Phase diagram for the cubic model in the Kikuchi approximation

Abstract

A variant of the three-state Potts model known as the cubic model has been used to describe the critical behavior of some rare-earth pnictides. In the mean-field (MF) approximation this model has a tricritical-like phase transition, while in the Bethe-Peierls-Weiss (BPW) approximation the transition is first order. To obtain a better estimate of the phase diagram of this model we have used the Kikuchi cluster variational method with a tetrahedron as the basic cluster. By using the symmetry properties of the cubic model we were able to reduce the 36 coupled equations with 1296 variational parameters to 12 coupled equations and 58 distinct variational parameters. We were able to solve these equations and find the surfaces in the phase diagram where transitions occurred. The qualitative features of the phase diagram for the cubic model do not change too much in going from the BPW to the Kikuchi approximation. However, there are quantitative changes which become large for certain points and along symmetry directions in the phase diagram, i.e., for those values of the anisotropy ($\stackrel{\ensuremath{\rightarrow}}{\mathrm{D}}$) for which several components of the spin become simultaneously critical. For these points the Kikuchi tetrahedron approximation takes much better account of the correlations between spins than the MF and BPW approximations. It follows that the estimates of the critical fields ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{D}}}_{c}$ and quadrupolar coupling ${\ensuremath{\eta}}_{c}$ necessary to drive the transition tricritical are substantially better in the Kikuchi tetrahedron approximation. At the origin $\stackrel{\ensuremath{\rightarrow}}{\mathrm{D}}=0$ we find a first-order phase transition occurs at a temperature 13% lower than that found with the MF approximation and 5% lower than the BPW approximation.