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Progress of Theoretical Physics

Renormalization Group Equations in Critical Dynamics. I: Isotropic Heisenberg Ferromagnet

Kyozi Kawasaki

Published Dec 1, 1975• 17 citations
Publisher

Abstract

A renormalization group equation for the inverse propagator is obtained for a stochastic model of isotropic Heisenberg ferromagnet, whose solution at a fixed point yields the dynamical scaling law with the correct dynamic critical exponent. For 6-ε dimensions (ε>0) we describe an iterative method to obtain an ε-expansion for the inverse propagator.

DOI
10.1143/ptp.54.1665

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A renormalization group equation for the inverse propagator is obtained for a stochastic model of isotropic Heisenberg ferromagnet, whose solution at a fixed point yields the dynamical scaling law with the correct dynamic critical exponent. For 6-ε dimensions (ε>0) we describe an iterative method to obtain an ε-expansion for the inverse propagator.
The critical dynamics of a model isotropic ferromagnet is studied in $6\ensuremath{-}\ensuremath{\epsilon}$ dimensions both above and below Tc{T}_{c}Tc​. The renormalization group and the characteristic frequency exponent zzz are determined to order $\ensuremath{\epsilon}$. Implications of the renormalization group are discussed. A direct calculation of the dynamic response function to order $\ensuremath{\epsilon}$ shows many interesting features, especially for T<TcT<{T}_{c}T<Tc​.
The dynamic critical behavior of isotropic Heisenberg ferromagnets with a planar free surface is investigated by means of field-theoretic renormalization group techniques and high-precision computer simulations. An appropriate semi-infinite extension of the stochastic model J is constructed. The relevant boundary terms of the action of the associated dynamic field theory are identified, the implied boundary conditions are derived, and the renormalization of the model in d<6d<6d<6 bulk dimensions is clarified. Two distinct renormalization schemes are utilized. The first is a massless one based on minimal subtraction of dimensional poles and the dimensionality expansion about d=6.d=6.d=6. To overcome its problems in going below d=4d=4d=4 dimensions, a massive one for fixed dimensions d< 4d<~4d< 4 is constructed. The resulting renormalization group (or Callan-Symanzik) equations are exploited to obtain the scaling forms of surface quantities like the dynamic structure factor. In conjunction with boundary operator expansions scaling relations follow that relate the critical indices of the dynamic and static infrared singularities of surface quantities to familiar static bulk and surface exponents. To test the predicted scaling forms and scaling-law expressions for the critical exponents involved, accurate computer-simulation data are presented for the dynamic surface structure factor. These are in conformity with our predictions.
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 qqq is the wave number and $\ensuremath{\xi}$ is the correlation length) and shows a narrow lydrodynamical width as $x\ensuremath{\rightarrow}0$.
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.