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Physical review. B, Condensed matter

Critical behavior of m -component magnets with correlated impurities

D. Boyanovsky, John Cardy

Published Jul 1, 1982• 165 citations
Publisher

Abstract

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

DOI
10.1103/physrevb.26.154

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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 .
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.
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.
We analyze the critical properties of the three-dimensional Ising model with linear parallel extended defects. Such a form of disorder produces two distinct correlation lengths, a parallel correlation length ${\ensuremath{\xi}}_{\ensuremath{\parallel}}$ in the direction along defects and a perpendicular correlation length ${\ensuremath{\xi}}_{\ensuremath{\perp}}$ in the direction perpendicular to the lines. Both ${\ensuremath{\xi}}_{\ensuremath{\parallel}}$ and ${\ensuremath{\xi}}_{\ensuremath{\perp}}$ diverge algebraically in the vicinity of the critical point, but the corresponding critical exponents ${\ensuremath{\nu}}_{\ensuremath{\parallel}}$ and ${\ensuremath{\nu}}_{\ensuremath{\perp}}$ take different values. This property is specific for anisotropic scaling and the ratio ${\ensuremath{\nu}}_{\ensuremath{\parallel}}/{\ensuremath{\nu}}_{\ensuremath{\perp}}$ defines the anisotropy exponent $\ensuremath{\theta}$. Until now, estimates of quantitative characteristics of the critical behavior for such systems have been obtained only within the renormalization group approach. We report a study of the anisotropic scaling in this system via Monte Carlo simulation of the three-dimensional system with Ising spins and nonmagnetic impurities arranged into randomly distributed parallel lines. Several independent estimates for the anisotropy exponent $\ensuremath{\theta}$ of the system are obtained, as well as an estimate of the susceptibility exponent $\ensuremath{\gamma}$. Our results corroborate the renormalization group predictions obtained earlier.
We study the crossover from pure to random critical behavior in ddd-dimensional ($d=4\ensuremath{-}\ensuremath{\epsilon}$) classical spin systems which is induced by quenched random fields with spatial correlations such that ${[h(0)h(R)]}_{\mathrm{av}}\ensuremath{\sim}\frac{{h}_{0}^{2}}{{R}^{d+\ensuremath{\theta}}}$, where $\ensuremath{\theta}=2\ensuremath{-}{\ensuremath{\epsilon}}_{d}$. An effective critical exponent for the magnetic susceptibility ${\ensuremath{\gamma}}_{\mathrm{eff}}=\frac{\ensuremath{-}\ensuremath{\partial}\mathrm{ln}\ensuremath{\chi}}{\ensuremath{\partial}\mathrm{ln}t}$ is calculated with the use of a renormalization-group approcah and a double $\ensuremath{\epsilon}$,$\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}$ expansion to linear order in $\ensuremath{\epsilon}$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}\ensuremath{\equiv}\ensuremath{\epsilon}+{\ensuremath{\epsilon}}_{d}$. The crossover scaling function for the susceptibility is obtained and the width of the crossover region is estimated.
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.