On the equivalence of different Landau expansions

The critical behavior of magnetic model systems in the mean-field approximation may be obtained from a standard Landau expansion, in terms of the order parameter m, of the functional ψ(T,H; m) = −Hm + f(T,m), where f is the Helmholtz free energy. As an alternative procedure, the Gibbs free energy, g(T,H), may be written as the minimum of a functional g(T,H; y) with respect to a continous spin field y, and the critical behavior can be analysed on the basis of an expansion of this functional in powers of y. We show that, in spite of displaying quite distinct coefficients, in zero ordering field, both expansions lead to the same critical and tricritical behavior. We obtain relationships between these sets of coefficients for a rather general case and discuss, as an illustration, the simple Ising ferromagnet. We also remark that these distinctions are already present in different strategies to obtain a suitable Landau-Ginzburg-Wilson Hamiltonian for performing a momentum-space renormalization-group calculation.