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arXiv:cond-mat/0301383 · Physical review. B, Condensed matter

Critical behavior of magnetic systems with extended impurities in general dimensions

Viktoria Blavatska, Christian von Ferber, Yurij V. Holovatch

Published Mar 7, 2003• 23 citations
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Abstract

We investigate the critical properties of d-dimensional magnetic systems with quenched extended defects, correlated in ${\ensuremath{\varepsilon}}_{d}$ dimensions (which can be considered as the dimensionality of the defects) and randomly distributed in the remaining $d\ensuremath{-}{\ensuremath{\varepsilon}}_{d}$ dimensions; both in the case of fixed dimension $d=3$ and when the space dimension continuously changes from the lower critical dimension to the upper one. Our renormalization-group calculations are performed in the minimal subtraction scheme. We analyze the two-loop renormalization-group functions for different fixed values of the parameters $d,{\ensuremath{\varepsilon}}_{d}.$ To this end, we apply the Chisholm-Borel resummation technique and report the numerical values of the critical exponents for the universality class of this system.

DOI
10.1103/physrevb.67.094404

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We investigate the critical properties of d-dimensional magnetic systems with quenched extended defects, correlated in \ensuremathεd{\ensuremath{\varepsilon}}_{d}\ensuremathεd​ dimensions (which can be considered as the dimensionality of the defects) and randomly distributed in the remaining d\ensuremath−\ensuremathεdd\ensuremath{-}{\ensuremath{\varepsilon}}_{d}d\ensuremath−\ensuremathεd​ dimensions; both in the case of fixed dimension d=3d=3d=3 and when the space dimension continuously changes from the lower critical dimension to the upper one. Our renormalization-group calculations are performed in the minimal subtraction scheme. We analyze the two-loop renormalization-group functions for different fixed values of the parameters d,\ensuremathεd.d,{\ensuremath{\varepsilon}}_{d}.d,\ensuremathεd​. To this end, we apply the Chisholm-Borel resummation technique and report the numerical values of the critical exponents for the universality class of this system.
We investigate the asymptotic and effective static and dynamic critical behavior of (d=3)(d=3)(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 AAA dynamics. We discuss different scenarios for static and dynamic effective critical behavior and give values for corresponding nonuniversal exponents.
We study nonequilibrium critical relaxation properties of systems with quenched extended defects, correlated in ${\ensuremath{\varepsilon}}_{d}$ dimensions and randomly distributed in the remaining $d\ensuremath{-}{\ensuremath{\varepsilon}}_{d}$ dimensions. Using a field-theoretic renormalization-group approach, we find the scaling behavior of the nonequilibrium response and correlation functions and calculate the initial slip exponents $\ensuremath{\theta}$ and ${\ensuremath{\theta}}^{\ensuremath{'}},$ which describe the growth of correlations during the initial stage of the critical relaxation, in the two-loop approximation.
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.
A field-theory description of the static and dynamic critical behavior of systems with quenched defects obeying power law correlations $\ensuremath{\sim}|\mathbf{x}{|}^{\ensuremath{-}a}$ for large separations x\mathbf{x}x is given. Directly, for three-dimensional systems and for different values of the correlation parameter, 2< a< 3,2<~a<~3,2< a< 3, a renormalization analysis of the scaling functions in the two-loop approximation is carried out, and the fixed points corresponding to the stability of various types of critical behavior are identified. The obtained results essentially differ from results evaluated by a double $\ensuremath{\varepsilon},\ensuremath{\delta}$ expansion. The static and dynamic critical exponents in the two-loop approximation are calculated with the use of the Pad\'e-Borel summation technique.
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 ${\ensuremath{\nu}}_{\ensuremath{\parallel}}^{\ensuremath{-}1}$.
A field-theoretic description of the critical behaviour of systems with quenched defects obeying a power law correlations ∼ |x | −a for large separations x is given. Directly for three-dimensional systems and different values of correlation parameter 2 ≤ a ≤ 3 a renormalization analysis of scaling function in the two-loop approximation is carried out, and the fixed points corresponding to stability of the various types of critical behaviour are identified. The obtained results essentially differ from results evaluated by double ε, δ- expansion. The critical exponents in the two-loop approximation are calculated with the use of the Pade-Borel summation technique. PACS: 64.60.Ak; 64.60.-i; 64.60.FrIn last years, much effort has been devoted to investigation of the critical behaviour of solids containing quenched defects. In most papers considerations have been restricted to the case of point defects with small concentrations so that the defects and corresponding random fields have been assumed to be Gaussian distributed and δ- correlated. For the first time in work of Weinrib and Halperin (WH) [1] have been offered the model of the critical behaviour of disordered system in which the correlation function of the random local transition temperature g(x −y) =<< Tc(x)Tc(y)>> − << Tc(x)>> 2