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Physical review. B, Solid state

Dynamics of spins interacting with quenched random impurities

G. Grinstein, Shang-keng Ma, Gene F. Mazenko

Published Jan 1, 1977• 98 citations
Publisher

Abstract

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 $z$ 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 ${T}_{c}$ the equation of state, static susceptibility $\ensuremath{\chi}$, and dynamic response function $G$, are studied. A new, purely static correlated function, ${C}^{(s)}$, whose existence is unique to the random system is introduced. The coexistence curve singularities of ${C}^{(s)}$, $\ensuremath{\chi}$, and $G$ in systems with continuously broken symmetry are explored. The connection of the quenched-impurity model with "model $C$" of Halperin, Hohenberg, and Ma is discussed.

DOI
10.1103/physrevb.15.258

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Physical review. B, Condensed matter•Jul 1, 1982•165 citations
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 \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ expansion. In contrast to the situation in pure systems, the dynamic critical exponent zzz deviates from its conventional value at first order in \ensuremathϵ\ensuremath≡4\ensuremath−d\ensuremath{\epsilon}\ensuremath{\equiv}4\ensuremath{-}d\ensuremathϵ\ensuremath≡4\ensuremath−d. The impurities cause an enhancement of the shape function fx(\ensuremathν){f}_{x}(\ensuremath{\nu})fx​(\ensuremathν) at small frequencies \ensuremathν\ensuremath{\nu}\ensuremathν; fx(\ensuremathν=0){f}_{x}(\ensuremath{\nu}=0)fx​(\ensuremathν=0) diverges as T\ensuremath→TcT\ensuremath{\rightarrow}{T}_{c}T\ensuremath→Tc​. Below Tc{T}_{c}Tc​ the equation of state, static susceptibility \ensuremathχ\ensuremath{\chi}\ensuremathχ, 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), \ensuremathχ\ensuremath{\chi}\ensuremathχ, 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.
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 zzz 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.z.z.
The effect of random impurities (quenched disorder) on the growth of correlations is studied for model A and model B after a sudden quench to Tc{\mathit{T}}_{\mathit{c}}Tc​ from the high-temperature phase (i.e., random initial conditions). Exponents and scaling functions of the nonequilibrium dynamic response function Gk{\mathit{G}}_{\mathit{k}}Gk​(t)=[〈\ensuremath{\partial}${\mathrm{\ensuremath{\varphi}}}_{\mathbf{k}}$(t)/\ensuremath{\partial}${\mathrm{\ensuremath{\varphi}}}_{\mathrm{\ensuremath{-}}\mathbf{k}}$(0)〉] and the structure factor Sk{\mathit{S}}_{\mathit{k}}Sk​(t)=[〈${\mathrm{\ensuremath{\varphi}}}_{\mathbf{k}}$(t)${\mathrm{\ensuremath{\varphi}}}_{\mathrm{\ensuremath{-}}\mathbf{k}}$(t)〉] are calculated to first order in \ensuremath{\epsilon} (\ensuremath{\epsilon}=4-d) for the O(n) model. For a nonconserved order parameter, the scaling form Gk{\mathit{G}}_{\mathit{k}}Gk​(t)=${\mathit{t}}^{\ensuremath{\lambda}/\mathit{z}}$f(kz{\mathit{k}}^{\mathit{z}}kzt) is obtained, with f(0)=const and \ensuremath{\lambda}=\ensuremath{\epsilon}/4+O(${\mathrm{\ensuremath{\epsilon}}}^{2}$), for 1n4. For n>4, random impurities are irrelevant, and \ensuremath{\lambda}=[(n+2)/2(n+8)]\ensuremath{\epsilon}+O(${\mathrm{\ensuremath{\epsilon}}}^{2}$), in agreement with calculations on the pure system. For a conserved order parameter \ensuremath{\lambda}=0, but the scaling function f(x) is nontrivial. For both conserved and nonconserved order parameter, disorder gives rise to algebraically decaying scaling functions.
Effects of quenched random magnetic fields on the critical relaxation of the time-dependent Ginzburg-Landau equation for m-component classical spin system are investigated by means of the dynamical renormalization group. In the static limit, the recursion relations near below 6 space dimensions found by Grinstein are recovered. Dynamic critical exponent for the characteristic frequency is obtained to be z = 2 + c η, c = 2 up to order ε2 (ε= 6 - d) on the basis of the dynamic scaling hypothesis. The static correlation exponent η is given by that of a pure system in d - 2 space dimensions. Since c does not agree with the result by Halperin-Hohenberg-Ma (c = 6 log (4/3) - 1), d →d - 2 rule does not hold for time-dependent phenomena.
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.
We study the critical behavior of an mmm-component classical spin system with quenched impurities correlated along an ${\ensuremath{\epsilon}}_{d}$-dimensional "line" and randomly distributed in $d\ensuremath{-}{\ensuremath{\epsilon}}_{d}$ dimensions ($d=4\ensuremath{-}\ensuremath{\epsilon}$). The presence of this line of impurities makes the system anisotropic and the interactions highly nonlocal. The renormalization group (RG) is used to approach the critical region and the quantities of interest are calculated in a double $\ensuremath{\epsilon}$, ${\ensuremath{\epsilon}}_{d}$ expansion. A two-loop calculation is needed to expose fully the divergent structure, and the theory is proved to be renormalizable up to this order. A consequence of the double $\ensuremath{\epsilon}$, ${\ensuremath{\epsilon}}_{d}$ expansion is the fact that the RG functions and consequently the critical exponents depend on the ratio $\frac{{\ensuremath{\epsilon}}_{d}}{(\ensuremath{\epsilon}+{\ensuremath{\epsilon}}_{d})}$. The solution of the RG equations leads to the existence of two correlation lengths: parallel to the "line" and perpendicular to it, with critical exponents ${\ensuremath{\nu}}_{\ensuremath{\parallel}}$ and ${\ensuremath{\nu}}_{\ensuremath{\perp}}$, respectively, with the relation ${\ensuremath{\nu}}_{\ensuremath{\parallel}}=z{\ensuremath{\nu}}_{\ensuremath{\perp}}$. The exponent zzz results from the presence of anisotropy in the system. New scaling laws are found for the critical exponents: $\ensuremath{\gamma}={\ensuremath{\nu}}_{\ensuremath{\perp}}(2\ensuremath{-}\ensuremath{\eta})$ and $\ensuremath{\alpha}=2\ensuremath{-}(d\ensuremath{-}{\ensuremath{\epsilon}}_{d}){\ensuremath{\nu}}_{\ensuremath{\perp}}\ensuremath{-}{\ensuremath{\epsilon}}_{d}{\ensuremath{\nu}}_{\ensuremath{\parallel}}$. We establish a relation between our model and a quantum model in one less dimension with random pointlike impurities. For this system we predict a quantum-to-classical crossover at finite temperature with cross-over exponent ${\ensuremath{\nu}}_{\ensuremath{\parallel}}^{\ensuremath{-}1}$.