Abstract: We generalize universal relations between the multifractal exponent alpha(0) for the scaling of the typical wave-function magnitude at a (Anderson) localization-delocalization transition in two dimensions and the corresponding critical finite-size-scaling (FSS) amplitude Lambda(c) of the typical localization length in quasi-one-dimensional (Q1D) geometry: (i) when open boundary conditions are imposed in the transverse direction of Q1D samples (strip geometry), we show that the corresponding critical FSS amplitude Lambda(o)(c) is universally related to the boundary multifractal exponent alpha(s)(0) for the typical wave-function amplitude along a straight boundary (surface). (ii) We further propose a generalization of these universal relations to those symmetry classes whose density of states vanishes at the transition. (iii) We verify our generalized relations [Eqs. (6) and (7)] numerically for the following four types of two-dimensional Anderson transitions: (a) the metal-to-(ordinary insulator) transition in the spin-orbit (symplectic) symmetry class, (b) the metal-to-(Z(2) topological insulator) transition which is also in the spin-orbit (symplectic) class, (c) the integer quantum-Hall plateau transition, and (d) the spin quantum-Hall plateau transition.
Abstract: We use a superspin Hamiltonian defined on an infinite-dimensional Fock space with positive definite scalar product to study localization and delocalization of noninteracting spinless quasiparticles in quasione-dimensional quantum wires perturbed by weak quenched disorder. Past works using this approach have considered a single chain. Here, we extend the formalism to treat a quasi-one-dimensional system: a quantum wire with an arbitrary number of channels coupled by random hopping amplitudes. The computations are carried out explicitly for the case of a chiral quasi-one-dimensional wire with broken time-reversal symmetry (chiral-unitary symmetry class). By treating the space direction along the chains as imaginary time, the effects of the disorder are encoded in the time evolution induced by a single site superspin (non-Hermitian) Hamiltonian. We obtain the density of states near the band center of an infinitely long quantum wire. Our results agree with those based on the Dorokhov-Mello-Pereyra-Kumar equation for the chiral-unitary symmetry class. (C) 2009 Elsevier B.V. All rights reserved.
Abstract: The von Neumann entanglement entropy is a useful measure to characterize a quantum phase transition. We investigate the nonanalyticity of this entropy at disorder-dominated quantum phase transitions in noninteracting electronic systems. At these critical points, the von Neumann entropy is determined by the single particle wave function intensity, which exhibits complex scale invariant fluctuations. We find that the concept of multifractality is naturally suited for studying von Neumann entropy of the critical wave functions. Our numerical simulations of the three dimensional Anderson localization transition and the integer quantum Hall plateau transition show that the entanglement at these transitions is well described using multifractal analysis.
Abstract: We investigate boundary multifractality of critical wave functions at the Anderson metal-insulator transition in two-dimensional disordered non-interacting electron systems with spin-orbit scattering. We show numerically that multifractal exponents at a corner with an opening angle theta = 3 pi/2 arc directly related to those near a straight boundary in the way dictated by conformal symmetry. This result extends our previous numerical results on corner multifractality obtained for theta < pi to theta > pi, and gives further supporting evidence for conformal invariance at criticality. We also propose a refinement of the validity of the symmetry relation of A.D. Mirlin et al. [Phys. Rev. Lett. 97 (2006) 046803] for corners. (c) 2007 Elsevier B.V. All rights reserved.
Abstract: Standard Schramm-Loewner evolution (SLE) is driven by a continuous Brownian motion which then produces a trace, a continuous fractal curve connecting the singular points of the motion. If jumps are added to the driving function, the trace branches. In a recent publication (Rushkin et al 2006 J. Stat. Mech. P01001 [cond-mat/ 0509187]) we introduced a generalized SLE driven by a superposition of a Brownian motion and a fractal set of jumps ( technically a stable Levy process). We then discussed the small scale properties of the resulting Levy-SLE growth process. Here we discuss the same model, but focus on the global scaling behavior which ensues as time goes to infinity. This limiting behavior is independent of the Brownian forcing and depends upon only a single parameter, alpha, which defines the shape of the stable Levy distribution. We learn about this behavior by studying a Fokker-Planck equation which gives the probability distribution for end points of the trace as a function of time. As in the short time case previously studied, we observe that the properties of this growth process change qualitatively and singularly at alpha = 1. We show both analytically and numerically that the growth continues indefinitely in the vertical direction for a > 1, goes as log t for alpha = 1, and saturates for alpha < 1. The probability density has two different scales corresponding to directions along and perpendicular to the boundary. In the former case, the characteristic scale is X( t) similar to t(1/alpha). In the latter case the scale is Y(t) similar to A + Bt(1-1/alpha) for alpha not equal 1, and Y ( t) similar to ln t for alpha = 1. Scaling functions for the probability density are given for various limiting cases.
Abstract: Fractal geometry of random curves appearing in the scaling limit of critical two-dimensional statistical systems is characterized by their harmonic measure and winding angle. The former is the measure of the jaggedness of the curves while the latter quantifies their tendency to form logarithmic spirals. We show how these characteristics are related to local operators of conformal field theory and how they can be computed using conformal invariance of critical systems with central charge c <= 1.
Abstract: Multifractal scaling of critical wave functions at a disorder-driven (Anderson) localization transition is modified near boundaries of a sample. Here this effect is studied for the example of the spin quantum Hall plateau transition using the supersymmetry technique for disorder averaging. Upon mapping the spin quantum Hall transition to the classical percolation problem with reflecting boundaries, a number of multifractal exponents governing wave-function scaling near a boundary are obtained exactly. Moreover, additional exact boundary scaling exponents of the localization problem are extracted, and the problem is analyzed in other geometries.
Abstract: We study multifractal spectra of critical wave functions at the integer quantum Hall plateau transition using the Chalker-Coddington network model. Our numerical results provide important new constraints which any critical theory for the transition will have to satisfy. We find a nonparabolic multifractal spectrum and determine the ratio of boundary to bulk multifractal exponents. Our results rule out an exactly parabolic spectrum that has been the centerpiece in a number of proposals for critical field theories of the transition. In addition, we demonstrate analytically exact parabolicity of the related boundary spectra in the two-dimensional chiral orthogonal "Gade-Wegner" symmetry class.
Abstract: 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.
Abstract: Boundary multifractality of electronic wave functions is studied analytically and numerically for the power-law random banded matrix (PRBM) model, describing a critical one-dimensional system with long-range hopping. The peculiarity of the Anderson localization transition in this model is the existence of a line of fixed points describing the critical system in the bulk. We demonstrate that the boundary critical theory of the PRBM model is not uniquely determined by the bulk properties. Instead, the boundary criticality is controlled by an additional parameter characterizing the hopping amplitudes of particles reflected by the boundary.
Abstract: We study the multifractality (MF) of critical wave functions at boundaries and corners at the metal-insulator transition (MIT) for noninteracting electrons in the two-dimensional (2D) spin-orbit (symplectic) universality class. We find that the MF exponents near a boundary are different from those in the bulk. The exponents at a corner are found to be directly related to those at a straight boundary through a relation arising from conformal invariance. This provides direct numerical evidence for conformal invariance at the 2D spin-orbit MIT. The presence of boundaries modifies the MF of the whole sample even in the thermodynamic limit.
Abstract: Standard stochastic Loewner evolution (SLE) is driven by a continuous Brownian motion, which then produces a continuous fractal trace. If jumps are added to the driving function, the trace branches. We consider a generalized SLE driven by a superposition of a Brownian motion and a stable Levy process. The situation is defined by the usual SLE parameter, kappa, as well as a, which defines the shape of the stable Levy distribution. The resulting behaviour is characterized by two descriptors: p, the probability that the trace self-intersects, and similar to (p) over tilde, the probability that it will approach arbitrarily close to doing so. Using Dynkin's formula, these descriptors are shown to change qualitatively and singularly at critical values of. and a. It is reasonable to call such changes 'phase transitions'. These transitions occur as kappa passes through four (a well-known result) and as a passes through one (a new result). Numerical simulations are then used to explore the associated touching and near-touching events.
Abstract: We develop the concept of surface multifractality for localization-delocalization (LD) transitions in disordered electronic systems. We point out that the critical behavior of various observables related to wave functions near a boundary at a LD transition is different from that in the bulk. We illustrate this point with a calculation of boundary critical and multifractal behavior at the 2D spin quantum Hall transition and in a 2D metal at scales below the localization length.
Abstract: Conformally invariant curves that appear at critical points in two-dimensional statistical mechanics systems and their fractal geometry have received a lot of attention in recent years. On the one hand, Schramm (2000 Israel J. Math. 118 221 (Preprint math.PR/9904022)) has invented a new rigorous as well as practical calculational approach to critical curves, based on a beautiful unification of conformal maps and stochastic processes, and by now known as Schramm-Loewner evolution (SLE). On the other hand, Duplantier ( 2000 Phys. Rev. Lett. 84 1363; Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot: Part 2 (Proc. Symp. Pure Math. vol 72) ( Providence, RI: American Mathematical Society) p 365 ( Preprint math-ph/0303034)) has applied boundary quantum gravity methods to calculate exact multifractal exponents associated with critical curves. In the first part of this paper, I provide a pedagogical introduction to SLE. I present mathematical facts from the theory of conformal maps and stochastic processes related to SLE. Then I review basic properties of SLE and provide practical derivation of various interesting quantities related to critical curves, including fractal dimensions and crossing probabilities. The second part of the paper is devoted to a way of describing critical curves using boundary conformal field theory (CFT) in the so-called Coulomb gas formalism. This description provides an alternative ( to quantum gravity) way of obtaining the multifractal spectrum of critical curves using only traditional methods of CFT based on free bosonic fields.
Abstract: Modes of light that contain topological defects such as screw dislocations can be focused into optical traps with interesting and useful properties. The way in which the intensity distribution within helical modes of light varies with topological charge is discussed, and new scaling predictions for their radial profiles that are consistent with experimental observations are introduced. (C) 2005 Optical Society of America.
Abstract: Localization and delocalization of noninteracting 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 pi 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.
Abstract: The stochastic Loewner evolution is a recent tool in the study of two-dimensional critical systems. We extend this approach to the case of critical systems with continuous symmetries, such as SU(2) Wess-Zumino-Witten models, where domain walls carry an additional spin-1/2 degree of freedom.
Abstract: Fractal geometry of critical curves appearing in 2D critical systems is characterized by their harmonic measure. For systems described by conformal field theories with central charge c <= 1, scaling exponents of the harmonic measure have been computed by Duplantier [Phys. Rev. Lett. 84, 1363 ( 2000)] by relating the problem to boundary two-dimensional gravity. We present a simple argument connecting the harmonic measure of critical curves to operators obtained by fusion of primary fields and compute characteristics of the fractal geometry by means of regular methods of conformal field theory. The method is not limited to theories with c <= 1.
Abstract: An approach called Schramm-Loewner evolution (SLE) provides a new method for dealing with a wide variety of scale-invariant problems in two dimensions. This approach is based upon an older method called Loewner Evolution ( LE), which connects analytic and geometrical constructions in the complex plane. In this paper, the bases of LE and SLE are described and some simple applications are discussed in relatively non-technical form. A bibliography of the subject is presented.
Abstract: We analyze the behavior of the density of states in a singlet s-wave superconductor with weak magnetic impurities in the clean limit. By using the method of optimal fluctuation and treating the order parameter self-consistently we show that the density of states is finite everywhere in the superconducting gap, and that it varies as lnN(E) proportional to -\E-Delta(0)\((7-d)/4) near the mean field gap edge Delta(0) in a d-dimensional superconductor. In contrast to most studied cases the optimal fluctuation is strongly anisotropic.
Abstract: We analyze the behavior of the density of states in a singlet s-wave superconductor with weak magnetic impurities in the clean limit by using the method of optimal fluctuation. We show that the density of states varies as In N(E) proportional to - \E - Delta(0)\((7-d)/4) near the mean field gap edge Delta(0) in a d-dimensional superconductor. The optimal fluctuation in d > 1 is strongly anisotropic. We compare the density of states with that obtained in other recent approaches. (C) 2002 Elsevier Science B.V. All rights reserved.
Abstract: We consider a classical random-bond Ising model (RBIM) with binary distribution of +/-K bonds on the square lattice at finite temperature. In the phase diagram of this model there is the so-called Nishimori line which intersects the phase boundary at a multicritical point. It is known that the correlation functions obey many exact identities on this line. We use a supersymmetry method to treat the disorder. In this approach the transfer matrices nf the model on the Nishimori line have an enhanced supersymmetry osp(2n+1\2n), in contrast to the rest of the phase diagram, when the symmetry is osp(2n/2n) (where n is an arbitrary positive integer). An anisotropic limit of the model leads to a one-dimensional quantum Hamiltonian describing a chain of interacting superspins, which are irreducible representations of the osp(2n + 1\2n) superalgebra. By generalizing this superspin chain, we embed it into a wider class of models. These include other models that have been studied previously in one and two dimensions. We suggest that the multicritical behavior in two dimensions of a class of these generalized models (possibly not including the multicritical point in the RBIM itself) may he governed by a single fixed point, at which the supersymmetry is enhanced still further to osp(2n +2/2n). This suggestion is supported by a calculation of the renormalization-group flows for the corresponding nonlinear sigma models at weak coupling.
Abstract: 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/root 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.
Abstract: Sire consider the spin quantum Wall 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. [S0031-9007(99)09270-4].
Abstract: We consider the directed network (DN) of edge states on the surface of a cylinder of length L and circumference C. By mapping it to a ferromagnetic superspin chain and using a scaling analysis we show its equivalence to a one-dimensional supersymmetric nonlinear sigma model in the scaling limit-for any value of the ratio L/C, except for short systems where L is less than of order C-1/2. For the sigma model, the universal crossover functions for the conductance and its variance have been determined previously. We also show that the DN model can be mapped directly onto the random matrix (Fokker-Planck) approach to disordered quasi-one-dimensional wires, which implies that the entire distribution of the conductance is' the same as in the latter system for any value of L/C in the same scaling limit. The results of Chalker and Dohmen [Phys. Rev. Lett. 75, 4496 (1995)] are explained quantitatively.
Abstract: The directed network model describing chiral edge states on the surface of a cylindrical three-dimensional quantum Hall system is known to map to a one-dimensional (1D) quantum ferromagnetic spin chain. Using the spin-wave expansion for this chain, we determine the universal functions for the crossovers between the two-dimensional chiral metallic and 1D metallic regimes in the mean and variance of the conductance along the cylinder, to first nontrivial order.
Abstract: We solve the O(n, 1) nonlinear vector model on the Bethe lattice and show that it exhibits a transition from ordered to disordered state for 0 less than or equal to n < 1. If the replica limit n --> 0 is taken carefully, the model is shown to reduce to the corresponding supersymmetric model. The latter was introduced by Zirnbauer as a toy model for the Anderson localization transition. We argue thus that the non-compact replica models describe correctly the Anderson transition features. This should be contrasted to their failure in the case of the level correlation problem.
