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, we generally 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 nonzero 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 finite-size 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.
It has previously been pointed out that the coexistence of infinite-range and short-range interactions causes a system to have a phase transition of the mean-field universality class, in which the cluster size is finite even at the critical point. In the present paper, we study this property in a model of bistable molecules, whose size changes depending on the bistable states. The molecules can move in space, interacting via an elastic interaction. It is known that due to the different sizes, an effective long-range interaction between the spins appears, and thus this model has a mean-field type of phase transition. It is found that the scaling properties of the shift of the critical temperature from the pure short-range limit in the model with infinite-range and short-range interactions hold also in the present model, regarding the ratio of the size of the two states as a control parameter for the strength of the long-range interaction. By studying the structure factor, it is shown that the dependence of the cluster size at the critical temperature also shows the same scaling properties as a previously studied model with both infinite-range and short-range interactions. We therefore conclude that these scaling relations hold universally in hybrid models with both short-range and weak long-range interactions.
We study domain wall propagation in systems with long-range interactions caused by lattice distortion. When the local bistable states of the molecular unit have different sizes, e.g., high-spin and low-spin states in spin-crossover systems, the elastic distortion causes an effective long-range interaction. In switching processes between the ordered states in such systems, there exist two degrees of freedom for the domain wall, that is, one with respect to lattice structure and the other to a bistable spin (electronic) state. The interface width of the lattice domain wall is proportional to the system size, which reflects the macroscopic structure due to the long-range interaction, while the interface width of the spin state shows different characteristics depending on the relative time scales of dynamics of the spin state and the lattice. When the change of the spin state is relatively fast and the lattice relaxation is insufficient, the roughness exponent α is 1/2, which has been commonly found in growth models of short-range interaction systems in two dimensions. In contrast, when the lattice relaxes sufficiently, α of the spin interface is the same as that of the lattice interface, i.e., α = 1. We analyze the pressure field of the systems and find that the equilibration of the field via the long-range interaction plays a key role in the mechanism of the crossover of α between 1/2 and 1.
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