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arXiv:cond-mat/9805247 · Physical review. B, Condensed matter

Dynamical properties of the Landau-Ginzburg model with long-range correlated quenched impurities

Elka Korutcheva, F. Javier de la Rubia

Published Sep 1, 1998• 29 citations
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Abstract

We investigate the critical dynamics of the time-dependent Landau-Ginzburg model with nonconserved n-component order parameter (model A) in the presence of long-range correlated quenched impurities. We use a special kind of long-range correlations $g(|r\ensuremath{-}{r}^{\ensuremath{'}}|)\ensuremath{\sim}|r\ensuremath{-}{r}^{\ensuremath{'}}{|}^{\ensuremath{-}a},$ $a=4\ensuremath{-}\ensuremath{\delta},$ previously introduced by Weinrib and Halperin. The parameter $\ensuremath{\delta}\ensuremath{\sim}\ensuremath{\epsilon},$ where $\ensuremath{\epsilon}=4\ensuremath{-}d$ is the usual small parameter within the Wilson-Fisher renormalization group theory. Using a double expansion in $\ensuremath{\epsilon}$ and $\ensuremath{\delta}$ we calculate the critical exponent $z$ up to second order on the small parameters. We show that quenched impurities of this kind affect the critical dynamics already in first order of $\ensuremath{\epsilon}$ and $\ensuremath{\delta},$ leading to a relevant correction for the mean field value of the exponent $z.$

DOI
10.1103/physrevb.58.5153

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The critical dynamics of the time-dependent Ginzburg-Landau model for a system with quenched random impurities and nonconserved order parameter is studied in the framework of the $\ensuremath{\epsilon}$ expansion. In contrast to the situation in pure systems, the dynamic critical exponent zzz deviates from its conventional value at first order in $\ensuremath{\epsilon}\ensuremath{\equiv}4\ensuremath{-}d$. The impurities cause an enhancement of the shape function ${f}_{x}(\ensuremath{\nu})$ at small frequencies $\ensuremath{\nu}$; ${f}_{x}(\ensuremath{\nu}=0)$ diverges as $T\ensuremath{\rightarrow}{T}_{c}$. Below Tc{T}_{c}Tc​ the equation of state, static susceptibility $\ensuremath{\chi}$, and dynamic response function GGG, are studied. A new, purely static correlated function, C(s){C}^{(s)}C(s), whose existence is unique to the random system is introduced. The coexistence curve singularities of C(s){C}^{(s)}C(s), $\ensuremath{\chi}$, and GGG in systems with continuously broken symmetry are explored. The connection of the quenched-impurity model with "model CCC" of Halperin, Hohenberg, and Ma is discussed.
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 theoretic renormalization-group approach is applied to the study of short-time dynamics of the d-dimensional n-component spin systems with long-range interactions r-(d+σ) and quenched disorder which has long-range correlations r-(d-ρ). Asymptotic scaling laws are obtained in a frame of double expansions in ∊=2σ-d and ρ with ρ of the order ∊. The static exponents are obtained exactly to all the order. The initial slip exponents θ′ for the order parameter and θ for the response function, as well as the dynamic exponent z, are calculated upto the first order in ∊. In d=2σ, in contrast to the unique logarithmic decay in the long-time regime which does not depend on σ, ρ, n and the disorder, we find rich scaling structures including logarithmic and exponential-logarithmic scalings in the short-time regime. Non-universal critical scalings of Ising systems are also discussed for d=2σ.
The theoretic renormalization-group approach is applied to the study of the critical behavior of the ddimensional Ising model with long-range correlated quenched impurities, which has a power-like correlations r-(d-ρ).The asymptotic scaling law is studied in the framework of the expansion in e = 4 - d. In d ＜ 4, the dynamic exponent z .is calculated up to the second order in ρ with ρ = O(e1/2). The shape function is obtained in one-loop calculation.When d = 4, the logarithmic corrections to the critical behavior are found. The finite size effect on the order parameter relaxation rate is also studied.
The theoretic renormalization-group approach is applied to the study of short-time critical behavior of the ddd-dimensional spin systems (model A) in the presence of quenched impurities with a long-range correlations decaying as ${r}^{\ensuremath{-}(d\ensuremath{-}\ensuremath{\rho})}$. The asymptotic scaling laws are studied in the frame of a double expansion in $ϵ=4\ensuremath{-}d$ and $\ensuremath{\rho}$ with $\ensuremath{\rho}$ of order ϵϵϵ. In d<4d<4d<4, the initial slip exponents ${\ensuremath{\theta}}^{\ensuremath{'}}$ of the magnetization and $\ensuremath{\theta}$ of the response function, are calculated up to two-loop order. The crossover between fixed points is obtained. The long-time limit of the fluctuation-dissipation ratio is found in the aging regime, and its connection to equilibrium quantities is discussed. The comparison of our results with those of other systems without long-range correlated quenched impurities is also investigated.
The theoretic renormalization-group approach is applied to the study of the critical behavior of the -dimensional Ising model with long-range correlated quenched impurities, which has a power-like correlations . The asymptotic scaling law is studied in the framework of the expansion in . In , the dynamic exponent is calculated up to the second order in with . The shape function is obtained in one-loop calculation. When , the logarithmic corrections to the critical behavior are found. The finite size effect on the order parameter relaxation rate is also studied.