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.
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.
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 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(q,\ensuremath{\omega}) displays an anomalous line shape.
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.
We study a model equation of motion describing the dynamics of isotropic antiferromagnets near the critical point. We perform a renormalization-group analysis of this equation of motion correct to $O(\ensuremath{\epsilon})$, where $\ensuremath{\epsilon}=4\ensuremath{-}d$, finding recursion relations and analyzing their fixed points and their stability. For the one stable fixed point we obtain the dynamical index $z=\frac{d}{2}=2\ensuremath{-}\frac{\ensuremath{\epsilon}}{2}$. We also calculate the response and correlation functions in the scaling region using perturbation theory correct to $O(\ensuremath{\epsilon})$. Our calculation gives analytical forms for the correlation functions for the staggered magnetization and the magnetization. These correlation functions can be written in the dynamical scaling form. The shape function that characterizes the frequency spectrum for the staggered magnetization is Lorentzian in the hydrodynamical regime, but shows fluctuation induced peaks at and near the N\ifmmode\dagger\else\textdagger\fi{}eel temperature. The shape function for the magnetization is essentially Lorentzian for all $x=q\ensuremath{\xi}$ (where q is the wave number and $\ensuremath{\xi}$ is the correlation length) and shows a narrow lydrodynamical width as $x\ensuremath{\rightarrow}0$.
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.
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 relaxation dynamics of two distinct non-equilibrium processes: relaxation of three-dimensional antiferromagnetic lattice spin models with Heisenberg interaction following a critical quench, and a one-dimensional exclusion process inspired by the gear-like motion of molecular motors. In a system of three-dimensional Heisenberg antiferromagnets the non-conserved staggered magnetization components couple non-trivially to the conserved magnetization densities inducing fully reversible terms that enter the Langevin dynamic equation. We simulate the exact microscopic dynamics of such a system of antiferromagnets by employing a hybrid simulation algorithm that combines the reversible spin precession implemented by the fourth-order Runge-Kutta integration method with the standard relaxational dynamics at finite temperatures using Monte Carlo updates. We characterize the dynamic universality class of this system by probing the early temporal window where the system exhibits aging scaling properties. We also verify an earlier renormalization group prediction that the temporal decay exponent in the two-time spin autocorrelation function exhibits non-universality, specifically it depends on the width of the initial spin orientation distribution. We employ a similar numerical technique to study the critical dynamics of an anisotropic Heisenberg antiferromagnet in the presence of an external field. The phase diagram of this system exhibits two critical lines that meet at a bicritical point. We study the aging scaling dynamics for the model C critical line, probe the model F critical line by investigating the system size dependence of the characteristic spin-wave frequencies near criticality, and measure the dynamic critical exponents for the order parameter including its aging scaling at the bicritical point. We introduce a one-dimensional non-equilibrium lattice gas model representing the processive motion of dynein molecular motors over the microtubule. We study both dynamical and stationary state properties for the model consisting of hardcore particles hopping on the lattice with variable step sizes. We find that the stationary state gap-distribution exhibits striking peaks around gap sizes that are multiples of the maximum step size, for both open and periodic boundary conditions, and verify this using a mean-field calculation. For open boundary conditions, we observe intriguing damped oscillator-like distribution of particles over the lattice with a periodicity equal to the maximum step size. To characterize transient dynamics, we measure the mean square displacement that shows weak superdiffusive growth with exponent γ≈ 1.34 for periodic boundary and ballistic growth ( γ≈ 2) for open boundary conditions at early times. We also study the effect of Langmuir dynamics on the density profile.