Crossover from Orthogonal to Unitary Symmetry for Ballistic Electron Transport in Chaotic Microstructures

We study the ensemble-averaged conductance as a function of applied magnetic field for ballistic electron transport across few-channel microstructures constructed in the shape of classically chaotic billiards. We analyse the results of recent experiments, which show suppression of weak localization due to magnetic field, in the framework of random-matrix theory. By analysing a random-matrix Hamiltonian for the billiard-lead system with the aid of Landauer's formula and Efetov's supersymmetry technique, we derive a universal expression for the weak-localization contribution to the mean conductance that depends only on the number of channels and the magnetic flux. We consequently gain a theoretical understanding of the continuous crossover from orthogonal symmetry to unitary symmetry arising from the violation of time-reversal invariance for generic chaotic systems.Comment: 49 pages, latex, 9 figures as tar-compressed uuencoded fil

A need-based theory of persuasion

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Conformal invariance and linear defects in the two-dimensional Ising model

Using conformal invariance, we show that the non-universal exponent eta_0 associated with the decay of correlations along a defect line of modified bonds in the square-lattice Ising model is related to the amplitude A_0=xi_n/n of the correlation length \xi_n(K_c) at the bulk critical coupling K_c, on a strip with width n, periodic boundary conditions and two equidistant defect lines along the strip, through A_0=(\pi\eta_0)^{-1}.Comment: Old paper, for archiving. 5 pages, 4 figures, IOP macro, eps

Critical phenomena on scale-free networks: logarithmic corrections and scaling functions

In this paper, we address the logarithmic corrections to the leading power laws that govern thermodynamic quantities as a second-order phase transition point is approached. For phase transitions of spin systems on d-dimensional lattices, such corrections appear at some marginal values of the order parameter or space dimension. We present new scaling relations for these exponents. We also consider a spin system on a scale-free network which exhibits logarithmic corrections due to the specific network properties. To this end, we analyze the phase behavior of a model with coupled order parameters on a scale-free network and extract leading and logarithmic correction-to-scaling exponents that determine its field- and temperature behavior. Although both non-trivial sets of exponents emerge from the correlations in the network structure rather than from the spin fluctuations they fulfil the respective thermodynamic scaling relations. For the scale-free networks the logarithmic corrections appear at marginal values of the node degree distribution exponent. In addition we calculate scaling functions, which also exhibit nontrivial dependence on intrinsic network properties.Comment: 15 pages, 4 figure

Characterization of the Local Density of States Fluctuations near the Integer Quantum Hall Transition in a Quantum Dot Array

We present a calculation for the second moment of the local density of states in a model of a two-dimensional quantum dot array near the quantum Hall transition. The quantum dot array model is a realistic adaptation of the lattice model for the quantum Hall transition in the two-dimensional electron gas in an external magnetic field proposed by Ludwig, Fisher, Shankar and Grinstein. We make use of a Dirac fermion representation for the Green functions in the presence of fluctuations for the quantum dot energy levels. A saddle-point approximation yields non-perturbative results for the first and second moments of the local density of states, showing interesting fluctuation behaviour near the quantum Hall transition. To our knowledge we discuss here one of the first analytic characterizations of chaotic behaviour for a two-dimensional mesoscopic structure. The connection with possible experimental investigations of the local density of states in the quantum dot array structures (by means of NMR Knight-shift or single-electron-tunneling techniques) and our work is also established.Comment: 11 LaTeX pages, 1 postscript figure, to appear in Phys.Rev.

Liouvillian Approach to the Integer Quantum Hall Effect Transition

We present a novel approach to the localization-delocalization transition in the integer quantum Hall effect. The Hamiltonian projected onto the lowest Landau level can be written in terms of the projected density operators alone. This and the closed set of commutation relations between the projected densities leads to simple equations for the time evolution of the density operators. These equations can be used to map the problem of calculating the disorder averaged and energetically unconstrained density-density correlation function to the problem of calculating the one-particle density of states of a dynamical system with a novel action. At the self-consistent mean-field level, this approach yields normal diffusion and a finite longitudinal conductivity. While we have not been able to go beyond the saddle point approximation analytically, we show numerically that the critical localization exponent can be extracted from the energetically integrated correlation function yielding $\nu=2.33 \pm 0.05$ in excellent agreement with previous finite-size scaling studies.Comment: 9 pages, submitted to PR

Fisher Renormalization for Logarithmic Corrections

For continuous phase transitions characterized by power-law divergences, Fisher renormalization prescribes how to obtain the critical exponents for a system under constraint from their ideal counterparts. In statistical mechanics, such ideal behaviour at phase transitions is frequently modified by multiplicative logarithmic corrections. Here, Fisher renormalization for the exponents of these logarithms is developed in a general manner. As for the leading exponents, Fisher renormalization at the logarithmic level is seen to be involutory and the renormalized exponents obey the same scaling relations as their ideal analogs. The scheme is tested in lattice animals and the Yang-Lee problem at their upper critical dimensions, where predictions for logarithmic corrections are made.Comment: 10 pages, no figures. Version 2 has added reference

Thouless energy and multifractality across the many-body localization transition

Thermal and many-body localized phases are separated by a dynamical phase transition of a new kind. We analyze the distribution of off-diagonal matrix elements of local operators across this transition in two different models of disordered spin chains. We show that the behavior of matrix elements can be used to characterize the breakdown of thermalization and to extract the many-body Thouless energy. We find that upon increasing the disorder strength the system enters a critical region around the many-body localization transition. The properties of the system in this region are: (i) the Thouless energy becomes smaller than the level spacing, (ii) the matrix elements show critical dependence on the energy difference, and (iii) the matrix elements, viewed as amplitudes of a fictitious wave function, exhibit strong multifractality. This critical region decreases with the system size, which we interpret as evidence for a diverging correlation length at the many-body localization transition. Our findings show that the correlation length becomes larger than the accessible system sizes in a broad range of disorder strength values and shed light on the critical behavior near the many-body localization transition

Bond-disordered Anderson model on a two dimensional square lattice - chiral symmetry and restoration of one-parameter scaling

Bond-disordered Anderson model in two dimensions on a square lattice is studied numerically near the band center by calculating density of states (DoS), multifractal properties of eigenstates and the localization length. DoS divergence at the band center is studied and compared with Gade's result [Nucl. Phys. B 398, 499 (1993)] and the powerlaw. Although Gade's form describes accurately DoS of finite size systems near the band-center, it fails to describe the calculated part of DoS of the infinite system, and a new expression is proposed. Study of the level spacing distributions reveals that the state closest to the band center and the next one have different level spacing distribution than the pairs of states away from the band center. Multifractal properties of finite systems furthermore show that scaling of eigenstates changes discontinuously near the band center. This unusual behavior suggests the existence of a new divergent length scale, whose existence is explained as the finite size manifestation of the band center critical point of the infinite system, and the critical exponent of the correlation length is calculated by a finite size scaling. Furthermore, study of scaling of Lyapunov exponents of transfer matrices of long stripes indicates that for a long stripe of any width there is an energy region around band center within which the Lyapunov exponents cannot be described by one-parameter scaling. This region, however, vanishes in the limit of the infinite square lattice when one-parameter scaling is restored, and the scaling exponent calculated, in agreement with the result of the finite size scaling analysis.Comment: 23 pages, 11 figures. RevTe

Diagrammatic analysis of the two-state quantum Hall system with chiral invariance

The quantum Hall system in the lowest Landau level with Zeeman term is studied by a two-state model, which has a chiral invariance. Using a diagrammatic analysis, we examine this two-state model with random impurity scattering, and find the exact value of the conductivity at the Zeeman energy $E = \Delta$. We further study the conductivity at the another extended state $E = E_1$ ($E_1 > \Delta$). We find that the values of the conductivities at $E = 0$ and $E = E_1$ do not depend upon the value of the Zeeman energy $\Delta$. We discuss also the case where the Zeeman energy $\Delta$ becomes a random field.Comment: 14P, Late