Multirange Ising model on the square lattice

Abstract

We study the Ising model on the square lattice $\left({\mathbb{Z}}^2\right)$ and show, via numerical simulation, that allowing interactions between spins separated by distances 1 and $m$ (two ranges), the critical temperature, $T_c\left(m\right)$, converges monotonically to the critical temperature of the Ising model on ${\mathbb{Z}}^4$ as $m\to \infty$. Only interactions between spins located in directions parallel to each coordinate axis are considered. We also simulated the model with interactions between spins at distances of 1, $m$, and $u$ (three ranges), with $u$ a multiple of $m$; in this case our results indicate that $T_c(m,u)$ converges to the critical temperature of the model on ${\mathbb{Z}}^6$. For percolation, analogous results were proven for the critical probability $p_c$ [B. N. B. de Lima, R. P. Sanchis, and R. W. C. Silva, Stochast. Process. Appl. 121, 2043 (2011)].