Critical dynamics of strong coupling paramagnetic systems exhibiting a paramagnetic–ferrimagnetic transition

The purpose of this work is the investigation of critical dynamic properties of two strongly coupled paramagnetic sublattices exhibiting a paramagnetic–ferrimagnetic transition. To go beyond the mean-field approximation, and in order to get a correct critical dynamic behavior, use is made of the renormalization-group (RG) techniques applied to a field model describing such a transition. The model is of Landau–Ginzburg type, whose free energy is a functional of two kinds of order parameters (local magnetizations) and ψ, which are scalar fields associated with these sublattices. This free energy involves, beside quadratic and quartic terms in both fields and ψ, a lowest-order coupling, −C0 ψ, where C0 is the coupling constant measuring the interaction between the two sublattices. Within the framework of mean-field theory, we first compute exactly the partial dynamic structure factors, when the temperature is changed from an initial value Ti to a final one Tf very close to the critical temperature Tc. The main conclusion is that, physics is entirely controlled by three kinds of lengths, which are the wavelength q−1, the static thermal correlation length ξ and an extra length Lt measuring the size of ordered domains at time t. Second, from the Langevin equations (with a Gaussian white noise), we derive an effective action allowing to compute the free propagators in terms of wave vector q and frequency ω. Third, through a supersymmetric formulation of this effective action and using the RG-techniques, we obtain all critical dynamic properties of the system. In particular, we derive a relationship between the relaxation time τ and the thermal correlation length ξ, i.e., τ ξz, with the exponent z=(4−η)/(2ν+1), where ν and η are the usual critical exponents of Ising-like magnetic systems. At two dimensions, we find the exact value . At three dimensions, and using the best values for exponents ν and η, we find z=1.7562±0.0027.