Crossover from Isotropic to Directed Percolation

Directed percolation is one of the generic universality classes for dynamic processes. We study the crossover from isotropic to directed percolation by representing the combined problem as a random cluster model, with a parameter $r$ controlling the spontaneous birth of new forest fires. We obtain the exact crossover exponent $y_{DP}=y_T-1$ at $r=1$ using Coulomb gas methods in 2D. Isotropic percolation is stable, as is confirmed by numerical finite-size scaling results. For $D \geq 3$, the stability seems to change. An intuitive argument, however, suggests that directed percolation at $r=0$ is unstable and that the scaling properties of forest fires at intermediate values of $r$ are in the same universality class as isotropic percolation, not only in 2D, but in all dimensions.Comment: 4 pages, REVTeX, 4 epsf-emedded postscript figure

Directed Percolation with a Wall or Edge

We examine the effects of introducing a wall or edge into a directed percolation process. Scaling ansatzes are presented for the density and survival probability of a cluster in these geometries, and we make the connection to surface critical phenomena and field theory. The results of previous numerical work for a wall can thus be interpreted in terms of surface exponents satisfying scaling relations generalising those for ordinary directed percolation. New exponents for edge directed percolation are also introduced. They are calculated in mean-field theory and measured numerically in 2+1 dimensions.Comment: 14 pages, submitted to J. Phys.

Kondo Effect in a Luttinger Liquid: Exact Results from Conformal Field Theory

We report on exact results for the low-temperature thermodynamics of a spin-$\frac{1}{2}$ magnetic impurity coupled to a one-dimensional interacting electron system. By using boundary conformal field theory, we show that there are only two types of critical behaviors consistent with the symmetries of the problem: {\em either} a local Fermi liquid, {\em or} a theory with an anomalous response identical to that recently proposed by Furusaki and Nagaosa. Suppression of back scattering off the impurity leads to the same critical properties as for the two-channel Kondo effect.Comment: 9 pages, REVTeX, uses amsfonts, accepted for publication in Phys. Rev. Let

Surface Critical Behavior in Systems with Non-Equilibrium Phase Transitions

We study the surface critical behavior of branching-annihilating random walks with an even number of offspring (BARW) and directed percolation (DP) using a variety of theoretical techniques. Above the upper critical dimensions d_c, with d_c=4 (DP) and d_c=2 (BARW), we use mean field theory to analyze the surface phase diagrams using the standard classification into ordinary, special, surface, and extraordinary transitions. For the case of BARW, at or below the upper critical dimension, we use field theoretic methods to study the effects of fluctuations. As in the bulk, the field theory suffers from technical difficulties associated with the presence of a second critical dimension. However, we are still able to analyze the phase diagrams for BARW in d=1,2, which turn out to be very different from their mean field analog. Furthermore, for the case of BARW only (and not for DP), we find two independent surface beta_1 exponents in d=1, arising from two distinct definitions of the order parameter. Using an exact duality transformation on a lattice BARW model in d=1, we uncover a relationship between these two surface beta_1 exponents at the ordinary and special transitions. Many of our predictions are supported using Monte-Carlo simulations of two different models belonging to the BARW universality class.Comment: 19 pages, 12 figures, minor additions, 1 reference adde

Two-Dimensional Semi-Infinite Multicritical Ising Models

The order-parameter correlation functions of the Z 2 -invariant multicritical points in the unitary minimal series of conformal field theory are derived for a semi-infinite plane constrained to fixed or free boundary conditions. These yield the corresponding universal surface exponents which distinguish the behaviour between even- and odd-critical models. We also make explicit the interplay between duality and boundary conditions. PACS numbers 64.60.Kw, 68.35.Rh, 75.40.Cx Typeset using REVT E X Present address: Department of Physics, FM-15, University of Washington, Seattle, WA 98195, USA 1. INTRODUCTION The implications of conformal invariance of a statistical mechanical system at criticality are very powerful in two dimensions. All so-called minimal conformal field theories are completely classified, the scaling dimensions of their operators are determined and one knows in principle how to calculate all correlation functions [1,2]. From extending this work to semi-infinite system..

Criticality and Frustration: Classical and Quantum Spin Models in Two Dimensions

We investigate two-dimensional spin models and begin with an introduction to critical phenomena with emphasis on classical systems with boundaries and conformal invariance. Then we introduce the antiferromagnetic quantum Heisenberg model with frustrating interactions. <p> Conformal invariance of a statistical mechanical system at criticality has powerful implications in two dimensions. With the use of conformal field theory, we investigate finite-size and boundary effects for a class of N-critical models that describe the simplest type of multicritical behaviour. In Papers I and II, correlation functions for the order parameter and other operators of these multicritical Ising models are derived for restricted domains (semi-infinite plane, infinite strip, disc and rectangle) with free or fixed boundary conditions. These yield the corresponding universal surface exponents and allow for a detailed analysis of correlations lengths, structure factors and response functions. The interplay between duality and boundary conditions is also made explicit. In Paper III, we analyse structure factors at criticality and show how apparent power- law scaling at intermediate momenta - induced by finite size and boundaries - depends on the boundary conditions applied. The predictions are confirmed by using correlation functions calculated in Papers I and II. <p> Two-dimensional Heisenberg antiferromagnets have attracted much attention because of their relation to high-T_c superconductors. The second part of the thesis (Papers IV and V) treats spin-1/2 antiferromagnets defined on square- and honeycomb lattices with frustrating interactions. By using a Schwinger-boson mean-field theory, we calculate low-temperature quantum corrections to thermodynamic parameters, such as spin-wave velocity and transverse susceptibility. Their dependence on frustration is explicated and used to estimate the stability of the Néel state via a mapping to the nonlinear .sigma. model that describes the low-energy limit of the antiferromagnet

Interacting Electrons and Localized Spins: Exact Results from Conformal Field Theory

Introduction Much of traditional condensed matter physics falls under the Fermi liquid paradigm. The notion goes back to Landau and implies that a liquid of fermions (such as 3 He or conduction electrons in a metal) can be treated as a system of essentially free particles [1]. Certain conditions apply: symmetries must remain unbroken and the energies probed should be &quot;small.&quot; But with these provisos the only price to pay for removing the interactions is to keep track on how certain parameters (quasiparticle masses, lifetimes, etc.) renormalize. The Fermi-liquid picture has proven enormously successful and explains how we can get away with single-particle quantum mechanics when doing elementary solid state physics. Fortunately --- to keep us busy! --- experimentalists have found a number of lab systems that violate the standard Fermi liquid picture. (An outstanding example is the metallic phase of the high-T c superconductors.) The des

Magnetic Impurity in a Luttinger Liquid: A View from Conformal Field Theory

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