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Physical review. B, Condensed matter

Static and dynamic critical behavior of extended-defectN-component systems in cubic anisotropic crystals

Yoshitake Yamazaki, Arno Holz, Moyuru Ochiai, Yoshiichi Fukuda

Published Mar 1, 1986• 24 citations
Publisher

Abstract

We study the static and dynamic critical behavior of N-component spin systems in crystals with cubic anisotropy in the presence of extended defects, namely ${\ensuremath{\epsilon}}_{d}$-dimensionally connected defects distributed randomly in d-${\ensuremath{\epsilon}}_{d}$ dimensions. These defects make the systems coordinate anisotropic. As a result of the presence of the cubic anisotropy and the coordinate anisotropy, various new types of critical behavior appear, e.g., an ${\ensuremath{\epsilon}}^{1/2}$ (\ensuremath{\epsilon}\ensuremath{\equiv}4-d) expansion for extended-defect XY systems, spiral trajectories, and oscillatory critical behavior for extended-defect systems, first-order phase transitions for extended-defect N-component systems with 2\ensuremath{\le}N<4 near the stability boundary of the free energy, and so on. The first two types of behavior will be investigated in detail.

DOI
10.1103/physrevb.33.3460

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We study the static and dynamic critical behavior of N-component spin systems in crystals with cubic anisotropy in the presence of extended defects, namely \ensuremathϵd{\ensuremath{\epsilon}}_{d}\ensuremathϵd​-dimensionally connected defects distributed randomly in d-\ensuremathϵd{\ensuremath{\epsilon}}_{d}\ensuremathϵd​ dimensions. These defects make the systems coordinate anisotropic. As a result of the presence of the cubic anisotropy and the coordinate anisotropy, various new types of critical behavior appear, e.g., an \ensuremathϵ1/2{\ensuremath{\epsilon}}^{1/2}\ensuremathϵ1/2 (\ensuremath{\epsilon}\ensuremath{\equiv}4-d) expansion for extended-defect XY systems, spiral trajectories, and oscillatory critical behavior for extended-defect systems, first-order phase transitions for extended-defect N-component systems with 2\ensuremath{\le}N<4 near the stability boundary of the free energy, and so on. The first two types of behavior will be investigated in detail.
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.
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.
The renormalization group approach is applied to the study of the short-time critical behavior of the d-dimensional n-component spin systems with cubic anisotropy. First, the system is quenched from high temperature to the critical temperature and then relaxes to equilibrium within model A dynamics. The asymptotic scaling laws and the initial slip exponents ${\ensuremath{\theta}}^{\ensuremath{'}}$ and $\ensuremath{\theta}$ of the order parameter and the response function, respectively, are calculated to the second order in $\ensuremath{\epsilon}=4\ensuremath{-}d.$ Logarithmic corrections to short-time behaviors of both the autocorrelation and the order parameter are found in d=4d=4d=4 dimension.
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 critical behavior in zero field above Tc{T}_{c}Tc​ of ferromagnets or ferroelectrics with a Hamiltonian of cubic symmetry is studied, to order ${\ensuremath{\epsilon}}^{2}$, by exact renormalization-group techniques in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions with nnn-component spins. For $\ensuremath{\epsilon}=1$, $n\ensuremath{\ge}3$, a crossover from isotropic (Heisenberg) to characteristic cubic behavior occurs, with the new value $2{v}^{C}=1+[\frac{(n\ensuremath{-}1)}{3n}]\ensuremath{\epsilon}+[\frac{(n\ensuremath{-}1)}{324{n}^{3}}](17{n}^{2}+290n\ensuremath{-}424){\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$, and cubic symmetry appearing in the four-spin correlation function. Experiments on structural phase transitions are considered briefly.
The critical behavior at the special surface transition and crossover bevavior from special to ordinary surface transition in semi-infinite n-component anisotropic cubic models are investigated by applying the field theoretic approach directly in d=3 dimensions up to the two-loop approximation. The crossover behavior for random semi-infinite Ising-like system, which is the nontrivial particular case of the cubic model in the limit n→0n\to 0n→0, is also investigated. The numerical estimates of the resulting two-loop series expansions for the critical exponents of the special surface transition, surface crossover critical exponent ΦΦΦ and the surface critical exponents of the layer, α1α_{1}α1​, and local specific heats, α11α_{11}α11​, are computed by means of Pade and Pade-Borel resummation techniques. For nncnn_{c}nnc​ the cubic fixed point is stable, where ncn_{c}nc​ is the marginal spin dimensionality of the cubic model. The obtained results indicate that the surface critical behavior of semi-infinite systems with cubic anisotropy is characterized by new set of surface critical exponents for n>ncn>n_{c}n>nc​.