. In terms of the average intergrain transmission and the interlayer tunnelling we find quantum Hall phases with spin Hall coefficients of 0 and 2 separated by a spin metal phase. We identify a spin metal-insulator localization exponent as well as a spin conductivity exponent of ~0.9. In presence of a Zeeman term an additional phase with spin Hall coefficient of 1 appears.Comment: 4 pages, 4 figure