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Communications in Theoretical Physics

Critical Phenomena in Systems with Pseudo-Long-Range is Disorder

Xintian Wu, Zheng‐Qing Huang

Published Apr 30, 1996• 0 citations
Publisher

Abstract

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.

DOI
10.1088/0253-6102/25/3/289

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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.
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.
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.
The aging behavior of the random n-vector model with long-range interaction decaying as r-(d+σ) (where d is the dimensionality), is investigated by the theoretic renormalization-group approach. The system initially disordered at a high temperature is firstly quenched to the critical temperature T c and then released to an evolution with model A dynamics. The aging properties are studied by the short-time expansions. The scaling behavior of two-time response and correlation functions are obtained in a frame of the expansion in ∊ = 2σ-d. In dimensions d < 2σ, the long-time limit of the critical fluctuation dissipation ratio X∞ is calculated up to one-loop order. The simulation of X∞ is discussed.
The replica symmetry breaking (RSB) is investigated in -component ferromagnetic spin systems with a long-range disorder, with a correlation function obeying a power law . The Landau–Ginzburg Hamiltonian with an RSB quartic interaction term is studied, with the renormalization-group expansion in and , where is the spatial dimension. It is shown that the long-range disorder fixed point found previously is unstable under the perturbation of RSB for and stable for . The RSB fixed points are calculated and there is no physical stable fixed point for .
The theoretical renormalization-group approach is applied to the study of the short-time critical behavior of the Ginzburg-Landau model with long-range random impurities which have a power-like correlation r—(d—ρ. The system initially at a high temperature is firstly quenched to the critical temperature Tc and then released to an evolution with a model A dynamics. The asymptotic scaling laws are studied in the frame of a double expansion in ϵ = 4 — d and δ = ϵ + ρ with δ of the order of ϵ. The initial slip exponent θ′ for the order parameter is calculated up to the first order in ϵ = 4 — d or in ϵ1/2 corresponding to different fixed points, respectively. For d > 4, the short-time behavior of the order parameter is investigated in one loop. Two different logarithmic and exponential-logarithmic corrections to short-time behaviors of both the autocorrelation and the order parameter are also solved in d = 4 dimension. Crossover between the nonrandom behavior and quenched behavior is found for n = 4 (where n is the spin dimensionality) in d = 4 dimension.
The effect of the long-range correlated random magnetic fields which behave like 〈h(x)h(y)〉〈h(x)h(y)〉〈h(x)h(y)〉 $\ensuremath{\sim}\frac{1}{{|x\ensuremath{-}y|}^{d\ensuremath{-}\ensuremath{\sigma}}}$ on the critical phenomena is discussed with the use of the nonlinear $\ensuremath{\sigma}$ model. The crossover between the two critical behaviors dominated by the long-range and short-range disorder fixed points is shown to occur at $(m\ensuremath{-}2)\ensuremath{\sigma}\ensuremath{-}d+4=0$ (mmm is the spin dimensionality). The critical exponents calculated at the long-range disorder fixed point around the lower critical dimensionality $4+\ensuremath{\sigma}$ are not the same as those in $d\ensuremath{-}2$ expansions in the pure system.