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) ϕ and ψ, which are scalar fields associated with these sublattices. This free energy involves, beside quadratic and quartic terms in both fields ϕ and ψ, a lowest-order coupling, −C0ϕψ, 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 z=54. At three dimensions, and using the best values for exponents ν and η, we find z=1.7562±0.0027.