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Physical review. B, Condensed matter

Critical dynamics of the O(n)-symmetric relaxational models below the transition temperature

Uwe Claus Täuber, Franz Schwabl

Published Aug 1, 1992• 36 citations
Publisher

Abstract

The critical dynamics of the O(n)-symmetric relaxational models with either nonconserved (model A) or conserved order parameter (model B) are studied below the transition temperature. As a consequence of Goldstone's theorem, the transverse modes are massless, implying infrared divergences in the theory along the entire coexistence curve. These Goldstone singularities can be treated within the field-theoretical formulation of the dynamical renormalization group by using the generalized regularization scheme as introduced by Amit and Goldschmidt, which has already been applied on the statics of the ${\mathrm{\ensuremath{\varphi}}}^{4}$ model below ${\mathit{T}}_{\mathit{c}}$ by Lawrie. We extend the formalism in several respects: (i) we generalize it to dynamical phenomena, (ii) taking advantage of the fact that the theory is exactly treatable in the coexistence limit, we do not use the \ensuremath{\epsilon} expansion; (iii) the flow equations are solved numerically, thus allowing for a detailed description of the crossover from the critical isotropic Heisenberg fixed point to the infrared-stable coexistence fixed point. We calculate the static susceptibilities as well as the dynamical correlation functions for models A and B within the complete crossover region, identifying the asymptotic coexistence anomalies and also a pronounced intermediate minimum of the effective critical exponents. Furthermore, the longitudinal dynamical correlation function ${\mathit{G}}_{\mathit{L}}$(q,\ensuremath{\omega}) displays an anomalous line shape.

DOI
10.1103/physrevb.46.3337

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The critical dynamics of the O(n)-symmetric relaxational models with either nonconserved (model A) or conserved order parameter (model B) are studied below the transition temperature. As a consequence of Goldstone's theorem, the transverse modes are massless, implying infrared divergences in the theory along the entire coexistence curve. These Goldstone singularities can be treated within the field-theoretical formulation of the dynamical renormalization group by using the generalized regularization scheme as introduced by Amit and Goldschmidt, which has already been applied on the statics of the \ensuremathφ4{\mathrm{\ensuremath{\varphi}}}^{4}\ensuremathφ4 model below Tc{\mathit{T}}_{\mathit{c}}Tc​ by Lawrie. We extend the formalism in several respects: (i) we generalize it to dynamical phenomena, (ii) taking advantage of the fact that the theory is exactly treatable in the coexistence limit, we do not use the \ensuremath{\epsilon} expansion; (iii) the flow equations are solved numerically, thus allowing for a detailed description of the crossover from the critical isotropic Heisenberg fixed point to the infrared-stable coexistence fixed point. We calculate the static susceptibilities as well as the dynamical correlation functions for models A and B within the complete crossover region, identifying the asymptotic coexistence anomalies and also a pronounced intermediate minimum of the effective critical exponents. Furthermore, the longitudinal dynamical correlation function GL{\mathit{G}}_{\mathit{L}}GL​(q,\ensuremath{\omega}) displays an anomalous line shape.
The critical behaviour along the coexistence curve of isotropic ferromagnets is characterized by two diverging correlation lengths since longitudinal and transverse fluctuations of the order parameter become critical. Therefore, a critical theory valid for the whole phase diagram has to incorporate the crossover between a region characterized by one correlation length to another region where two different correlation lengths dominate. Treating this crossover problem entirely within the framework of the trajectory integral method. The author has calculated the equation of state and the susceptibilities. It is shown that coexistence behaviour is governed by a coexistence fixed point which is characterized by the vanishing interaction among the Goldstone modes. Taking the coupling of the critical transverse modes to the longitudinal modes into account, it is shown that coexistence behaviour is characterized by a Fisher renormalization. This central result sheds new light on the symmetry-broken phase in isotropic systems. The asymptotic forms of the equation of state and the susceptibilities at the coexistence curve are expressed by the specific heat exponent alpha t =1/2 for d>or=3 and for all spin dimensions. These results are in perfect agreement with the nonlinear sigma -model. The crossover from coexistence curve behaviour to ordinary critical behaviour is calculated for the equation of state and for the susceptibilities to O( in )( in =4-d).
We study the near-equilibrium critical dynamics of the O(3) nonlinear sigma model describing isotropic antiferromagnets with a nonconserved order parameter reversibly coupled to the conserved total magnetization. To calculate response and correlation functions, we set up a description in terms of Langevin stochastic equations of motion, and their corresponding Janssen-De Dominicis response functional. We find that in equilibrium, the dynamics is well-separated from the statics, at least to one-loop order in a perturbative treatment with respect to the static and dynamical nonlinearities. Since the static nonlinear sigma model must be analyzed in a dimensional d=2+ɛ expansion about its lower critical dimension d_{lc}=2, whereas the dynamical mode-coupling terms are governed by the upper critical dimension d_{c}=4, a simultaneous perturbative dimensional expansion is not feasible, and the reversible critical dynamics for this model cannot be accessed at the static critical renormalization group fixed point. However, in the coexistence limit addressing the long-wavelength properties of the low-temperature ordered phase, we can perform an ε=4-d expansion near d_{c}. This yields anomalous scaling features induced by the massless Goldstone modes, namely subdiffusive relaxation for the conserved magnetization density with asymptotic scaling exponent z_{Γ}=d-2, which may be observable in neutron scattering experiments. Intriguingly, if initialized near the critical point, the renormalization group flow for the effective dynamical exponents recovers their universal critical values z_{c}=d/2 in an intermediate crossover region.
Phase transitions in non-equilibrium steady states of O(n)-symmetric models with reversible mode couplings are studied using dynamic field theory and the renormalization group. The systems are driven out of equilibrium by dynamical anisotropy in the noise for the conserved quantities, i.e., by constraining their diffusive dynamics to be at different temperatures T^\par and T^\perp in d_\par- and d_\perp-dimensional subspaces, respectively. In the case of the Sasv'ari-Schwabl-Sz'epfalusy (SSS) model for planar ferro- and isotropic antiferromagnets, we assume a dynamical anisotropy in the noise for the non-critical conserved quantities that are dynamically coupled to the non-conserved order parameter. We find the equilibrium fixed point (with isotropic noise) to be stable with respect to these non-equilibrium perturbations, and the familiar equilibrium exponents therefore describe the asymptotic static and dynamic critical behavior. Novel critical features are only found in extreme limits, where the ratio of the effective noise temperatures T^\par/T^\perp is either zero or infinite. On the other hand, for model J for isotropic ferromagnets with a conserved order parameter, the dynamical noise anisotropy induces effective long-range elastic forces, which lead to a softening only of the d_\perp-dimensional sector in wavevector space with lower noise temperature T^\perp < T^\par. The ensuing static and dynamic critical behavior is described by power laws of a hitherto unidentified universality class, which, however, is not accessible by perturbational means for d_\par \geq 1. We obtain formal expressions for the novel critical exponents in a double expansion about the static and dynamic upper critical dimensions and d_\par, i.e., about the equilibrium theory.
In this central and essential chapter, we develop the dynamic renormalization group approach to time-dependent critical phenomena. Again, we base our exposition on the simple O ( n )-symmetric relaxational models A and B; the generalization to other dynamical systems is straightforward. We begin with an analysis of the infrared and ultraviolet singularities appearing in the dynamic perturbation expansion. Although we are ultimately interested in the infrared critical region, we first take care of the ultraviolet divergences. Below and at the critical dimension d c = 4, only a finite set of Feynman diagrams carries ultraviolet singularities, which we evaluate by means of the dimensional regularization prescription, and then eliminate via multiplicative as well as additive renormalization (the latter takes into account the fluctuation-induced shift of the critical temperature). The renormalization group equation then permits us to explore the ensuing scaling behavior of the correlation and vertex functions of the renormalized theory upon varying the arbitrary renormalization scale. Of fundamental importance is the identification of an infrared-stable renormalization group fixed point, which describes scale invariance and hence allows the derivation of the critical power laws in the infrared limit from the renormalization constants determined in the ultraviolet regime. This program is explicitly carried through for the relaxational models A and B. For model B, we derive a scaling relation connecting the dynamic exponent to the Fisher exponent η . The critical exponents ν , η , and z are computed to first non-trivial order in an ∊ expansion around the upper critical dimension d c = 4.
We investigate the critical dynamics of the n-component relaxational models C and D, which incorporate the coupling of a nonconserved and conserved order parameter S, respectively, to the conserved energy density rho, under nonequilibrium conditions by means of the dynamical renormalization group. Detailed balance violations can be implemented isotropically by allowing for different effective temperatures for the heat baths coupling to the slow modes. In the case of model D with conserved order parameter, the energy density fluctuations can be integrated out, leaving no trace of the nonequilibrium perturbations in the asymptotic regime. For model C with scalar order parameter, in equilibrium governed by strong dynamic scaling (z(S)=z(rho)), we find no genuine nonequilibrium fixed point either. The nonequilibrium critical dynamics of model C with n=1 thus follows the behavior of other systems with nonconserved order parameter wherein detailed balance becomes effectively restored at the phase transition. For n> or =4, the energy density generally decouples from the order parameter. However, for n=2 and n=3, in the weak dynamic scaling regime (z(S)< or =z(rho)) entire lines of genuine nonequilibrium model C fixed points emerge to one-loop order, which are characterized by continuously varying static and dynamic critical exponents. Similarly, the nonequilibrium model C with spatially anisotropic noise and n<4 allows for continuously varying exponents, yet with strong dynamic scaling. Subjecting model D to anisotropic nonequilibrium perturbations leads to genuinely different critical behavior with softening only in subsectors of momentum space and correspondingly anisotropic scaling exponents. Similar to the two-temperature model B (randomly driven diffusive systems) the effective theory at criticality can be cast into an equilibrium model D dynamics, albeit incorporating long-range interactions of the uniaxial dipolar or ferroelastic type.
We calculate the relaxational dynamical critical behavior of systems of O(n_{ parallel}) plus sign in circleO(n_{ perpendicular}) symmetry including conservation of magnetization by renormalization group theory within the minimal subtraction scheme in two-loop order. Within the stability region of the Heisenberg fixed point and the biconical fixed point, strong dynamical scaling holds, with the asymptotic dynamical critical exponent z=2varphinu-1 , where varphi is the crossover exponent and nu the exponent of the correlation length. The critical dynamics at n_{ parallel}=1 and n_{ perpendicular}=2 is governed by a small dynamical transient exponent leading to nonuniversal nonasymptotic dynamical behavior. This may be seen, e.g., in the temperature dependence of the magnetic transport coefficients.