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Physical review. B, Condensed matter

Critical properties of systems with anisotropic long-range correlated quenched disorder

L. De Cesare

Published May 1, 1994• 22 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.49.11742

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A renormalization-group treatment is used for investigating the critical properties of d-dimensional systems with anisotropic quenched disorder totally correlated in \ensuremathϵd{\mathrm{\ensuremath{\epsilon}}}_{\mathit{d}}\ensuremathϵd​ dimensions and randomly distributed in the remaining d-\ensuremathϵd{\mathrm{\ensuremath{\epsilon}}}_{\mathit{d}}\ensuremathϵd​ 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},\ensuremathϵd{\mathrm{\ensuremath{\epsilon}}}_{\mathit{d}}\ensuremathϵd​,\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.
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 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 $a\ensuremath{\nu}\ensuremath{-}2>0$, while if a>da>da>d we recover the usual Harris criterion: The disorder is irrelevant if $d\ensuremath{\nu}\ensuremath{-}2=\ensuremath{-}\ensuremath{\alpha}>0$. An mmm-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.
A renormalization-group approach is used for investigating the effects of the Parisi replica symmetry breaking (``spin-glass effects'') on critical properties of d-dimensional systems with extended quenched disorder along ${\ensuremath{\varepsilon}}_{d}$ directions. Within a double expansion scheme to first order in $(\ensuremath{\varepsilon}=4\ensuremath{-}d, {\ensuremath{\varepsilon}}_{d})$ it is found that for $d<~4+{\ensuremath{\varepsilon}}_{d}$ and order-parameter dimensionalities n< 4,n<~4,n< 4, a full fixed-point instability with a consequent runaway towards a strong-coupling regime occurs in the parameter space while, for d<4d<4d<4 and n>4,n>4,n>4, compatibly with the modified Harris criterion, an additional marginally stable fixed point exists which might control a new anisotropic random critical behavior. In any case, the traditional replica-symmetric Dorogovtsev-like solution for classical and quantum systems is unstable against the replica symmetry-breaking modes.
We studied the statics and dynamics of elastic manifolds in disordered media with long-range correlated disorder using functional renormalization group (FRG). We identified different universality classes and computed the critical exponents and universal amplitudes describing geometric and velocity-force characteristics. In contrast to uncorrelated disorder, the statistical tilt symmetry is broken resulting in a nontrivial response to a transverse tilting force. For instance, the vortex lattice in disordered superconductors shows a new glass phase whose properties interpolate between those of the Bragg and Bose glasses formed by pointlike and columnar disorder, respectively. Whereas there is no response in the Bose glass phase (transverse Meissner effect), the standard linear response expected in the Bragg-glass gets modified to a power law response in the presence of disorder correlations. We also studied the long distance properties of the O(N) spin system with random fields and random anisotropies correlated as 1/x^{d-sigma}. Using FRG we obtained the phase diagram in (d,sigma,N)-parameter space and computed the corresponding critical exponents. We found that below the lower critical dimension 4+sigma, there can exist two different types of quasi-long-range-order with zero order-parameter but infinite correlation length.
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.
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.
Critical phenomena in systems with long(but finite)-range-correlated disorder of "random temperature" are studied. The disorder with correlation function (d is the spatial dimension) is considered. The critical behavior in an -vector spin system with such a disorder is investigated by using renormalization-group expansion in and . The recursion relations of coupling constants for are obtained. It is shown that critical phenomena in systems with such a pseudo-long-range disorder will exhibit crossover from tricritical to critical behavior for . In the crossover regime the scaling relations are expected to break down.