On Sak's criterion for statistical models with long-range interaction

TL;DR

This study investigates the critical threshold separating long-range and short-range universality classes in 2D statistical models, providing strong evidence that this crossover occurs at  = 2 across multiple models.

Contribution

The paper offers comprehensive large-scale simulations of the 2D LR-Ising model, confirming the universality boundary at  = 2 and unifying the understanding of LR to SR crossover in various models.

Findings

Universality crossover at  = 2 confirmed for multiple models

Large-scale Monte Carlo simulations up to L=8192 support the results

Evidence suggests a sharp transition in universality class at  = 2

Abstract

Determining the threshold value $\sigma_*$ that separates the short-range (SR) and long-range (LR) universality classes in phase transitions remains a controversial issue. While Sak's criterion, $\sigma_* = 2 - \eta_{\mathrm{SR}}$, has been widely accepted, recent studies of two-dimensional (2D) models with long-range interactions have challenged it. In this work, we focus on the crossover between LR and SR criticality in several classical 2D statistical models, including the XY, Heisenberg, percolation, and Ising models, whose interactions decay as $1/r^{2+\sigma}$. Our previous simulations for the XY, Heisenberg, and percolation models consistently indicate a universal boundary at $\sigma_* = 2$. Here, we complete the picture by performing large-scale Monte Carlo simulations of the 2D LR-Ising model, reaching lattice sizes up to $L = 8192$. By analyzing the Fortuin-Kasteleyn critical polynomial $R_p$, the Binder ratio $Q_m$, and the anomalous dimension $\eta$, we obtain convergent and self-consistent evidence that the universality class already changes sharply at $\sigma = 2$. Taken together, these results establish a unified scenario for LR interacting systems: across all studied models, the crossover from LR to SR universality occurs at $\sigma_* = 2$.