A new method incorporating the effects of many-body static spin correlations into an effective-field theory is discussed. The method is based on the introduction of a differential operator and the concept of the correlated effective field into two exact Callen identities of the Ising model. The resulting statistical theory is shown to have an accuracy equivalent to that of the Bethe-Peierls method. It is shown that the correlated-effective-field parameter at the transition temperature has a universal function of $\frac{1}{(z\ensuremath{-}1)}$, where
z is the number of nearest neighbors.
A new type of effective-field theory that has recently been used with success for many applications concerning the Ising model is, through the generalization of Callen identities, herein extended to the
q-state Potts model. Although mathematically simple, it yields results quite superior to those currently obtained within the molecular-field approximation. In order to test its reliability we check the following properties: (a) The critical temperature
Tc associated with a linear chain vanishes for all
q, (b) the value of
Tc associated with a
z-fold-coordinated lattice (
z>2) exhibits a qualitatively (and, to a certain extent, quantitatively) satisfactory
q dependence, (c) the $z\ensuremath{\rightarrow}\ensuremath{\infty}$ limit reproduces the exact value for
Tc, and (d) for fixed
z, all values of
q>2 provide first-order phase transitions, which are exact for
d>4. Furthermore, the procedures for obtaining, as functions of temperature and for any values of
q and
z, the order parameter, internal energy, and specific heat are outlined, and some typical illustrations are presented.
Within a new type of correlated-effective-field theory we calculate the effective-field parameters (including their response to external field), critical temperatures, susceptibility, short-range order parameters, and specific heat associated with an anisotropic, square-lattice, spin-\textonehalf{} Ising ferromagnet. The present formalism, which is shown to recover the Bethe-Peierls thermodynamics for the pure
Z-coordinated-lattice, spin-\textonehalf{} Ising model, yields satisfactory results; in particular, the crossover from a pure Ising ferromagnet on a square lattice to that on a linear chain is reasonably exhibited in the behavior of the susceptibility and the specific heat.
The partition function of the Ising model with
N spins, written as a functional integral, is approximated by replacing the field ${\ensuremath{\psi}}_{l}$ at the lattice site
l by the root-mean-square value of these fields, and evaluating the integral by a saddle-point method. A phase transition, whose mechanism is mathematically similar to that of the spherical model, is predicted at a temperature lower than the Weiss critical temperature. After parametrizing the mean-square field, one may apply a general variational principle to choose the parameters which minimize the difference between the approximate free energy and the exact Ising-model free energy. This optimum choice leads to the Horwitz-Callen theory.