  Phase diagrams and hysteresis loops were obtained by Monte Carlo simulations
and a mean-field method for a simplified model of a spin-crossover material
with a two-step transition between the high-spin and low-spin states. This
model is a mapping onto a square-lattice $S=1/2$ Ising model with
antiferromagnetic nearest-neighbor and ferromagnetic Husimi-Temperley
(equivalent-neighbor) long-range interactions. Phase diagrams obtained by the
two methods for weak and strong long-range interactions are found to be
similar. However, for intermediate-strength long-range interactions, the Monte
Carlo simulations show that tricritical points decompose into pairs of critical
endpoints and mean-field critical points surrounded by horn-shaped regions of
metastability. Hysteresis loops along paths traversing the horn regions are
strongly reminiscent of thermal two-step transition loops with hysteresis,
recently observed experimentally in several spin-crossover materials. We
believe analogous phenomena should be observable in experiments and simulations
for many systems that exhibit competition between local antiferromagnetic-like
interactions and long-range ferromagnetic-like interactions caused by elastic
distortions.

Materials Science;Statistical Mechanics