Stability conditions for the generalized Ising model:S=1

Abstract

We consider the generalized Ising model for a spin-1 system, i.e., the most general static Hamiltonian with pair interactions in a spin-1 subspace. This Hamiltonian has previously been used to model lattice-gas systems, ternary alloys, and normal liquid mixtures, as well as other threestate systems. As special cases one obtains the Blume-Emery-Griffiths model for $^{3}\mathrm{He}$-$^{4}\mathrm{He}$ mixtures and the Blume-Capel model for singlet-ground-state systems. Here we use exact hightemperature expansions in field, to the linear term in $\ensuremath{\beta}={({k}_{B}T)}^{\ensuremath{-}1}$ for the pair-correlation functions, to construct the generalized static susceptibility. The singularity in the susceptibility as the system is lowered in temperature yields an expression for the temperature at which the system becomes unstable with respect to fluctuations in the order parameters. This stability temperature is a function of the order parameters and the interactions. Our result is equivalent to the mean-field approximation (with its incumbent limitations) and yields a surface in thermodynamic space which separates regions of stability (or metastability) and instability. We look specifically at the stability surfaces and stability temperatures for a number of special cases as well as the general result.