Alternative variational approach to cactus lattices

In this paper, I will present an alternative approach to the Bethe or cactus lattice approximation, widely employed in the theory of cooperative phenomena. This approach relies on a variational free energy, which is equivalent to the Bethe free energy in that it has the same stationary points, but allows one to simplify analytical calculations, since it is a function of only single-site probability distributions, in the same way as an ordinary mean-field (Bragg-Williams) free energy. As an application, I shall discuss a derivation of closed-form equations for critical points in Ising-like models. Moreover, I will suggest a rule of thumb to choose the cactus lattice connectivity yielding the best approximation for the corresponding model defined on an ordinary lattice.