Cubic rare-earth compounds: Variants of the three-state Potts model

In appropriate cubic fields rare-earth ions have sixfold degenerate ground states. When the angular momentum of the rare earth is large, the six levels are characterized by states that are directed along the cube edges. Within these states the angular momentum operators $J_x$, $J_y$, and $J_z$ have particularly simple matrix representations. The projection of an isotropic pair coupling between the rare earths onto these sixfold degenerate states leads to an interaction Hamiltonian $\mathcal{H}=-\mathcal{I}\Sigma {〈\mathrm{ij}〉}^{}{\sigma }_i{\sigma }_j{\delta }_{l_i{l}_j}$, where $\sigma$ takes on the values ± 1 and $l$ the values $x$, $y$, and $z$. This interaction is a variant of the three-state Potts model. We add magnetic and quadrupolar anisotropy field terms to the Hamiltonian and determine the symmetry properties of the phase diagram associated with this model. For nonzero quadrupolar anisotropy fields, the model is shown to have the thermodynamic behavior of an Ising model. However, for zero fields we find a new symmetry appears and in the mean-field approximation the model has tricritical-like exponents. This simple model is able to account for the large specific-heat critical exponent ${\alpha }^{\prime }=\frac{1}{2}$ which has been observed for holmium antimonide in zero external fields. To the extent that the mean-field approximation is an accurate guide, we predict there are many cubic rare-earth compounds which exhibit tricritical-like behavior in zero field. In addition we find that for pure quadrupole coupling between rare earths in the sixfold degenerate states, the interaction Hamiltonian is exactly the three-state Potts model. In the mean-field approximation this system has a first-order phase transition. However, a small quadrupolar anisotropy field is sufficient to drive the system to a wing critical point. The specific heat has a critical exponent of $\alpha =\frac{2}{3} or 1$ depending on the path taken to approach this critical point.