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arXiv:cond-mat/9610159 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Critical behavior of O(n)-symmetric systems with reversible mode-coupling terms: Stability against detailed-balance violation

Uwe Claus Täuber, Zoltàn Ràcz

Published Apr 1, 1997• 15 citations
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Abstract

We investigate nonequilibrium critical properties of O(n)-symmetric models with reversible mode-coupling terms. Specifically, a variant of the model of Sasv\'ari, Schwabl, and Sz\'epfalusy (SSS) is studied, where violation of detailed balance is incorporated by allowing the order parameter and the dynamically coupled conserved quantities to be governed by heat baths of different temperatures ${\mathrm{T}}_{\mathrm{S}}$ and ${\mathrm{T}}_{\mathrm{M}}$, respectively. Dynamic perturbation theory and the field-theoretic renormalization group are applied to one-loop order, and yield two new fixed points in addition to the equilibrium ones. The first fixed point corresponds to \ensuremath{\Theta}=${\mathrm{T}}_{\mathrm{S}}$/${\mathrm{T}}_{\mathrm{M}}$=\ensuremath{\infty} and leads to model A critical behavior for the order parameter and to anomalous noise correlations for the generalized angular momenta; the second one is at \ensuremath{\Theta}=0 and is characterized by mean-field behavior of the conserved quantities, by a dynamic exponent z=d/2 equal to that of the equilibrium SSS model, and by modified static critical exponents. However, both these new fixed points are unstable, and upon approaching the critical point detailed balance is restored, and the equilibrium static and dynamic critical properties are recovered.

DOI
10.1103/physreve.55.4120

Related papers

Non-equilibrium critical behavior of O(n)-symmetric systems: Effect of reversible mode-coupling terms and dynamical anisotropy

Uwe Claus Täuber, Jaime E. Santos, Zoltàn Ràcz

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.

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Hydrodynamic modes and reversible mode couplings

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Equipped with the field theory representation of non-linear Langevin equations, the tools of dynamic perturbation theory, and the dynamic renormalization group introduced in Chapters 4 and 5, we are now in the position to revisit models for dynamic critical behavior that entail reversible mode couplings and other conserved hydrodynamic modes. We have already encountered some of these in Section 3.3. In models C and D, respectively, a non-conserved or conserved n -component order parameter is coupled to a conserved scalar field, the energy density. Through a systematic renormalization group analysis, we may critically assess the earlier predictions from scaling theory, and discuss the stability of fixed points characterized by strong dynamic scaling, wherein the order parameter and conserved non-critical mode fluctuate with equal rates, and weak dynamic scaling regimes, where these characteristic time scales differ. Next we investigate isotropic ferromagnets (model J), with the conserved spin density subject to reversible precession in addition to diffusive relaxation. Exploiting rotational invariance, we can now firmly establish the scaling relation z = ( d + 2 − η )/2. Similar symmetry arguments yield a scaling relation for the dynamic exponents associated with the order parameter and the non-critical fields in the O ( n )-symmetric SSS model that encompasses model E for planar ferromagnets and superfluid helium 4 (for n = 2), and model G for isotropic antiferromagnets ( n = 3). There exist competing strong- and weak-scaling fixed points, with the former stable to one-loop order, and characterized by z = d /2 for all slow modes.

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Effects of Violating Detailed Balance on Critical Dynamics

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We present an overview of the effects of detailed-balance violating perturbations on the universal static and dynamic scaling behavior near a critical point. It is demonstrated that the standard critical dynamics universality classes are generally quite robust: In systems with nonconserved order parameter, detailed balance is effectively restored at criticality. This also holds for models with conserved order parameter, and isotropic nonequilibrium perturbations. Genuinely novel features are found only for models with conserved order parameter and spatially anisotropic noise correlations.

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

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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 ${\mathrm{\ensuremath{\varphi}}}^{4}$ 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.

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Critical dynamics of the antiferromagnetic O(3) nonlinear sigma model with conserved magnetization

Louie Hong Yao, Uwe Claus Täuber

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.

arXiv:2204.11145•Physical review. E•Jun 27, 2022•3 citations
We investigate nonequilibrium critical properties of O(n)-symmetric models with reversible mode-coupling terms. Specifically, a variant of the model of Sasv'ari, Schwabl, and Sz'epfalusy (SSS) is studied, where violation of detailed balance is incorporated by allowing the order parameter and the dynamically coupled conserved quantities to be governed by heat baths of different temperatures TS{\mathrm{T}}_{\mathrm{S}}TS​ and TM{\mathrm{T}}_{\mathrm{M}}TM​, respectively. Dynamic perturbation theory and the field-theoretic renormalization group are applied to one-loop order, and yield two new fixed points in addition to the equilibrium ones. The first fixed point corresponds to \ensuremath{\Theta}=TS{\mathrm{T}}_{\mathrm{S}}TS​/TM{\mathrm{T}}_{\mathrm{M}}TM​=\ensuremath{\infty} and leads to model A critical behavior for the order parameter and to anomalous noise correlations for the generalized angular momenta; the second one is at \ensuremath{\Theta}=0 and is characterized by mean-field behavior of the conserved quantities, by a dynamic exponent z=d/2 equal to that of the equilibrium SSS model, and by modified static critical exponents. However, both these new fixed points are unstable, and upon approaching the critical point detailed balance is restored, and the equilibrium static and dynamic critical properties are recovered.
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.
Equipped with the field theory representation of non-linear Langevin equations, the tools of dynamic perturbation theory, and the dynamic renormalization group introduced in Chapters 4 and 5, we are now in the position to revisit models for dynamic critical behavior that entail reversible mode couplings and other conserved hydrodynamic modes. We have already encountered some of these in Section 3.3. In models C and D, respectively, a non-conserved or conserved n -component order parameter is coupled to a conserved scalar field, the energy density. Through a systematic renormalization group analysis, we may critically assess the earlier predictions from scaling theory, and discuss the stability of fixed points characterized by strong dynamic scaling, wherein the order parameter and conserved non-critical mode fluctuate with equal rates, and weak dynamic scaling regimes, where these characteristic time scales differ. Next we investigate isotropic ferromagnets (model J), with the conserved spin density subject to reversible precession in addition to diffusive relaxation. Exploiting rotational invariance, we can now firmly establish the scaling relation z = ( d + 2 − η )/2. Similar symmetry arguments yield a scaling relation for the dynamic exponents associated with the order parameter and the non-critical fields in the O ( n )-symmetric SSS model that encompasses model E for planar ferromagnets and superfluid helium 4 (for n = 2), and model G for isotropic antiferromagnets ( n = 3). There exist competing strong- and weak-scaling fixed points, with the former stable to one-loop order, and characterized by z = d /2 for all slow modes.
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 present an overview of the effects of detailed-balance violating perturbations on the universal static and dynamic scaling behavior near a critical point. It is demonstrated that the standard critical dynamics universality classes are generally quite robust: In systems with nonconserved order parameter, detailed balance is effectively restored at criticality. This also holds for models with conserved order parameter, and isotropic nonequilibrium perturbations. Genuinely novel features are found only for models with conserved order parameter and spatially anisotropic noise correlations.
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 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.
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.