We study the critical behavior of the O(n) O ( n ) model under steady shear flow using a dynamical renormalization group (RG) method. Incorporating the strong anisotropy in scaling ansatz, which has been neglected in earlier RG analyses, we identify a new stable Gaussian fixed point. This fixed point reproduces the anisotropic scaling of static and dynamical critical exponents for both non-conserved (Model A) and conserved (Model B) order parameters. Notably, the upper critical dimensions are d_{\text{up}} = 2 d up = 2 for the non-conserved order parameter (Model A) and d_{\text{up}} = 0 d up = 0 for the conserved order parameter (Model B), implying that the mean-field critical exponents are observed even in both d=2 d = 2 and 3 3 dimensions. Furthermore, the scaling exponent of the order parameter is negative for all dimensions d ≥ 2 d ≥ 2 , indicating that shear flow stabilizes the long-range order associated with continuous symmetry breaking even in d = 2 d = 2 . In other words, the lower critical dimensions are d_{\rm low} < 2 for both types of order parameters. This contrasts with equilibrium systems, where the Hohenberg—Mermin—Wagner theorem prohibits continuous symmetry breaking in d = 2 d = 2 .
Dynamical renormalization group analysis of O(n) model in steady shear flow
We study the critical behavior of the O(n) O ( n ) model under steady shear flow using a dynamical renormalization group (RG) method. Incorporating the strong anisotropy in scaling ansatz, which has been neglected in earlier RG analyses, we identify a new stable Gaussian fixed point. This fixed point reproduces the anisotropic scaling of static and dynamical critical exponents for both non-conserved (Model A) and conserved (Model B) order parameters. Notably, the upper critical dimensions are d_{\text{up}} = 2 d up = 2 for the non-conserved order parameter (Model A) and d_{\text{up}} = 0 d up = 0 for the conserved order parameter (Model B), implying that the mean-field critical exponents are observed even in both d=2 d = 2 and 3 3 dimensions. Furthermore, the scaling exponent of the order parameter is negative for all dimensions d ≥ 2 d ≥ 2 , indicating that shear flow stabilizes the long-range order associated with continuous symmetry breaking even in d = 2 d = 2 . In other words, the lower critical dimensions are d_{\rm low} < 2 for both types of order parameters. This contrasts with equilibrium systems, where the Hohenberg—Mermin—Wagner theorem prohibits continuous symmetry breaking in d = 2 d = 2 .
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.
The critical behavior of a model describing phase transitions in cubic and tetragonal anti-ferromagnets with 2N-component (N>1) real order parameters as well as the structural transition in NbO 2 crystal is studied within the field-theoretical renormalization-group (RG) approach in three and (4-∊)-dimensions. Perturbative expansions for RG functions are calculated up to three-loop order and resummed, in 3D, by means of the generalized Padé–Borel procedure which is shown to preserve the specific symmetry properties of the model. It is found that a stable fixed point does exist in the three-dimensional RG flow diagram for N>1, in accordance with predictions obtained earlier within the ∊-expansion. Fixed-point coordinates and critical-exponent values are presented for physically interesting cases N=2 and N=3. In both cases critical exponents are found to be numerically close to those of the 3DXY model. The analysis of the results given by the ∊-expansion and by the RG approach in three dimensions is performed resulting in a conclusion that the latter provides much more accurate numerical estimates.
This article concludes a series of papers [Folk, Holovatch, and Moser, Phys. Rev. E 78, 041124 (2008); 78, 041125 (2008); 79, 031109 (2009)] where the tools of the field theoretical renormalization group were employed to explain and quantitatively describe different types of static and dynamic behavior in the vicinity of multicritical points. Here we give the complete two-loop calculation and analysis of the dynamic renormalization-group flow equations at the multicritical point in anisotropic antiferromagnets in an external magnetic field. We find that the time scales of the order parameters characterizing the parallel and perpendicular ordering with respect to the external field scale in the same way. This holds independent whether the Heisenberg fixed point or the biconical fixed point in statics is the stable one. The nonasymptotic analysis of the dynamic flow equations shows that due to cancellation effects the critical behavior is described, in distances from the critical point accessible to experiments, by the critical behavior qualitatively found in one-loop order. Although one may conclude from the effective dynamic exponents (taking almost their one-loop values) that weak scaling for the order parameter components is valid, the flow of the time-scale ratios is quite different, and they do not reach their asymptotic values.
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.
The critical behaviour of a system, subjected to strongly anisotropic turbulent mixing, is studied by means of the field-theoretic renormalization group. Specifically, the relaxational stochastic dynamics of a non-conserved multicomponent order parameter of the Ashkin–Teller–Potts model, coupled to a random velocity field with prescribed statistics, is considered. The velocity is taken to be Gaussian, white in time, with a correlation function of the form ∝δ( t − t ')/| k ⊥ | d − 1 + ξ , where k ⊥ is the component of the wave vector, perpendicular to the distinguished direction ('direction of the flow')—the d -dimensional generalization of the ensemble was introduced by Avellaneda and Majda (1990 Commun. Math. Phys. 131 381) within the context of passive scalar advection. This model can describe a rich class of physical situations. It is shown that, depending on the values of the parameters that define the self-interaction of the order parameter and the relation between the exponent ξ and the space dimension d , the system exhibits various types of large-scale scaling behaviour, associated with different infrared attractive fixed points of the renormalization-group equations. In addition to known asymptotic regimes (critical dynamics of the Potts model and passively advected field without self-interaction), the existence of a new, non-equilibrium and strongly anisotropic, type of critical behaviour (universality class) is established, and the corresponding critical dimensions are calculated to the leading order of the double expansion in ξ and ε = 6 − d (one-loop approximation). The scaling appears to be strongly anisotropic in the sense that the critical dimensions related to the directions parallel and perpendicular to the flow are essentially different.
The critical behavior of a model describing phase transitions in 3D antiferromagnets with 2N-component real order parameters is studied within the renormalization-group (RG) approach. The RG functions are calculated in the three-loop order and resummed by the generalized Pad\'e-Borel procedure preserving the specific symmetry properties of the model. An anisotropic stable fixed point is found to exist in the RG flow diagram for N>2 and lies near the Bose fixed point; corresponding critical exponents are close to those of the XY model. The accuracy of the results obtained is discussed and estimated.