PHYSICAL REVIEW B VOLUME 12, NUMBER 1 DECEMBER 1975 Field-theoretic techniques and critical dynamics. I. Ginzburg-Landau stochastic models without energy conservation C. De Dominicis, E. Brezin, and J. Zinn-Justin Service de Physique Theorique, Centre d'Etudes Nucleaires de Saclay, BP. 2-91190 Gif-sur-Yvette, France (Received 11 March 1975) 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 e and so is the new exponent ~&, 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. I. INTRODUCTION Wilson's ' theory of static critical phenomena has been recently reformulated3 using the tech- niques of field renormalization. This approach, although less general in scope, has the advantages that capitalizing upon the known results of field- renormalization theory, yields simple expressions for the renormalization-group equations and allows a simple and complete derivation of all scaling laws, of the critical exponent expressions and cor- rections thereof (irrelevant variables of Wilson and Wegner ). We wish to show here that this ap- proach can be extended to time-dependent Landau- Qinzburg stochastic models studied by Halperin, Hohenberg, and Ma" (HHM) yielding a derivation of dynamical scaling and an expression for the dynamical exponents. We exhibit the method by considering the simplest of the dynamical models discussed by HHM and others, ' i. e. , a model which contains no external (energy) conserved field. More precisely the (complex) order param- eter y» '(f) satisfies the Landau-Ginzburg time- dependent equation aft» (t) = — f 0(i++0) "squared mass" (or a bare inverse susceptibility) and the critical region of interest is characterized by k«A, F «A, A being a cutoff and z the renor- malized "squared mass. " The space-time corre- lation function G»(t) of the order parameter is cal- culated by solving (1) for y&» ' I'0, &], averaging over g with a Qaussian weight and taking the linear de- pendence in h, 5 G, (f) = „„( )(0', "'(f)) with (q, & '(f)) =x-' d$qfd(q+] x exp — 210 dt's~ t g*„ t p~ t (4) X= d q d g* exp — 2I'0 dt's„ t g~ t (5) The use of standard renormalization techniques is difficult with the above equations, but it has been shown" that the system (1) — (3) is strictly equivalent to a standard Lagrangian system, in its classical limit, namely, we compute instead g&n)(~) (p&(M&yg&(M)) where the effective Qinzburg-Landau-Wilson Ham- iltonian is as usual, &I. V'~ P*]=Q (&o+ & )0'» ' &(&'» d q d y* expZg, q q d p d p* exp 2 p, p* with (6) +&, ' d"x g(x) y*(x)]' . (2) z(p p (=P ( ( b +o ~)( 'p q&» ' is the o(th component (o(= l, 2, .. . , n) of the order parameter of the external field and g~ ' of the Langevin noise source governed by a Qaussian probability distribution; I'0(1+Qo) is an inverse complex time scale. ' Finally, xo is a bare +~'P d'x(q (x) y* (x)') (y„, (x) y*, „, , „, (x)) . (8) 4945