Crossover between a Short-range and a Long-range Ising model

TL;DR

This paper explores the transition between short-range and long-range Ising models, deriving a formula for the critical temperature and analyzing the correlation length, confirmed through Monte Carlo simulations.

Contribution

It introduces a comprehensive analysis of the crossover from short-range to long-range interactions in Ising models, including a new formula for critical temperature and a scaling form for correlation length.

Findings

Critical temperature increases with long-range interaction strength.

Correlation length remains finite at criticality when long-range interactions are present.

Monte Carlo simulations confirm theoretical predictions.

Abstract

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.