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Physical review. B, Condensed matter

Critical phenomena in systems with long-range-correlated quenched disorder

Abel Weinrib, B. I. Halperin

Published Jan 1, 1983• 437 citations
Publisher

Abstract

As a model for a phase transition in an inhomogeneous system, we consider a system where the local transition temperature varies in space, with a correlation function obeying a power law $\ensuremath{\sim}{x}^{\ensuremath{-}a}$ for large separations $x$. We extend the Harris criterion for this case, finding that for $a<d$ (where $d$ is the spatial dimension) the disorder is irrelevant if $a\ensuremath{\nu}\ensuremath{-}2>0$, while if $a>d$ we recover the usual Harris criterion: The disorder is irrelevant if $d\ensuremath{\nu}\ensuremath{-}2=\ensuremath{-}\ensuremath{\alpha}>0$. An $m$-vector system of this type is studied with the use of a renormalization-group expansion in $\ensuremath{\epsilon}=4\ensuremath{-}d$ and $\ensuremath{\delta}=4\ensuremath{-}a$. We find a new long-range-disorder fixed point in addition to the short-range-disorder and pure fixed points found previously. The crossover between fixed points is found to follow the extended Harris criterion. The new fixed point has complex eigenvalues, leading to oscillating corrections to scaling, and has a correlation-length exponent $\ensuremath{\nu}=\frac{2}{a}$. We argue that this new scaling relation is exact and applies more generally than just to the specific model. We show that the extended Harris criterion also applies to percolation with long-range-correlated site or bond-occupation probabilities, so that the scaling law should be obeyed by such systems. Results for the percolation properties of the triangular Ising model are in agreement with these predictions.

DOI
10.1103/physrevb.27.413

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Motivated by the Harris criterion, the absorbing phase transitions of the conserved lattice gas (CLG) model were studied on lattices with a quenched disorder, i.e., on infinite percolation networks, in two and three dimensions. The Harris criterion suggests that, in a magnetic system, the pure fixed point will be unstable if the specific-heat exponent is positive. For the CLG model, the specific-heat exponent α calculated by the hyperscaling relation α=2-dν, where ν is the spatial correlation length exponent in d dimensions, will be positive in two dimensions if the value of ν obtained earlier by Lübeck and Heger is employed. On the other hand, it will be close to 0 if the more recent value by Lee and Lee is used and it is positive in three dimensions with the available value of ν. Extensive numerical simulations showed that, when the concentration of disordered sites is less than the critical concentration, the critical exponents were similar to those on a regular lattice both in two and three dimensions. When the concentration of disordered sites becomes critical, the density of active particles showed nonuniversal power-law behavior for all particle densities considered in both dimensions. These results were in contrast to the results for the diluted contact process. The cause of such a nonuniversal behavior was addressed.

Physical Review E•Oct 18, 2011•8 citations
As a model for a phase transition in an inhomogeneous system, we consider a system where the local transition temperature varies in space, with a correlation function obeying a power law \ensuremath∼x\ensuremath−a\ensuremath{\sim}{x}^{\ensuremath{-}a}\ensuremath∼x\ensuremath−a for large separations xxx. We extend the Harris criterion for this case, finding that for a<da<da<d (where ddd is the spatial dimension) the disorder is irrelevant if a\ensuremathν\ensuremath−2>0a\ensuremath{\nu}\ensuremath{-}2>0a\ensuremathν\ensuremath−2>0, while if a>da>da>d we recover the usual Harris criterion: The disorder is irrelevant if d\ensuremathν\ensuremath−2=\ensuremath−\ensuremathα>0d\ensuremath{\nu}\ensuremath{-}2=\ensuremath{-}\ensuremath{\alpha}>0d\ensuremathν\ensuremath−2=\ensuremath−\ensuremathα>0. An mmm-vector system of this type is studied with the use of a renormalization-group expansion in \ensuremathϵ=4\ensuremath−d\ensuremath{\epsilon}=4\ensuremath{-}d\ensuremathϵ=4\ensuremath−d and \ensuremathδ=4\ensuremath−a\ensuremath{\delta}=4\ensuremath{-}a\ensuremathδ=4\ensuremath−a. We find a new long-range-disorder fixed point in addition to the short-range-disorder and pure fixed points found previously. The crossover between fixed points is found to follow the extended Harris criterion. The new fixed point has complex eigenvalues, leading to oscillating corrections to scaling, and has a correlation-length exponent \ensuremathν=2a\ensuremath{\nu}=\frac{2}{a}\ensuremathν=a2​. We argue that this new scaling relation is exact and applies more generally than just to the specific model. We show that the extended Harris criterion also applies to percolation with long-range-correlated site or bond-occupation probabilities, so that the scaling law should be obeyed by such systems. Results for the percolation properties of the triangular Ising model are in agreement with these predictions.
This paper is a study of the percolation problem with long-range correlations in the site or bond occupations. An extension of the Harris criterion for the relevance of the correlations is derived for the case that the correlations decay as ${x}^{\ensuremath{-}a}$ for large distances xxx.. For a<da<da<d the correlations are relevant if $a\ensuremath{\nu}\ensuremath{-}2<0$ (where $\ensuremath{\nu}$ is the percolation correlation-length exponent), while for a>da>da>d the correlations are relevant if $d\ensuremath{\nu}\ensuremath{-}2<0$.. Applying this criterion to the behavior that results when the correlations are relevant, we argue that the new behavior will have ${\ensuremath{\nu}}_{\mathrm{long}}=\frac{2}{a}$.. It is shown that the correlated bond percolation problem is equivalent to a qqq-state Potts model with quenched disorder in the limit $q\ensuremath{\rightarrow}1$.. With the use of this result, a renormalization-group study of the problem is presented, expanding in $\ensuremath{\epsilon}=6\ensuremath{-}d$ and in $\ensuremath{\delta}=4\ensuremath{-}a$.. In addition to the normal percolation fixed point, we find a new long-range fixed point. The crossover to this new fixed point follows the extended Harris criterion, and the fixed point has exponents ${\ensuremath{\nu}}_{\mathrm{long}}=\frac{2}{a}$ (as predicted) and ${\ensuremath{\eta}}_{\mathrm{long}}=\frac{1}{11}(\ensuremath{\delta}\ensuremath{-}\ensuremath{\epsilon})$. Finally, several results on the percolation properties of the Ising model at its critical point are shown to be in agreement with the predictions of this paper.
A renormalization-group treatment is used for investigating the critical properties of d-dimensional systems with anisotropic quenched disorder totally correlated in ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{d}}$ dimensions and randomly distributed in the remaining d-${\mathrm{\ensuremath{\epsilon}}}_{\mathit{d}}$ space directions (d=4-\ensuremath{\epsilon}) with long-range correlation which decays with the distance as a power law involving a characteristic parameter \ensuremath{\sigma}. Within a triple (\ensuremath{\epsilon},${\mathrm{\ensuremath{\epsilon}}}_{\mathit{d}}$,\ensuremath{\sigma}) expansion it is shown that the recursion relations possess a novel focus-type random fixed point leading to new anisotropic critical properties. Under certain conditions, different critical regimes and crossovers are found to occur, consistent with the modified version of the Harris criterion obtained for the random problem under study.
Quenched disorder--in the sense of the Harris criterion--is generally a relevant perturbation at an absorbing state phase transition point. Here using a strong disorder renormalization group framework and effective numerical methods we study the properties of random fixed points for systems in the directed percolation universality class. For strong enough disorder the critical behavior is found to be controlled by a strong disorder fixed point, which is isomorph with the fixed point of random quantum Ising systems. In this fixed point dynamical correlations are logarithmically slow and the static critical exponents are conjecturedly exact for one-dimensional systems. The renormalization group scenario is confronted with numerical results on the random contact process in one and two dimensions and satisfactory agreement is found. For weaker disorder the numerical results indicate static critical exponents which vary with the strength of disorder, whereas the dynamical correlations are compatible with two possible scenarios. Either they follow a power-law decay with a varying dynamical exponent, like in random quantum systems, or the dynamical correlations are logarithmically slow even for a weak disorder. For models in the parity conserving universality class there is no strong disorder fixed point according to our renormalization group analysis.
Critical properties are studied in systems with quenched bond disorder that is correlated along d1{d}_{1}d1​ of d dimensions. A renormalization-group scheme (based on the Migdal-Kadanoff method) which follows the full distribution of the random bonds and which gives correctly the modified Harris criterion \ensuremath{\varphi}=\ensuremath{\alpha}+d1{d}_{1}d1​\ensuremath{\nu} is used. For d1{d}_{1}d1​, we find fixed distributions at finite temperatures, yielding new ``random'' exponents for various q-state Potts models. For d1{d}_{1}d1​=d-1 there is no long-range order if there is a finite weight to zero coupling. Otherwise, we find a novel zero-temperature fixed distribution, for which all the moments diverge to infinity with finite ratios among them. This fixed distribution has a magnetic eigenvalue equal to d, indicating a first-order transition in the magnetization and possible related essential singularities. Thus, by analogy, the possibility of a magnetization jump is raised for the McCoy-Wu transition on a square lattice. The results for d1{d}_{1}d1​=1 are relevant to random quantum systems.
We describe the critical behavior of the Anderson-like transition predicted to occur in the 2D tight-binding model with isotropic scale-free long-range correlated disorder characterized by a power-law spectral density. We explore the scale invariance of the participation function relative fluctuation at the critical point to locate the mobility edge as a function of the power-law spectral density exponent α 2D . The states near the band center, which exhibit power-law localization for uncorrelated disorder, become delocalized for α 2D >2. In addition, we consider the finite-size scaling hypothesis to estimate the correlation length critical exponent. We find that the critical exponent ν depends on α 2D , thus indicating that correlations in the disorder distribution are indeed relevant in this regime, in agreement with the extended Harris criterion.
Motivated by the Harris criterion, the absorbing phase transitions of the conserved lattice gas (CLG) model were studied on lattices with a quenched disorder, i.e., on infinite percolation networks, in two and three dimensions. The Harris criterion suggests that, in a magnetic system, the pure fixed point will be unstable if the specific-heat exponent is positive. For the CLG model, the specific-heat exponent α calculated by the hyperscaling relation α=2-dν, where ν is the spatial correlation length exponent in d dimensions, will be positive in two dimensions if the value of ν obtained earlier by Lübeck and Heger is employed. On the other hand, it will be close to 0 if the more recent value by Lee and Lee is used and it is positive in three dimensions with the available value of ν. Extensive numerical simulations showed that, when the concentration of disordered sites is less than the critical concentration, the critical exponents were similar to those on a regular lattice both in two and three dimensions. When the concentration of disordered sites becomes critical, the density of active particles showed nonuniversal power-law behavior for all particle densities considered in both dimensions. These results were in contrast to the results for the diluted contact process. The cause of such a nonuniversal behavior was addressed.