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Physical review. B, Solid state

Dynamics of a coupled-mode system: Explicit analysis at order ε2

K. K. Murata

Published Mar 1, 1976• 24 citations
Publisher

Abstract

The renormalization-group equations for critical behavior in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions are generalized to the discussion of dynamic exponents at $T={T}_{c}$ and then applied to a situation recently considered by Hohenberg, Halperin, and Ma (HHM); namely, one in which an $n$-component primary field $\ensuremath{\psi}$ is coupled to a secondary field $\ensuremath{\phi}$ via a ${\ensuremath{\lambda}}_{0}(\ensuremath{\psi}\ensuremath{\psi})\ensuremath{\phi}$ coupling. In this work their analysis of relaxational fields is explicitly considered at order ${\ensuremath{\epsilon}}^{2}$. For $n<4$, we find the dynamic exponents differ in one case at $O({\ensuremath{\epsilon}}^{2})$ from that given by HHM. In contrast to HHM, the dynamic scaling postulate is also preserved by the second-order fixed points in all instances and detailed analysis is given. The fixed-point values of $R$, a dimensionless ratio of relaxation rates for the $\ensuremath{\phi}$ and $\ensuremath{\psi}$ fields, in certain cases, is found to go as ${R}^{*}\ensuremath{\sim}\frac{1}{\ensuremath{\epsilon}}$ which is a new result. This large value where it occurs is found essential to understanding dynamic scaling. Analysis of the case of a propagating conserved $\ensuremath{\phi}$ field also is given. It is found that its dynamic critical properties can be characterized by its damping term alone.

DOI
10.1103/physrevb.13.2028

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The relaxational models introduced in a previous treatment of critical dynamics are studied in detail using renormalization-group methods. The earlier results are justified by an analysis to all order in $\ensuremath{\epsilon}=4\ensuremath{-}d$, where ddd is the dimensionality. The diagrammatic formalism of the full dynamic renormalization group is presented, and applied to the earlier models. A generalization of Wilson's Feynman-graph expansion method is used to calculate the exponents to second order in $\ensuremath{\epsilon}$. In model CCC, where a nonconserved order parameter is coupled to a conserved energy field, ambiguities were found in the earlier recursion-relation treatment for 2<n<42<n<42<n<4, $d\ensuremath{\approx}4$ (nnn is the number of components of the order parameter). These ambiguities are discussed in the present work, but are not fully resolved.
We review the progress made in dynamic bulk critical behaviour in equilibrium in the last 25 years since the review of Halperin and Hohenberg. We unify the presentation of the theoretical background by restricting ourselves to the field-theoretic renormalization group method. The main results obtained in the different universality classes are presented. This contains the critical dynamics near the gas–liquid transition in pure fluids (model H), the plait point and consolute point in mixtures (model H � ), the superfluid transition in 4 He (model F) and 4 He– 3 He mixtures (model F � ), the Curie point (model J) and Neel point (model G) in Heisenberg magnets and the superconducting transition. In comparison with experimental results, it became clear that in most cases one has to consider apart from the universal asymptotic critical behaviour also the non-universal effective behaviour. Either because it turned out to be inevitable due to a small dynamical transient exponent inhibiting the system to reach the asymptotics (e.g., at the superfluid transition) or because one is interested in the region further away from the phase transition like in pure fluids and mixtures at their gas–liquid or demixing transition. The calculation of the critical dynamics is adequate in most cases only in two-loop order. We review these results and present the solution to unreasonable features found for some models. Thus, we consider model C where relaxational and diffusive dynamics are coupled and the scaling properties and the limit to a purely relaxational model (model A) have not been understood. In general for models where the order parameter couples to other conserved densities time scale ratios between the kinetic coefficients of the order parameter and the conserved densities play an important role. Their fixed-point values and the approach to the fixed point are changed considerably in two-loop order compared to their values in one-loop order. These considerations are relevant for the explanation of the dynamical critical shape functions of systems such as superfluid helium (model F) and the isotropic antiferromagnet (model G). As far as possible, the comparison of results obtained by the renormalization group theory with numerical simulations has been made.
The critical dynamics at elastic phase transitions of second order is studied by the renormalization-group theory. The dynamical theory is based on a stochastic equation of motion for damped phonons. For systems with one-dimensional soft sectors usual dynamical scaling holds with the critical dynamical exponent z=2z=2z=2. For two-dimensional sectors logarithmic corrections appear. Novel features are found for an isotropic-phonon model. The dynamical susceptibility and the characteristic frequency depend singularly on an irrelevant parameter, which, however, diverges at the fixed point. Consequently dynamical scaling breaks down; e.g., the characteristic frequency is no longer a homogeneous function of the wave number kkk and the inverse correlation length ${\ensuremath{\xi}}^{\ensuremath{-}1}$. Instead it is a homogeneous function of these variables and an irrelevant parameter. Consequently, the critical dynamics can be characterized by the relaxational exponent $z=2+c\ensuremath{\eta}$ at Tc{T}_{c}Tc​, while in the hydrodynamic region the sound frequency is characterized by $z=2\ensuremath{-}\frac{1}{2}\ensuremath{\eta}$ and the damping by yet another exponent. This breakdown of scaling is also reflected by the fact that different fixing conditions, i.e., different choices of the frequency, lead to different values of zzz. All of these apparently different transformations lead to the same modified dynamical scaling relations.
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 study the critical dynamics of stochastic models appropriate for describing bicritical and tetracritical points in anisotropic antiferromagnetic systems. The dynamic exponents and the transient exponents are calculated by renormalized field theory up to two-loop order. In the presence of reversible mode-coupling terms, two-loop contributions establish bicritical dynamic scaling in the restricted sense and invalidate recent predictions based on mode-coupling arguments. In the case of an n-component relaxational model total dynamic scaling is found to O (ε2) both at the bicritical (n≲3) and the tetracritical (4≲n<11) points.