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Physical review. B, Solid state

Renormalization-group methods for critical dynamics: II. Detailed analysis of the relaxational models

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

Published May 1, 1976• 75 citations
Publisher

Abstract

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 $d$ 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 $C$, where a nonconserved order parameter is coupled to a conserved energy field, ambiguities were found in the earlier recursion-relation treatment for $2<n<4$, $d\ensuremath{\approx}4$ ($n$ is the number of components of the order parameter). These ambiguities are discussed in the present work, but are not fully resolved.

DOI
10.1103/physrevb.13.4119

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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 \ensuremathϵ=4\ensuremath−d\ensuremath{\epsilon}=4\ensuremath{-}d\ensuremathϵ=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 \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ. 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≈4d\ensuremath{\approx}4d\ensuremath≈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 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 FeF2{\mathrm{F}}_{2}F2​ is discussed.
The renormalization-group equations for critical behavior in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions are generalized to the discussion of dynamic exponents at T=TcT={T}_{c}T=Tc​ and then applied to a situation recently considered by Hohenberg, Halperin, and Ma (HHM); namely, one in which an nnn-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<4n<4n<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 RRR, 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.
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.
The real-space renormalization-group method for the dynamics of critical phenomena introduced in the previous paper is applied to two calculations within the two-dimensional kinetic Ising model on a triangular lattice. In both cases a cumulant expansion in the intercell couplings is performed. In the first calculation are found the most general first-order recursion relations coupling neighboring cells. From these recursion relations is extracted the dynamic critical exponent z=1.70z=1.70z=1.70. While this is a reasonable first-order result, the results for the statics are not as satisfactory. In the second calculation it is shown how the terms that are generated at second order are important in determining the static results. We obtain at second order z=2.22z=2.22z=2.22, together with a critical coupling Kc=0.251{K}_{c}=0.251Kc​=0.251 and a thermal critical index $\ensuremath{\nu}=0.950$.
As discussed in the previous chapter, the phenomenological theory of critical phenomena finds a sound basis in the logical sequence of universality, scaling, relevant and irrelevant variables. The difficult problem of a large number of degrees of freedom strongly correlated within the coherence distance, which diverges at criticality, is thereby, in principle, reduced to the determination of the homogeneous form of the relevant response functions and of the evaluation of a few critical indices related to the relevant parameters, whereas the details specifying each system do not matter. In order to give a microscopic foundation to universality and to the scaling theory, an exact transformation is needed, which changes into a scaling transformation asymptotically near the critical point. This is the renormalization group (RG) approach, recently presented in a huge number of reviewpapers and books.Historically, the first comprehensive overview was given by the sixth volume of the series edited by Domb and Green (1976) and by the classic books by Patashinskij and Pokrovskij (1979) and by Ma (1976). Along with scaling theory, the field-theoretic renormalization group (RG) approach (Gell- Mann and Low (1954); Bogoliubov and Shirkov (1959); Bonch-Bruevich and Tyablikov (1962)) was introduced in critical phenomena (Di Castro and Jona-Lasinio (1969)) by showing how the RG equations generalize the universality relations in the sense that they relate one model system to another by varying the coupling and suitably rescaling the other variables and the correlation functions. At the same time a detailed analysis of the pertubative diagrammatic structure of the correlation functions together with the use of Ward identities yielded a renormalized theory (Migdal (1969); Polyakov (1968, 1969)). By iterating the RG transformation the coupling disappears from the equations near to the critical point, the homogeneous form of the static order-parameter correlation function is obtained and a microscopic definition of the critical indices is given. The scaling picture then acquires a theoretical basis. Wilson (Wilson (1971a, b); Wilson and Kogut (1974)) gave a great contribution to the understanding of the physics underlying the RG procedure and to the explicit calculation of the critical indices. He proposed a simple mathematical realization of the other very physical idea of universality due to Kadanoff of grouping together degrees of freedom on the scale s associated with larger and larger cells.
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.