Dynamic scaling functions and amplitude ratios of stochastic models with energy conservation above T c M. Dudka, 1,2, * R. Folk, 2,† and G. Moser 3,‡ 1 Institute for Condensed Matter Physics, National Academy of Sciences of Ukraine, 1 Svientsitskii Str., UA-79011 Lviv, Ukraine 2 Institute for Theoretical Physics, Johannes Kepler University Linz, Altenbergerstrasse 69, A-4040 Linz, Austria 3 Department for Material Research and Physics, Paris Lodron University Salzburg, Hellbrunnerstrasse 34, A-5020 Salzburg, Austria Received 9 April 2009; revised manuscript received 15 June 2009; published 17 September 2009 Dynamical scaling functions above T c for the characteristic frequencies and the dynamical correlation functions of the order parameter and the conserved density of model C are calculated in one loop order. By a proper exponentiation procedure these results can be extended in order to consider the changes in these functions using the fixed point values and exponents in two loop order. The dynamical amplitude ratio R of the characteristic frequencies is generalized to the critical region. Surprisingly the decay of the shape functions at large scaled frequency does not behave as expected from applying scaling arguments. The exponent  of the decay does not change when going from the critical to the hydrodynamic region although the shape functions change. The value of  for the order parameter is in agreement with its value in the critical region, whereas for the conserved density it is equal to 2, the value in the hydrodynamic region. DOI: 10.1103/PhysRevE.80.031124 PACS numbers: 05.70.Jk, 64.60.ae, 64.60.Ht I. INTRODUCTION Critical dynamics is described by the slow variables of a system containing the order parameter OP due to critical slowing down and the secondary densities of the conserved variables CD1. If the dynamics of these quantities are coupled the dynamical behavior of the OP is changed. The simplest case is realized when the dynamical coupling is due to a static coupling only of these slow variables, which are described by a relaxation equation for the OP and a diffusion equation for the CD. This model has been set up by Halperin et al. 2,3 and later has been called model C 1. From renormalization-group theory it is known that the static asymmetric coupling between the OP and the CD is irrel- evant if the specific heat of the system considered is not diverging. Thus, for a system with the specific-heat exponent   0 except logarithmic divergence in dynamics one re- covers model A 2. If, however, the specific heat is diverg- ing the situation is more complicated. The asymmetric cou- pling is relevant and the dynamical behavior may be characterized by strong dynamical scaling both OP and CD scale with the same time scale or weak dynamical scaling there are two different time scales, one for the OP and one for the CD depending on the dimension of space d =4-  and the number n of components of the OP for more details see Refs. 4,5. For n = 1 strong scaling is realized, whereas for n = 2 the asymptotics is described by model A, but non- asymptotic effects might be present due to a small dynamical transient exponent. The most prominent example where the model C dynam- ics and strong scaling will be realized is the anisotropic an- tiferromagnet in an external magnetic field 6,7. There exists a whole line of critical points with the z component of the staggered magnetization as OP and the z component of the magnetization as CD. The advantage of this example is also that both, the OP and the CD, are experimentally accessible. Thus the dynamical correlation function can be measured by neutron scattering and the transport coefficients, the relax- ation coefficient for the OP and the diffusion coefficient of the CD are in principle measurable. There are also other physical systems where the critical dynamical behavior is described by model C such as systems with quenched impu- rities 8; for more examples see 9. The paper is organized as follows: in the next section we summarize the results expected by the dynamical scaling theory for the scaling functions to be calculated above T c . Then the dynamical model is defined and in Sec. IV the one loop results for the correlation functions are presented. Then in Sec. V the scaling functions for the width of the OP and CD are given. These results are then used in the next section to evaluate the shape functions of the correlation functions. Section VII contains the limiting behavior of these shape functions “shape crossover” going from the critical into the hydrodynamic region above T c . Section VIII considers the dynamical amplitude ratio of model C and generalizes this expression in order to be valid in the hydrodynamic and critical region above T c . The calculations are based on the validity of strong scaling. Short remarks if weak scaling is valid are made in Sec. IX followed by the conclusions. In the Appendix details of the one loop integrals are given. II. DYNAMICAL SCALING The dynamical scaling hypothesis states that dynamical critical phenomena are described by one time scale. The time scale is defined by the wave vector dependence of the char- acteristic frequency of the OP at the phase transition. This dependence is a power law defining the dynamical critical exponent z. However this only holds in the case of strong dynamical scaling, whereas in the case of weak dynamical scaling several dynamical critical exponents for the OP, , * maxdudka@icmp.lviv.ua † folk@tphys.uni-linz.ac.at ‡ guenter.moser@sbg.ac.at PHYSICAL REVIEW E 80, 031124 2009 1539-3755/2009/803/03112415 ©2009 The American Physical Society 031124-1