Dipole and Quadrupole Phase Transitions in Spin-1 Models

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Abstract
A spin-1 Hamiltonian with arbitrary bilinear and biquadratic pair interactions has been studied in the molecular-field approximation by using one- and two-sublattice models. Various types of orderings and transitions are found for the Hamiltonians with different symmetries. For a system described by a one-sublattice Hamiltonian with Ising, isotropic, or cubic symmetry, only one phase transition is found; either from the paramagnetic phase to a ferromagnetic or to a ferroquadrupolar phase. (Parallel alignment of quadrupoles is denoted ferroquadrupolar.) For the one-sublattice Hamiltonian with axial symmetry, we find two separate phase transitions; first from the paramagnetic phase to a ferroquadrupolar phase and then to a ferromagnetic phase. A two-sublattice system described by an Ising or isotropic Hamiltonian can have a transition either from the paramagnetic phase to a ferromagnetic phase, or to a ferriquadrupolar phase. Additional transitions are also found from the ferromagnetic to a ferrimagnetic phase and then back to the ferromagnetic phase. The system can have as many as three successive transitions. For a two-sublattice Hamiltonian with cubic symmetry, besides the transitions found in Ising systems, there are "reorientation" phase transitions; i.e., transitions from a quadrupole ordering along the cube edge to an ordering along the cube diagonal. This system can have more than three successive phase transitions.