We consider the spin quantum Hall transition which may occur in disordered superconductors with unbroken SU(2) spin-rotation symmetry but broken time-reversal symmetry. Using supersymmetry, we map a model for this transition onto the two-dimensional percolation problem. The anisotropic limit is an sl(2|1) supersymmetric spin chain. The mapping gives exact values for critical exponents associated with disorder-averages of several observables in good agreement with recent numerical results.PACS numbers: 73.40. Hm, 73.20.Fz, 72.15.Rn Noninteracting electrons with disorder, and the ensuing metal-insulator transitions, have been studied for several decades, and are usually divided into just three classes by symmetry considerations. Recently, the ideas have been extended to quasiparticles in disordered superconductors, for which the particle number is not conserved at the mean field level. Several more symmetry classes have been found [1]. One of these, denoted class C in Ref. [1], is of particular interest [2][3][4][5]. This is the case in which time-reversal symmetry is broken but global SU(2) spin-rotation symmetry is not, and spin transport can be studied. In two dimensions (2D) it can occur in d-wave superconductors. Within class C, a delocalization transition is possible in which the quantized Hall conductivity for spin changes by two units, resembling the usual quantum Hall (QH) transition but in a different universality class. When a Zeeman term is introduced which breaks the SU(2) symmetry down to U(1), the transition splits into two that are each in the usual QH universality class.In this paper we present exact results for a recent model [4,5] for the spin QH transition, in class C, in a system of noninteracting quasiparticles in 2D. We use a supersymmetry (SUSY) representation of such models, considered previously [6], to obtain a mapping onto the 2D classical bond percolation transition, from which we obtain three independent critical exponents, and universal ratios, exactly. An anisotropic version of the model is also mapped onto an antiferromagnetic sl(2|1) SUSY [7] quantum spin chain. The results are in very good agreement with recent numerical simulations [4,5].We study the spin QH transition in an alternative description that is obtained from the superconductor after a particle-hole transformation on the down-spin particles [2], which interchanges the roles of particle number and z-component of spin, and so particle number is conserved rather than spin. This makes it possible to use a single-particle description, at the cost of obscuring the SU(2) symmetry. The single-particle energy (E) spectrum has a particle-hole symmetry [1] under which E → −E, so when states are filled up to E = 0, the The model [4,5] is a network (generalizing Ref.[8]), in which a particle of either spin and with E = 0, represented by a doublet of complex fluxes, can propagate in one direction along each link (Fig. 1). The propagation on each link is described by a random SU(2) scattering matrix (the black dot), with a unifor...

We present Fokker-Planck equations that describe transport of heat and spin in dirty unconventional superconducting quantum wires. Four symmetry classes are distinguished, depending on the presence or absence of time-reversal and spin-rotation invariance. In the absence of spin-rotation symmetry, heat transport is anomalous in that the mean conductance decays like 1/sqrt[L] instead of exponentially fast for large enough length L of the wire. The Fokker-Planck equations in the presence of time-reversal symmetry are solved exactly and the mean conductance for quasiparticle transport is calculated for the crossover from the diffusive to the localized regime.

Localization and delocalization of non-interacting quasiparticle states in a superconducting wire are reconsidered, for the cases in which spin-rotation symmetry is absent, and time-reversal symmetry is either broken or unbroken; these are referred to as symmetry classes BD and DIII, respectively. We show that, if a continuum limit is taken to obtain a Fokker-Planck (FP) equation for the transfer matrix, as in some previous work, then when there are more than two scattering channels, all terms that break a certain symmetry are lost. It was already known that the resulting FP equation exhibits critical behavior. The additional symmetry is not required by the definition of the symmetry classes; terms that break it arise from non-Gaussian probability distributions, and may be kept in a generalized FP equation. We show that they lead to localization in a long wire. When the wire has more than two scattering channels, these terms are irrelevant at the short distance (diffusive or ballistic) fixed point, but as they are relevant at the long-distance critical fixed point, they are termed dangerously irrelevant. We confirm the results in a supersymmetry approach for class BD, where the additional terms correspond to jumps between the two components of the sigma model target space. We consider the effect of random π fluxes, which prevent the system localizing. We show that in one dimension the transitions in these two symmetry classes, and also those in the three chiral symmetry classes, all lie in the same universality class.

We develop a classification of composite operators without gradients at Anderson-transition critical points in disordered systems. These operators represent correlation functions of the local density of states (or of wave-function amplitudes). Our classification is motivated by the Iwasawa decomposition for the field of the pertinent supersymmetric σ-model: the scaling operators are represented by "plane waves" in terms of the corresponding radial coordinates. We also present an alternative construction of scaling operators by using the notion of highest-weight vector. We further argue that a certain Weyl-group invariance associated with the σ-model manifold leads to numerous exact symmetry relations between the scaling dimensions of the composite operators. These symmetry relations generalize those derived earlier for the multifractal spectrum of the leading operators.

We consider critical curves -conformally invariant curves that appear at critical points of two-dimensional statistical mechanical systems. We show how to describe these curves in terms of the Coulomb gas formalism of conformal field theory (CFT). We also provide links between this description and the stochastic (Schramm-) Loewner evolution (SLE). The connection appears in the long-time limit of stochastic evolution of various SLE observables related to CFT primary fields. We show how the multifractal spectrum of harmonic measure and other fractal characteristics of critical curves can be obtained.