Perturbation theory with variational parameter. Inequalities and estimates for the free energy

A perturbation theory scheme is proposed on the basis of a representation of the free energy as a sequence $F_k(\sigma_k)$ ($\sigma_k$ is the ordering parameter). From the condition of a minimum of $F_k(\sigma_k)$ an equation of state is obtained and a phase transition temperature $T_c^{(k)}$ is determined. For the Heisenberg and Ising models $F_2$ is calculated ($F_1$ is the well-known molecular-field approximation) and the inequality $F_1>F_2>F$ (for the Istng model) is obtained, which shows that $F_2$ is a better approximation than $F_1$. The temperature $T_c^{(2)}$ is also determined for both models. The behavior of the expansion for the free energy as $T\to0$ is investigated.