arXiv:2512.04805

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

Xiao, Tianning, Liu, Ziyu, Fan, Zhijie, Youjin Deng

Published Dec 4, 2025• 0 citations

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 $σ_* = 2$.

DOI: 10.48550/arxiv.2512.04805

Related papers

Two-dimensional percolation model with long-range interaction

Ziyu Liu, Tianning Xiao, Zhijie Fan +1 more

arXiv:2509.18035Sep 22, 20250 citations

Analysis of the low-temperature phase in the two-dimensional long-range diluted XY model

Cescatti Fabiana, Miguel Ibáñez-Berganza, A. Vezzani +1 more

arXiv:1905.06688Physical review. B./Physical review. BAug 22, 201914 citations

Spontaneous Symmetry Breaking in Two-dimensional Long-range Heisenberg Model

Tianning Xiao, Dingyun Yao, Polle, Lode +2 more

arXiv:2512.01956Dec 1, 20250 citations

The $6-ε$ Expansion for Long-Range Lee--Yang and Percolation Criticality

Zhiyi Li, Kun Chen, Zhijie Fan +1 more

arXiv:2608.15120Aug 15, 20260 citations

Two-dimensional percolation with algebraically decaying interactions II: Critical exponents in the long-range regime

Ziyu Liu, Tianning Xiao, Zhijie Fan +1 more

arXiv:2608.20750Aug 21, 20260 citations

Universality Diagram of Phase Transitions in Long-range Statistical Systems

Tianning Xiao, Zhijie Fan, Youjin Deng

arXiv:2512.02948Dec 2, 20250 citations

arXiv:2512.04805

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

Xiao, Tianning, Liu, Ziyu, Fan, Zhijie, Youjin Deng

Published Dec 4, 2025• 0 citations

arXiv HTML PDF

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

DOI: 10.48550/arxiv.2512.04805

Related papers

Two-dimensional percolation model with long-range interaction

Ziyu Liu, Tianning Xiao, Zhijie Fan +1 more

We perform large-scale simulations of the two-dimensional long-range bond percolation model with algebraically decaying percolation probabilities $\sim 1/r^{2+σ}$ , using both conventional ensemble and event-based ensemble methods for system sizes up to $L=16384$ . We accurately determine the critical points, the universal values of several dimensionless quantities, and the corresponding critical exponents. Our results provide compelling evidence that the system undergoes a crossover from short-range to long-range universality at $σ= 2$ , in contradiction to Sak's criterion. Notably, we observe a pronounced jump in the universal values and critical exponents at $σ= 2$ , a feature absent from previous studies.We perform large-scale simulations of the two-dimensional long-range bond percolation model with algebraically decaying percolation probabilities $\sim 1/r^{2+σ}$, using both conventional ensemble and event-based ensemble methods for system sizes up to $L=16384$. We accurately determine the critical points, the universal values of several dimensionless quantities, and the corresponding critical exponents. Our results provide compelling evidence that the system undergoes a crossover from short-range to long-range universality at $σ= 2$, in contradiction to Sak's criterion. Notably, we observe a pronounced jump in the universal values and critical exponents at $σ= 2$, a feature absent from previous studies.

arXiv:2509.18035Sep 22, 20250 citations

Analysis of the low-temperature phase in the two-dimensional long-range diluted XY model

Cescatti Fabiana, Miguel Ibáñez-Berganza, A. Vezzani +1 more

The critical behavior of statistical models with long-range interactions exhibits distinct regimes as a function of $\ensuremath{\rho}$, the power of the interaction strength decay. For large enough $\ensuremath{\rho}, \ensuremath{\rho}>{\ensuremath{\rho}}_{\mathrm{sr}}$, the critical behavior is observed to coincide with that of the short-range model. However, there are controversial aspects regarding this picture, one of which is the value of the short-range threshold ${\ensuremath{\rho}}_{\mathrm{sr}}$ in the case of the long-range XY model in two dimensions. We study the 2D XY model on the diluted graph, a sparse graph obtained from the 2D lattice by rewiring links with probability decaying with the Euclidean distance of the lattice as ${|r|}^{\ensuremath{-}\ensuremath{\rho}}$, which is expected to feature the same critical behavior of the long-range model. Through Monte Carlo sampling and finite-size analysis of the spontaneous magnetization and of the Binder cumulant, we present numerical evidence that ${\ensuremath{\rho}}_{\mathrm{sr}}=4$. According to such a result, one expects the model to belong to the Berezinskii-Kosterlitz-Thouless universality class for $\ensuremath{\rho}\ensuremath{\ge}4$, and to present a second-order transition for $\ensuremath{\rho}<4$.The critical behavior of statistical models with long-range interactions exhibits distinct regimes as a function of $\ensuremath{\rho}$, the power of the interaction strength decay. For large enough $\ensuremath{\rho}, \ensuremath{\rho}>{\ensuremath{\rho}}_{\mathrm{sr}}$, the critical behavior is observed to coincide with that of the short-range model. However, there are controversial aspects regarding this picture, one of which is the value of the short-range threshold ${\ensuremath{\rho}}_{\mathrm{sr}}$ in the case of the long-range XY model in two dimensions. We study the 2D XY model on the diluted graph, a sparse graph obtained from the 2D lattice by rewiring links with probability decaying with the Euclidean distance of the lattice as ${|r|}^{\ensuremath{-}\ensuremath{\rho}}$, which is expected to feature the same critical behavior of the long-range model. Through Monte Carlo sampling and finite-size analysis of the spontaneous magnetization and of the Binder cumulant, we present numerical evidence that ${\ensuremath{\rho}}_{\mathrm{sr}}=4$. According to such a result, one expects the model to belong to the Berezinskii-Kosterlitz-Thouless universality class for $\ensuremath{\rho}\ensuremath{\ge}4$, and to present a second-order transition for $\ensuremath{\rho}<4$.

arXiv:1905.06688Physical review. B./Physical review. BAug 22, 201914 citations

Spontaneous Symmetry Breaking in Two-dimensional Long-range Heisenberg Model

Tianning Xiao, Dingyun Yao, Polle, Lode +2 more

Algebraically decaying interactions $\sim 1/r^{d+σ}$ can lead to nontrivial universality beyond short-range (SR) theories and spontaneous symmetry breaking in low-dimensional systems. We perform large-scale Monte Carlo simulations for the classical long-range (LR) Heisenberg model in two dimensions (2D) up to linear size $L=8192$ . We show that the system enters a long-range-ordered phase through a single continuous phase transition for all $σ\leq 2$ , including the marginal case $σ=2$ . In contrast, for $σ> 2$ it recovers the SR asymptotically free behavior with no finite-temperature transition. This places the LR--SR crossover threshold at $σ_* = 2$ . To characterize the ordered phase, we introduce an LR simple random walk with a fixed total length $\mathcal{L} \sim\mathcal{O}(L^d)$ . This fixed- $\mathcal L$ walk reproduces the finite-size scaling of the Goldstone-mode fluctuations in the LR Heisenberg model in both two and three dimensions, including the logarithmic scaling at $σ= 2$ . These results further motivate a general criterion for the existence of finite-temperature long-range order in LR systems with continuous symmetry in any spatial dimension.Algebraically decaying interactions $\sim 1/r^{d+σ}$ can lead to nontrivial universality beyond short-range (SR) theories and spontaneous symmetry breaking in low-dimensional systems. We perform large-scale Monte Carlo simulations for the classical long-range (LR) Heisenberg model in two dimensions (2D) up to linear size $L=8192$. We show that the system enters a long-range-ordered phase through a single continuous phase transition for all $σ\leq 2$, including the marginal case $σ=2$. In contrast, for $σ> 2$ it recovers the SR asymptotically free behavior with no finite-temperature transition. This places the LR--SR crossover threshold at $σ_* = 2$. To characterize the ordered phase, we introduce an LR simple random walk with a fixed total length $\mathcal{L} \sim\mathcal{O}(L^d)$. This fixed-$\mathcal L$ walk reproduces the finite-size scaling of the Goldstone-mode fluctuations in the LR Heisenberg model in both two and three dimensions, including the logarithmic scaling at $σ= 2$. These results further motivate a general criterion for the existence of finite-temperature long-range order in LR systems with continuous symmetry in any spatial dimension.

arXiv:2512.01956Dec 1, 20250 citations

The $6-ε$ Expansion for Long-Range Lee--Yang and Percolation CriticalityThe $6-ε$ Expansion for Long-Range Lee--Yang and Percolation Criticality

Zhiyi Li, Kun Chen, Zhijie Fan +1 more

The crossover from long-range (LR) to short-range (SR) criticality in percolation has remained unsettled because previous renormalization-group (RG) analysis within the $ε'=3σ-d$ expansion fixes the anomalous dimension at $η=2-σ$ , whereas SR percolation has $η_{\rm SR}<0$ near $d=6$ . Sak's matching condition then places the crossover above $σ=2$ , outside the regime in which the LR interaction dominates. In spatial dimension $d=6-ε$ , we formulate a perturbative expansion for the LR $ϕ^3$ field theory and perform a one-loop RG analysis throughout the perturbatively accessible nonclassical regime $0<δ<ε/3$ , where $δ= 2-σ$ . We derive the one-loop corrections to the critical exponents $η$ and $ν$ , which acquire nontrivial dependence on $ε$ and $δ$ . They reduce to their mean-field values at the LR upper critical line and continuously recover the SR $6-ε$ results as $σ\to2$ . These results support a crossover threshold $σ_*=2$ and remove the apparent discontinuity of $η$ between the LR and SR values within this framework. The same approach also yields the anomalous and edge exponents of the LR Lee--Yang universality class and $q$ -state Potts universality classes with $q<2$ .The crossover from long-range (LR) to short-range (SR) criticality in percolation has remained unsettled because previous renormalization-group (RG) analysis within the $ε'=3σ-d$ expansion fixes the anomalous dimension at $η=2-σ$, whereas SR percolation has $η_{\rm SR}<0$ near $d=6$. Sak's matching condition then places the crossover above $σ=2$, outside the regime in which the LR interaction dominates. In spatial dimension $d=6-ε$, we formulate a perturbative expansion for the LR $ϕ^3$ field theory and perform a one-loop RG analysis throughout the perturbatively accessible nonclassical regime $0<δ<ε/3$, where $δ= 2-σ$. We derive the one-loop corrections to the critical exponents $η$ and $ν$, which acquire nontrivial dependence on $ε$ and $δ$. They reduce to their mean-field values at the LR upper critical line and continuously recover the SR $6-ε$ results as $σ\to2$. These results support a crossover threshold $σ_*=2$ and remove the apparent discontinuity of $η$ between the LR and SR values within this framework. The same approach also yields the anomalous and edge exponents of the LR Lee--Yang universality class and $q$-state Potts universality classes with $q<2$.

arXiv:2608.15120Aug 15, 20260 citations

Two-dimensional percolation with algebraically decaying interactions II: Critical exponents in the long-range regime

Ziyu Liu, Tianning Xiao, Zhijie Fan +1 more

We present a comprehensive Monte Carlo study of two-dimensional bond percolation with algebraically decaying connection probabilities $p(r)\propto 1/r^{2+σ}$ , establishing the universality diagram in the long-range (LR) regime for $σ\le2$ . Using the event-based ensemble method, we simulate systems with linear sizes up to $L=16384$ and investigate three universality regimes: LR Wilson--Fisher (WF) A ( $1<σ\le2$ ), LR Wilson--Fisher B ( $2/3<σ\le1$ ), and LR mean-field (MF) ( $0<σ\le2/3$ ). In the LR-WF-B regime, the anomalous dimension is consistent with $η=2-σ$ , in agreement with mathematical results for $2/3<σ<1$ , while the correlation-length exponent $ν(σ)$ exhibits nontrivial, non-Gaussian variation. In the LR-WF-A regime, although $η$ remains close to $2-σ$ for smaller $σ$ , statistically resolvable deviations $δη(σ)=η-(2-σ)>0$ start to appear near $σ\simeq3/2$ and grow toward the short-range crossover at $σ=2$ . Finally, by complementing the event-based simulations with conventional ensemble simulations, we reveal the coexistence of complete-graph asymptotics and LR Gaussian-fixed-point scaling in the LR-MF regime. These results further clarify the critical properties in long-range percolation and provide crucial benchmarks for long-range statistical systems.We present a comprehensive Monte Carlo study of two-dimensional bond percolation with algebraically decaying connection probabilities $p(r)\propto 1/r^{2+σ}$, establishing the universality diagram in the long-range (LR) regime for $σ\le2$. Using the event-based ensemble method, we simulate systems with linear sizes up to $L=16384$ and investigate three universality regimes: LR Wilson--Fisher (WF) A ($1<σ\le2$), LR Wilson--Fisher B ($2/3<σ\le1$), and LR mean-field (MF) ($0<σ\le2/3$). In the LR-WF-B regime, the anomalous dimension is consistent with $η=2-σ$, in agreement with mathematical results for $2/3<σ<1$, while the correlation-length exponent $ν(σ)$ exhibits nontrivial, non-Gaussian variation. In the LR-WF-A regime, although $η$ remains close to $2-σ$ for smaller $σ$, statistically resolvable deviations $δη(σ)=η-(2-σ)>0$ start to appear near $σ\simeq3/2$ and grow toward the short-range crossover at $σ=2$. Finally, by complementing the event-based simulations with conventional ensemble simulations, we reveal the coexistence of complete-graph asymptotics and LR Gaussian-fixed-point scaling in the LR-MF regime. These results further clarify the critical properties in long-range percolation and provide crucial benchmarks for long-range statistical systems.

arXiv:2608.20750Aug 21, 20260 citations

Universality Diagram of Phase Transitions in Long-range Statistical Systems

Tianning Xiao, Zhijie Fan, Youjin Deng

The percolation, Ising, and O( $n$ ) models constitute fundamental systems in statistical and condensed matter physics. For short-range-interacting cases, the nature of their phase transitions is well established by renormalization-group theory. However, the universality of the transitions in these models remains elusive when algebraically decaying long-range interactions $\sim 1/r^{d+σ}$ are introduced, where $d$ is the dimensionality and $σ$ is the decay exponent. Building upon insights from Lévy flight, i.e., long-range simple random walk, we propose three universality diagrams in the $(d,σ)$ plane for the percolation model, the O( $n$ ) model, and the Fortuin-Kasteleyn Ising model, respectively. The conjectured universality diagrams are consistent with recent high-precision numerical studies and rigorous mathematical results, offering a unified perspective on critical phenomena in systems with long-range interactions.The percolation, Ising, and O($n$) models constitute fundamental systems in statistical and condensed matter physics. For short-range-interacting cases, the nature of their phase transitions is well established by renormalization-group theory. However, the universality of the transitions in these models remains elusive when algebraically decaying long-range interactions $\sim 1/r^{d+σ}$ are introduced, where $d$ is the dimensionality and $σ$ is the decay exponent. Building upon insights from Lévy flight, i.e., long-range simple random walk, we propose three universality diagrams in the $(d,σ)$ plane for the percolation model, the O($n$) model, and the Fortuin-Kasteleyn Ising model, respectively. The conjectured universality diagrams are consistent with recent high-precision numerical studies and rigorous mathematical results, offering a unified perspective on critical phenomena in systems with long-range interactions.

arXiv:2512.02948Dec 2, 20250 citations

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}}

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

We perform large-scale simulations of the two-dimensional long-range bond percolation model with algebraically decaying percolation probabilities

\sim 1/r^{2+σ}

, using both conventional ensemble and event-based ensemble methods for system sizes up to

L=16384

. We accurately determine the critical points, the universal values of several dimensionless quantities, and the corresponding critical exponents. Our results provide compelling evidence that the system undergoes a crossover from short-range to long-range universality at

σ= 2

, in contradiction to Sak's criterion. Notably, we observe a pronounced jump in the universal values and critical exponents at

σ= 2

, a feature absent from previous studies.

Algebraically decaying interactions

\sim 1/r^{d+σ}

can lead to nontrivial universality beyond short-range (SR) theories and spontaneous symmetry breaking in low-dimensional systems. We perform large-scale Monte Carlo simulations for the classical long-range (LR) Heisenberg model in two dimensions (2D) up to linear size

L=8192

. We show that the system enters a long-range-ordered phase through a single continuous phase transition for all

σ\leq 2

, including the marginal case

σ=2

. In contrast, for

σ> 2

it recovers the SR asymptotically free behavior with no finite-temperature transition. This places the LR--SR crossover threshold at

σ_* = 2

. To characterize the ordered phase, we introduce an LR simple random walk with a fixed total length

\mathcal{L} \sim\mathcal{O}(L^d)

. This fixed-

\mathcal L

walk reproduces the finite-size scaling of the Goldstone-mode fluctuations in the LR Heisenberg model in both two and three dimensions, including the logarithmic scaling at

σ= 2

. These results further motivate a general criterion for the existence of finite-temperature long-range order in LR systems with continuous symmetry in any spatial dimension.

The

6-ε

Expansion for Long-Range Lee--Yang and Percolation Criticality

The crossover from long-range (LR) to short-range (SR) criticality in percolation has remained unsettled because previous renormalization-group (RG) analysis within the

ε'=3σ-d

expansion fixes the anomalous dimension at

η=2-σ

, whereas SR percolation has

η_{\rm SR}<0

near

d=6

. Sak's matching condition then places the crossover above

σ=2

, outside the regime in which the LR interaction dominates. In spatial dimension

d=6-ε

, we formulate a perturbative expansion for the LR

ϕ^3

field theory and perform a one-loop RG analysis throughout the perturbatively accessible nonclassical regime

0<δ<ε/3

, where

δ= 2-σ

. We derive the one-loop corrections to the critical exponents

η

and

ν

, which acquire nontrivial dependence on

ε

and

δ

. They reduce to their mean-field values at the LR upper critical line and continuously recover the SR

6-ε

results as

σ\to2

. These results support a crossover threshold

σ_*=2

and remove the apparent discontinuity of

η

between the LR and SR values within this framework. The same approach also yields the anomalous and edge exponents of the LR Lee--Yang universality class and

q

-state Potts universality classes with

q<2

We present a comprehensive Monte Carlo study of two-dimensional bond percolation with algebraically decaying connection probabilities

p(r)\propto 1/r^{2+σ}

, establishing the universality diagram in the long-range (LR) regime for

σ\le2

. Using the event-based ensemble method, we simulate systems with linear sizes up to

L=16384

and investigate three universality regimes: LR Wilson--Fisher (WF) A (

1<σ\le2

), LR Wilson--Fisher B (

2/3<σ\le1

), and LR mean-field (MF) (

0<σ\le2/3

). In the LR-WF-B regime, the anomalous dimension is consistent with

η=2-σ

, in agreement with mathematical results for

2/3<σ<1

, while the correlation-length exponent

ν(σ)

exhibits nontrivial, non-Gaussian variation. In the LR-WF-A regime, although

η

remains close to

2-σ

for smaller

σ

, statistically resolvable deviations

δη(σ)=η-(2-σ)>0

start to appear near

σ\simeq3/2

and grow toward the short-range crossover at

σ=2

. Finally, by complementing the event-based simulations with conventional ensemble simulations, we reveal the coexistence of complete-graph asymptotics and LR Gaussian-fixed-point scaling in the LR-MF regime. These results further clarify the critical properties in long-range percolation and provide crucial benchmarks for long-range statistical systems.

The percolation, Ising, and O(

n

) models constitute fundamental systems in statistical and condensed matter physics. For short-range-interacting cases, the nature of their phase transitions is well established by renormalization-group theory. However, the universality of the transitions in these models remains elusive when algebraically decaying long-range interactions

\sim 1/r^{d+σ}

are introduced, where

d

is the dimensionality and

σ

is the decay exponent. Building upon insights from Lévy flight, i.e., long-range simple random walk, we propose three universality diagrams in the

(d,σ)

plane for the percolation model, the O(

n

) model, and the Fortuin-Kasteleyn Ising model, respectively. The conjectured universality diagrams are consistent with recent high-precision numerical studies and rigorous mathematical results, offering a unified perspective on critical phenomena in systems with long-range interactions.