  Recently, it has been found that an effective long-range interaction is
realized among local bistable variables (spins) in systems where the elastic
interaction causes ordering of the spins. In such systems, generally we expect
both long-range and short-range interactions to exist. In the short-range Ising
model, the correlation length diverges at the critical point. In contrast, in
the long-range interacting model the spin configuration is always uniform and
the correlation length is zero. As long as a system has non-zero long-range
interactions, it shows criticality in the mean-field universality class, and
the spin configuration is uniform beyond a certain scale. Here we study the
crossover from the pure short-range interacting model to the long-range
interacting model. We investigate the infinite-range model (Husimi-Temperley
model) as a prototype of this competition, and we study how the critical
temperature changes as a function of the strength of the long-range
interaction. This model can also be interpreted as an approximation for the
Ising model on a small-world network. We derive a formula for the critical
temperature as a function of the strength of the long-range interaction. We
also propose a scaling form for the spin correlation length at the critical
point, which is finite as long as the long-range interaction is included,
though it diverges in the limit of the pure short-range model. These properties
are confirmed by extensive Monte Carlo simulations.
