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Physical review. B, Solid state

Renormalization-group methods for critical dynamics: I. Recursion relations and effects of energy conservation

Bertrand I. Halperin, Pierre C. Hohenberg, Shang-keng Ma

Published Jul 1, 1974• 550 citations
Publisher

Abstract

The renormalization-group method for studying critical phenomena is generalized to a class of dynamical systems---the time-dependent Ginzburg-Landau models. The effects of conservation laws on the critical dynamics are investigated through the study of models with different conservation properties for the energy and the space integral of the order parameter. Dynamic critical exponents near four dimensions ($d\ensuremath{\approx}4$) are obtained from recursion relations, analogous to those of Wilson and Fisher. The physical significance of the time-dependent Ginzburg-Landau models is explored and the applicability of the results to experiments on the NMR linewidth of Fe${\mathrm{F}}_{2}$ is discussed.

DOI
10.1103/physrevb.10.139

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This paper presents an extension of Wilson's renormalization-group calculation of Ising-model critical exponents to include calculation of the critical exponent $\ensuremath{\eta}$. New recursion formulas are derived using the simplest set of consistent approximations which allow a nonzero $\ensuremath{\eta}$. They are intended to demonstrate, qualitatively, how nonzero values for $\ensuremath{\eta}$ are consistent with the renormalization-group approach; they do not represent systematic, quantitative improvements to Wilson's earlier calculation of the exponents $\ensuremath{\nu}$ and $\ensuremath{\gamma}$. The equations are solved both by $\ensuremath{\epsilon}$ expansion about four dimensions and by numerical integration in three dimensions. To order ${\ensuremath{\epsilon}}^{2}$ we obtain $\ensuremath{\eta}=0.05{\ensuremath{\epsilon}}^{2}$. Numerical results in three dimension are $\ensuremath{\eta}=0.058$, $\ensuremath{\nu}=0.588$, and $\ensuremath{\gamma}=1.14$. The relation $\ensuremath{\gamma}=(2\ensuremath{-}\ensuremath{\eta})\ensuremath{\nu}$ is confirmed.

Physical review. B, Solid state•Jul 1, 1973•36 citations
The renormalization-group method for studying critical phenomena is generalized to a class of dynamical systems---the time-dependent Ginzburg-Landau models. The effects of conservation laws on the critical dynamics are investigated through the study of models with different conservation properties for the energy and the space integral of the order parameter. Dynamic critical exponents near four dimensions (d\ensuremath≈4d\ensuremath{\approx}4d\ensuremath≈4) are obtained from recursion relations, analogous to those of Wilson and Fisher. The physical significance of the time-dependent Ginzburg-Landau models is explored and the applicability of the results to experiments on the NMR linewidth of FeF2{\mathrm{F}}_{2}F2​ is discussed.
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.
The main idea of Wilson's renormalization group theory of critical phenomena is extended to the dynamical case. In the first place, a concept of dynamical fixed point is proposed. The dynamic scaling law is then derived from our new fixed-point hypothesis. Wilson's fixed-point hypothesis is also deduced from ours. Further, a method of renormalizing the master equation by eliminating the short wavelength modes is found. This is performed by means of the projection operator method. As an illustration, the dynamic critical exponents are calculated for a purely dissipative spin system. Related work based on the time-dependent Ginzburg-Landau equation is examined.
We discuss renormalization-group and mode-coupling theories of critical dynamics for several models. We show that to lowest order in $\ensuremath{\epsilon}={d}_{c}\ensuremath{-}d$, where for d>dcd>{d}_{c}d>dc​ conventional theory is valid, the renormalizationgroup differential equations are related by a simple transformation to those obtained from mode-coupling theory. Furthermore, the values of the dynamical fixed-point parameters essentially coincide with the critical amplitudes of the diverging transport coefficients, as determined by mode-coupling theory. In addition, we present a correct renormalization-group treatment for dynamical systems with more than one time scale, as for the binary liquid. We stress the fact that there are two distinct classes of critical dynamics, as illustrated by the results for the models studied here. We also discuss the instability of the conventional fixed point (time-dependent Ginzburg-Landau like) against mode-coupling perturbations for d<dcd<{d}_{c}d<dc​, with particular emphasis on a model for superfluid helium.
Renormalization techniques of field theory are used to prove (order by order to all orders) dynamical scaling laws on a Ginzburg-Landau stochastic model studied by Halperin, Hohenberg, and Ma. The dynamical exponent is calculated to order ${\ensuremath{\epsilon}}^{3}$ and so is the new exponent ${\ensuremath{\omega}}_{b}$, which governs the vanishing of the imaginary part of the (renormalized) kinetic coefficient, and appears among the corrections to scaling. Difficulties of previous calculations taking a microscopic approach to the critical dynamics of a Bose system are commented upon.
This paper presents an extension of Wilson's renormalization-group calculation of Ising-model critical exponents to include calculation of the critical exponent $\ensuremath{\eta}$. New recursion formulas are derived using the simplest set of consistent approximations which allow a nonzero $\ensuremath{\eta}$. They are intended to demonstrate, qualitatively, how nonzero values for $\ensuremath{\eta}$ are consistent with the renormalization-group approach; they do not represent systematic, quantitative improvements to Wilson's earlier calculation of the exponents $\ensuremath{\nu}$ and $\ensuremath{\gamma}$. The equations are solved both by $\ensuremath{\epsilon}$ expansion about four dimensions and by numerical integration in three dimensions. To order ${\ensuremath{\epsilon}}^{2}$ we obtain $\ensuremath{\eta}=0.05{\ensuremath{\epsilon}}^{2}$. Numerical results in three dimension are $\ensuremath{\eta}=0.058$, $\ensuremath{\nu}=0.588$, and $\ensuremath{\gamma}=1.14$. The relation $\ensuremath{\gamma}=(2\ensuremath{-}\ensuremath{\eta})\ensuremath{\nu}$ is confirmed.