  The elastic interaction, induced by the lattice distortion due to the
difference of the molecular size, causes an effective long-range interaction.
In spin-crossover (SC) compounds, local bistable states, i.e., high-spin and
low-spin states have different molecular sizes, and the elastic interaction is
important. In bipartite lattices, e.g., the square lattice, the ground state
can be two types of phases: ferromagnetic-like (FM) and antiferromagnetic-like
(AFM) phases. In systems like SC compounds, the former phase consists of all
small or large molecules, and the latter phase has the configuration of
alternating small and large molecules. In fact, both cases are observed in SC
systems. In this paper, we have studied the effect of the short-range
interaction in the elastic system on the properties of those order-disorder
phase transitions. We have obtained a phase diagram in the coordinates of the
temperature and the strength of the short-range interaction, including the
metastable structures. We show that effects of the short-range interaction are
essentially different for FM and AFM phase transitions. In the FM transition,
the long-range interaction of elasticity is relevant, and the system exhibits a
phase transition in the mean-filed universality class. In this case, the
long-range interaction strongly enhances the FM order, and it works
cooperatively with the short-range interaction. In contrast, in the AFM
transition, the elastic interaction slightly enhances the AFM order, but
essentially it does not contribute to the ordering, and the system shows a
transition in the Ising universality class. We have found that in the border
region between FM and AFM phases, the AFM phase has an advantage at finite
temperatures. We discuss the critical properties of two-step SC transitions
with comparison between the elastic interaction model and conventional SC
models (Ising-like models).
