New effective-field theory for the Potts model

Abstract

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 ${T}_{c}$ associated with a linear chain vanishes for all $q$, (b) the value of ${T}_{c}$ 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 ${T}_{c}$, 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.