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

TL;DR

This work revisits the longstanding question of the threshold separating long-range and short-range critical behavior in 2D systems with interactions decaying as $1/r^{2+σ}$. Using high-precision Monte Carlo simulations of the 2D LR-Ising model up to $L=8192$ and leveraging Fortuin–Kasteyn–Pollack observables like the critical polynomial $R_p$, Binder ratio $Q_m$, and the anomalous dimension $η$, the authors demonstrate a sharp LR–SR crossover located at $σ=2$, contradicting Sak's criterion. The study also analyzes η and $ν$ (via $y_t$) across LR and SR regimes, finding a discontinuity in $η$ at $σ=2$ while $y_t$ remains near unity; similar LR–SR boundary behavior is observed in XY, Heisenberg, percolation, and UF models, supporting a unified LR universality scenario. The results challenge Sak's criterion and motivate a revised theoretical framework for LR criticality in 2D, with implications for experimental platforms and future RG analyses.

Abstract

Determining the threshold value $σ_*$ that separates the short-range (SR) and long-range (LR) universality classes in phase transitions remains a controversial issue. While Sak's criterion, $σ_* = 2 - η_{\mathrm{SR}}$, 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 $1/r^{2+σ}$. Our previous simulations for the XY, Heisenberg, and percolation models consistently indicate a universal boundary at $σ_* = 2$. 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 $η$, we obtain convergent and self-consistent evidence that the universality class already changes sharply at $σ= 2$. 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 $σ_* = 2$.

Figures (11)

• Figure 1: Sketch of the anomalous dimension $\eta$ as a function of $\sigma$ for four scenarios. For $\sigma > 2$, it is uncontroversial that the value belongs to the SR case, as indicated by the black solid line. As for $\sigma \leq 2$, (i) the prediction by Fisher, $\eta = 2 - \sigma$fisher1972, is shown with a sparse dashed line. A jump in $\eta$ occurs at $\sigma = 2$, marking the transition point of the universality class at $\sigma_* = 2$. (ii) Sak's criterion, $\eta = \max(2 - \sigma, \eta_{\rm SR})$sak1973, is shown with a dense dashed line. Here the transition point of the universality class is at $\sigma_* = 2 - \eta_{\rm SR}$ (for the Ising model, $\eta_{\rm SR} = 1/4$). (iii) Picco proposed a third scenario based on his numerical results picco2012, where the transition point is at $\sigma_* = 2$, smoothly connecting to the short-range region, shown by the blue solid line. (iv) We propose a fourth scenario, where although the curve smoothly connects to $\sigma = 2$, there is still a jump at $\sigma = 2$, indicating that the nature of the phase transition at $\sigma = 2$ is already different from the SR case, shown by the red solid line.
• Figure 2: Phase diagrams of the 2D O($n$) spin models: Ising ($n=1$), XY ($n=2$) cpl_42_7_070002yao2024nonclassical, and Heisenberg ($n=3$) 2dlrheisen, with long-range (LR) interactions decaying as $1/r^{2+\sigma}$. In all panels, $T_c^{\text{CG}} = 4/n$ denotes the complete-graph critical temperature kirkpatrick2017 under the normalization in Eq. \ref{['eq:norm']}, and $T_c^{\text{NN}}$ indicates the NN limit PhysRev.65.117komura2012a. The classical regime ($\sigma \leq 1$, purple line) follows mean-field behavior. (a) For the Ising case, the system exhibits a second-order transition throughout. The universality class changes at $1 < \sigma \leq 2$ (red line), and for $\sigma > 2$ (green line), it recovers the SR Ising class. The dashed line indicates that the geometric properties of the low-temperature phase differ between $\sigma < 2$ and $\sigma > 2$, in the FK representation of the Ising model. (b) For the XY model, the SR regime ($\sigma > 2$, brown line) undergoes a Bereinzinskii-Koterlitz-Thouless (BKT) topological phase transition into a quasi-long-range-order (QLRO) phase, while for $1 < \sigma \leq 2$, a direct second-order transition into an LRO phase appears. (c) For the Heisenberg model, finite-temperature transitions exist only for $\sigma \leq 2$. All three models consistently exhibit a universality boundary at $\sigma_* = 2$.
• Figure 3: Extrapolation of the crossing points $\beta_L$ for $R_p$ (blue) and $Q_m$ (red) at different $\sigma$. Both quantities yield consistent estimates of the critical inverse temperature $\beta_c$ for $\sigma = 1.75$ (a), $1.875$ (b), and $2$ (c). The horizontal axis represents the correction term $L^{-\omega}$, with $\omega = 1.34$ for $R_p$ and $1.29$ for $Q_m$ in panel (a), $\omega = 1.54$ and $1.59$ in (b), and $\omega = 1.48$ and $1.46$ in (c). The shaded bands indicate the extrapolated $\beta_c$ as $L \to \infty$.
• Figure 4: Extrapolation of the crossing points of $R_p$ and $Q_m$ for different system sizes. The data points represent the crossing value $\mathcal{O}_L$ between $(L/2, L)$, fitted using Eq. (\ref{['eq:universal_extrap']}). Deviations from the SR universal values, 0 for the critical polynomial $R_p$ and 0.856216(1) for the Binder ratio $Q_m$, are observed for $\sigma \leq 2$. The gray bands indicate the converged values obtained from extrapolation.
• Figure 5: Extrapolated universal values of $R_p$ and $Q_m$ as functions of $\sigma$. The black lines denote the SR universal values ($R_p = 0$, $Q_m = 0.856216$). A clear jump is observed at $\sigma = 2$ for $R_p$, marking the crossover from SR to LR universality. For $\sigma < 2$, both quantities continuously decrease from the SR values, while for $\sigma > 2$, they remain consistent with the SR values. Previous studies Luijten2002horita2017 are also shown for comparison.