Anisotropic generalization of Stinchcombe's solution for conductivity of random resistor network on a Bethe lattice

Our study is based on the work of Stinchcombe [1974 \emph{J. Phys. C} \textbf{7} 179] and is devoted to the calculations of average conductivity of random resistor networks placed on an anisotropic Bethe lattice. The structure of the Bethe lattice is assumed to represent the normal directions of the regular lattice. We calculate the anisotropic conductivity as an expansion in powers of inverse coordination number of the Bethe lattice. The expansion terms retained deliver an accurate approximation of the conductivity at resistor concentrations above the percolation threshold. We make a comparison of our analytical results with those of Bernasconi [1974 \emph{Phys. Rev. B} \textbf{9} 4575] for the regular lattice.Comment: 14 pages, 2 figure

Flux lattice melting and depinning in the weakly frustrated 2D XY model

Monte Carlo simulations of the frustrated 2D XY model were carried out at small commensurate values of the frustration $f$. For $f=1/30$ a single transition was observed at which phase coherence (finite helicity modulus) and vortex lattice orientational order vanish together. For $f=1/56$ a new phase in which phase coherence is absent but orientational order persists was observed. Where comparison is possible, the results are in detailed agreement with the behavior of the lattice Coulomb gas model of vortices. It is argued that the helicity modulus of the frustrated 2D XY model vanishes for any finite temperature in the limit of weak frustration $f$.Comment: 4 pages, RevTeX, 3 figures in separate uuencoded file The manuscript will appear in Phys. Rev.

The grand canonical ABC model: a reflection asymmetric mean field Potts model

We investigate the phase diagram of a three-component system of particles on a one-dimensional filled lattice, or equivalently of a one-dimensional three-state Potts model, with reflection asymmetric mean field interactions. The three types of particles are designated as $A$, $B$, and $C$. The system is described by a grand canonical ensemble with temperature $T$ and chemical potentials $T\lambda_A$, $T\lambda_B$, and $T\lambda_C$. We find that for $\lambda_A=\lambda_B=\lambda_C$ the system undergoes a phase transition from a uniform density to a continuum of phases at a critical temperature $\hat T_c=(2\pi/\sqrt3)^{-1}$. For other values of the chemical potentials the system has a unique equilibrium state. As is the case for the canonical ensemble for this $ABC$ model, the grand canonical ensemble is the stationary measure satisfying detailed balance for a natural dynamics. We note that $\hat T_c=3T_c$, where $T_c$ is the critical temperature for a similar transition in the canonical ensemble at fixed equal densities $r_A=r_B=r_C=1/3$.Comment: 24 pages, 3 figure

The Critical Behaviour of Potts models with symmetry breaking fields

The $Q$-state Potts model in two dimensions in the presence of external magnetic fields is studied. For general $Q\geq3$ special choices of these magnetic fields produce effective models with smaller $Z(Q')$ symmetry $(Q'< Q)$. The phase diagram of these models and their critical behaviour are explored by conventional finite-size scaling and conformal invariance. The possibility of multicritical behavior, for finite values of the symmetry breaking fields, in the cases where $Q>4$ is also analysed. Our results indicate that for effective models with $Z(Q')$ symmetry $(Q'\leq4)$ the multicritical point occurs at zero field. This last result is also corroborated by Monte Carlo simulations.Comment: 15 pages (standart LaTex), 2 figure (PostScript) available by request to [email protected]

A white-light trap for Bose-Einstein condensates

We propose a novel method for trapping Bose-condensed atoms using a white-light interference fringe. Confinement frequencies of tens of kHz can be achieved in conjunction with trap depths of only a few micro-K. We estimate that lifetimes on the order of 10 s can be achieved for small numbers of atoms. The tight confinement and shallow depth permit tunneling processes to be used for studying interaction effects and for applications in quantum information.Comment: 10 pages with 3 figure

Breaking a one-dimensional chain: fracture in 1 + 1 dimensions

The breaking rate of an atomic chain stretched at zero temperature by a constant force can be calculated in a quasiclassical approximation by finding the localized solutions ("bounces") of the equations of classical dynamics in imaginary time. We show that this theory is related to the critical cracks of stressed solids, because the world lines of the atoms in the chain form a two-dimensional crystal, and the bounce is a crack configuration in (unstable) mechanical equilibrium. Thus the tunneling time, Action, and breaking rate in the limit of small forces are determined by the classical results of Griffith. For the limit of large forces we give an exact bounce solution that describes the quantum fracture and classical crack close to the limit of mechanical stability. This limit can be viewed as a critical phenomenon for which we establish a Levanyuk-Ginzburg criterion of weakness of fluctuations, and propose a scaling argument for the critical regime. The post-tunneling dynamics is understood by the analytic continuation of the bounce solutions to real time.Comment: 15 pages, 5 figure

The shape of invasion perclation clusters in random and correlated media

The shape of two-dimensional invasion percolation clusters are studied numerically for both non-trapping (NTIP) and trapping (TIP) invasion percolation processes. Two different anisotropy quantifiers, the anisotropy parameter and the asphericity are used for probing the degree of anisotropy of clusters. We observe that in spite of the difference in scaling properties of NTIP and TIP, there is no difference in the values of anisotropy quantifiers of these processes. Furthermore, we find that in completely random media, the invasion percolation clusters are on average slightly less isotropic than standard percolation clusters. Introducing isotropic long-range correlations into the media reduces the isotropy of the invasion percolation clusters. The effect is more pronounced for the case of persisting long-range correlations. The implication of boundary conditions on the shape of clusters is another subject of interest. Compared to the case of free boundary conditions, IP clusters of conventional rectangular geometry turn out to be more isotropic. Moreover, we see that in conventional rectangular geometry the NTIP clusters are more isotropic than TIP clusters

Transport on percolation clusters with power-law distributed bond strengths: when do blobs matter?

The simplest transport problem, namely maxflow, is investigated on critical percolation clusters in two and three dimensions, using a combination of extremal statistics arguments and exact numerical computations, for power-law distributed bond strengths of the type $P(\sigma) \sim \sigma^{-\alpha}$. Assuming that only cutting bonds determine the flow, the maxflow critical exponent \ve is found to be \ve(\alpha)=(d-1) \nu + 1/(1-\alpha). This prediction is confirmed with excellent accuracy using large-scale numerical simulation in two and three dimensions. However, in the region of anomalous bond capacity distributions ($0\leq \alpha \leq 1$) we demonstrate that, due to cluster-structure fluctuations, it is not the cutting bonds but the blobs that set the transport properties of the backbone. This ``blob-dominance'' avoids a cross-over to a regime where structural details, the distribution of the number of red or cutting bonds, would set the scaling. The restored scaling exponents however still follow the simplistic red bond estimate. This is argued to be due to the existence of a hierarchy of so-called minimum cut-configurations, for which cutting bonds form the lowest level, and whose transport properties scale all in the same way. We point out the relevance of our findings to other scalar transport problems (i.e. conductivity).Comment: 9 pages + Postscript figures. Revtex4+psfig. Submitted to PR

Glassiness Vs. Order in Densely Frustrated Josephson Arrays

We carry out extensive Monte Carlo simulations on the Coulomb gas dual to the uniformly frustrated two dimensional XY model, for a sequence of frustrations f converging to the irraltional (3-sqrt 5)/2. We find in these systems a sharp first order equilibrium phase transition to an ordered vortex structure at a T_c which varies only slightly with f. This ordered vortex structure remains in general phase incoherent until a lower pinning transition T_p(f) that varies with f. We argue that the glassy behaviors reported for this model in earlier simulations are dynamic effects.Comment: 4 pages, 4 eps figure