Multirange Ising model on the square lattice

TL;DR

This paper investigates how adding multiple interaction ranges in the 2D Ising model affects the critical temperature, showing it converges to higher-dimensional Ising models as the interaction range increases.

Contribution

The study demonstrates that multi-range interactions in the 2D Ising model cause the critical temperature to approach that of higher-dimensional models, revealing a connection between interaction range and effective dimensionality.

Findings

Critical temperature converges to that of 4D Ising model with two-range interactions.

Critical temperature converges to that of 6D Ising model with three-range interactions.

Numerical simulations support the monotonic convergence of critical temperature with increasing interaction ranges.

Abstract

We study the Ising model on $\mathbb{Z}^{2}$ and show, via numerical simulation, that allowing interactions between spins separated by distances $1$ and $m$ (two ranges), the critical temperature, $ T_c (m) $, 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, Stochastic Process. Appl. {\bf 121}, 2043 (2011)].