Journal of Physics C Solid State Physics

On the critical dynamics of disordered spin systems with extended defects

Vladimir V. Prudnikov

Published Jul 10, 1983• 28 citations

Abstract

The critical dynamics is studied for a spin system with quenched impurities correlated along an epsilon d -dimensional 'line' and randomly distributed in dl- epsilon d dimensions (d=4- epsilon ). The presence of these line defects makes the system anisotropic and the interactions non-local. The renormalisation group (RG) is used to approach the critical region and the dynamical exponents are calculated to the lowest order in a double epsilon , epsilon d expansion. The existence of two correlation lengths leads to the existence of two dynamical exponents Z 1 and Z t , which are connected by the relation (2- eta t )/Z t =(2- eta l )/Z l .

DOI: 10.1088/0022-3719/16/19/009

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Journal of Physics C Solid State Physics

On the critical dynamics of disordered spin systems with extended defects

Vladimir V. Prudnikov

Published Jul 10, 1983• 28 citations

Publisher

Abstract

DOI: 10.1088/0022-3719/16/19/009

Related papers

Static and dynamic properties of systems with extended defects: two-loop approximation

Ian D. Lawrie, Vladimir V. Prudnikov

Static and dynamic critical behaviour is studied for spin systems with quenched impurities that are correlated along an epsilon d -dimensional 'line' and randomly distributed in d- epsilon d dimensions. These impurities make the system anisotropic and modify its critical properties. Critical exponents are calculated at the second order of an expansion in epsilon and epsilon d (with epsilon =4-d). Although renormalisation-group functions depend on the ratio epsilon d /( epsilon + epsilon d ), critical exponents are found not to involve this ratio. Pade-like approximants are used to give numerical estimates for the exponents.Static and dynamic critical behaviour is studied for spin systems with quenched impurities that are correlated along an epsilon d -dimensional 'line' and randomly distributed in d- epsilon d dimensions. These impurities make the system anisotropic and modify its critical properties. Critical exponents are calculated at the second order of an expansion in epsilon and epsilon d (with epsilon =4-d). Although renormalisation-group functions depend on the ratio epsilon d /( epsilon + epsilon d ), critical exponents are found not to involve this ratio. Pade-like approximants are used to give numerical estimates for the exponents.

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We study the critical behavior of an $m$ -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 $z$ 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 ${\ensuremath{\nu}}_{\ensuremath{\parallel}}^{\ensuremath{-}1}$.We study the critical behavior of an $m$-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 $z$ 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 ${\ensuremath{\nu}}_{\ensuremath{\parallel}}^{\ensuremath{-}1}$.

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We investigate the asymptotic and effective static and dynamic critical behavior of $(d=3)$ -dimensional magnets with quenched extended defects, correlated in ${\ensuremath{\epsilon}}_{d}$ dimensions (which can be considered as the dimensionality of the defects) and randomly distributed in the remaining $d\ensuremath{-}{\ensuremath{\epsilon}}_{d}$ dimensions. The field-theoretical renormalization-group perturbative expansions being evaluated naively do not allow for the reliable numerical data. We apply the Chisholm-Borel resummation technique to restore convergence of the two-loop expansions and report the numerical values of the asymptotic critical exponents for the model $A$ dynamics. We discuss different scenarios for static and dynamic effective critical behavior and give values for corresponding nonuniversal exponents.We investigate the asymptotic and effective static and dynamic critical behavior of $(d=3)$-dimensional magnets with quenched extended defects, correlated in ${\ensuremath{\epsilon}}_{d}$ dimensions (which can be considered as the dimensionality of the defects) and randomly distributed in the remaining $d\ensuremath{-}{\ensuremath{\epsilon}}_{d}$ dimensions. The field-theoretical renormalization-group perturbative expansions being evaluated naively do not allow for the reliable numerical data. We apply the Chisholm-Borel resummation technique to restore convergence of the two-loop expansions and report the numerical values of the asymptotic critical exponents for the model $A$ dynamics. We discuss different scenarios for static and dynamic effective critical behavior and give values for corresponding nonuniversal exponents.

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Europhysics Letters (EPL)Oct 1, 199511 citations

Short-time dynamics of spin systems with long-range correlated quenched impurities

Yuan Chen, Zhibing Li

The theoretic renormalization-group approach is applied to the study of short-time critical behavior of the $d$ -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 $ϵ$ . In $d<4$ , 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 short-time critical behavior of the $d$-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 $ϵ$. In $d<4$, 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.

Physical Review BMay 27, 200515 citations

We study the critical behavior of an

m

-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

z

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 ${\ensuremath{\nu}}_{\ensuremath{\parallel}}^{\ensuremath{-}1}$.

We investigate the asymptotic and effective static and dynamic critical behavior of

(d=3)

-dimensional magnets with quenched extended defects, correlated in ${\ensuremath{\epsilon}}_{d}$ dimensions (which can be considered as the dimensionality of the defects) and randomly distributed in the remaining $d\ensuremath{-}{\ensuremath{\epsilon}}_{d}$ dimensions. The field-theoretical renormalization-group perturbative expansions being evaluated naively do not allow for the reliable numerical data. We apply the Chisholm-Borel resummation technique to restore convergence of the two-loop expansions and report the numerical values of the asymptotic critical exponents for the model

A

dynamics. We discuss different scenarios for static and dynamic effective critical behavior and give values for corresponding nonuniversal exponents.

The theoretic renormalization-group approach is applied to the study of short-time critical behavior of the

d

-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

ϵ

. In

d<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.