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Journal of Physics C Solid State Physics

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

Ian D. Lawrie, Vladimir V. Prudnikov

Published Apr 10, 1984• 84 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3719/17/10/007

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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.
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 .
By means of the renormalisation-group approach, static and dynamic critical behaviours are studied for the XY spin system with cubic anisotropy and with quenched random nonmagnetic impurities. The stability of the quenched-random-impurity fixed point in d ( identical to 4- epsilon ) dimensions, the flow diagram of the interactions and the critical exponents are investigated, within the two-loop approximation, in terms of the epsilon 1/2 expansion. It is shown that the transition that appears is a first-order transition.
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.
A field-theoretic description of static and purely relaxational dynamic critical behaviour of systems with quenched defects obeying power law correlations ~| x | - a for large separations x is given. Directly, for three-dimensional systems and for different values of the correlation parameter, 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 behaviour are identified. The obtained results essentially differ from results evaluated by a double , -expansion. The static and dynamic critical exponents in the two-loop approximation are calculated with the use of the Padé-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 ${\ensuremath{\nu}}_{\ensuremath{\parallel}}^{\ensuremath{-}1}$.
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=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 $d,{\ensuremath{\varepsilon}}_{d}.$ 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.