The Loewner equation: maps and shapes

In the last few years, new insights have permitted unexpected progress in the study of fractal shapes in two dimensions. A new approach, called Schramm-Loewner evolution, or SLE, has arisen through analytic function theory and probability theory, and given a new way of calculating fractal shapes in critical phenomena, the theory of random walks, and of percolation. We present a non-technical discussion of this development aimed to attract the attention of condensed matter community to this fascinating subject

Finite Size Effects and Irrelevant Corrections to Scaling near the Integer Quantum Hall Transition

We present a numerical finite size scaling study of the localization length in long cylinders near the integer quantum Hall transition (IQHT) employing the Chalker-Coddington network model. Corrections to scaling that decay slowly with increasing system size make this analysis a very challenging numerical problem. In this work we develop a novel method of stability analysis that allows for a better estimate of error bars. Applying the new method we find consistent results when keeping second (or higher) order terms of the leading irrelevant scaling field. The knowledge of the associated (negative) irrelevant exponent $y$ is crucial for a precise determination of other critical exponents, including multifractal spectra of wave functions. We estimate $|y| > 0.4$, which is considerably larger than most recently reported values. Within this approach we obtain the localization length exponent $2.62 \pm 0.06$ confirming recent results. Our stability analysis has broad applicability to other observables at IQHT, as well as other critical points where corrections to scaling are present.Comment: 6 pages and 3 figures, plus supplemental material

Wave function correlations and the AC conductivity of disordered wires beyond the Mott-Berezinskii law

In one-dimensional disordered wires electronic states are localized at any energy. Correlations of the states at close positive energies and the AC conductivity $\sigma(\omega)$ in the limit of small frequency are described by the Mott-Berezinskii theory. We revisit the instanton approach to the statistics of wave functions and AC transport valid in the tails of the spectrum (large negative energies). Applying our recent results on functional determinants, we calculate exactly the integral over gaussian fluctuations around the exact two-instanton saddle point. We derive correlators of wave functions at different energies beyond the leading order in the energy difference. This allows us to calculate corrections to the Mott-Berezinskii law (the leading small frequency asymptotic behavior of $\sigma(\omega)$) which approximate the exact result in a broad range of $\omega$. We compare our results with the ones obtained for positive energies.Comment: 7 pages, 3 figure

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

Conductance and its universal fluctuations in the directed network model at the crossover to the quasi-one-dimensional regime

The directed network model describing chiral edge states on the surface of a cylindrical 3D quantum Hall system is known to map to a one-dimensional quantum ferromagnetic spin chain. Using the spin wave expansion for this chain, we determine the universal functions for the crossovers between the 2D chiral metallic and 1D metallic regimes in the mean and variance of the conductance along the cylinder, to first nontrivial order.Comment: 10 pages, REVTeX, uses epsf, 2 .eps figures included. Newly written Introduction and small changes to other section

Conformal invariance and multifractality at Anderson transitions in arbitrary dimensions

Electronic wave functions at Anderson transitions exhibit multifractal scaling characterized by a continuum of generalized multifractal exponents $\Delta_\gamma$ with vector indices $\gamma = (q_1,\ldots,q_n)$. In a field theory description of the transitions, there are corresponding multifractal operators $\mathcal{O}_\gamma$ with scaling dimensions $\Delta_\gamma$. Assuming conformal invariance and using the conformal bootstrap framework, we derive a constraint that implies that the generalized multifractal spectrum $\Delta_\gamma$ must be quadratic in all $q_i$ in any dimension $d > 2$. As several numerical studies have shown deviations from parabolicity, we argue that conformal invariance is likely absent at Anderson transitions in dimensions $d > 2$