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arXiv:2411.01811 · Physical review. B./Physical review. B

Nonclassical regime of the two-dimensional long-range XY model: A comprehensive Monte Carlo study

Dingyun Yao, Tianning Xiao, Chao Zhang, Youjin Deng, Zhijie Fan

Published Sep 24, 2025• 5 citations
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Abstract

The two-dimensional (2D) XY model plays a crucial role in statistical and condensed-matter physics. With the introduction of long-range interactions, the system exhibits a richer set of physical phenomena and a crossover between nonclassical and short-range universality classes. In this work, we investigate the 2D XY model with algebraically decaying interactions $\ensuremath{\sim}1/{r}^{2+\ensuremath{\sigma}}$ and provide a comprehensive numerical analysis of its thermodynamic properties. We demonstrate that for $\ensuremath{\sigma}\ensuremath{\le}2$, the system undergoes a second-order phase transition into a ferromagnetic phase characterized by the emergence of long-range order. In the low-temperature phase, due to the presence of the Goldstone mode, the correlation function saturates to a nonzero constant in the form of a power law for $\ensuremath{\sigma}<2$, with decaying exponent $2\ensuremath{-}\ensuremath{\sigma}$, and in the form of the inverse logarithm of distance for $\ensuremath{\sigma}=2$. Moreover, the critical points and exponents are also determined for various $\ensuremath{\sigma}$. We provide compelling evidence that the crossover between nonclassical and short-range regimes occurs at $\ensuremath{\sigma}=2$. This work presents a detailed account of the simulation methodology, extensive numerical data, and new insights into the physics of long-range interacting systems.

DOI
10.1103/yghs-p5mg

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The two-dimensional (2D) XY model plays a crucial role in statistical and condensed-matter physics. With the introduction of long-range interactions, the system exhibits a richer set of physical phenomena and a crossover between nonclassical and short-range universality classes. In this work, we investigate the 2D XY model with algebraically decaying interactions \ensuremath∼1/r2+\ensuremathσ\ensuremath{\sim}1/{r}^{2+\ensuremath{\sigma}}\ensuremath∼1/r2+\ensuremathσ and provide a comprehensive numerical analysis of its thermodynamic properties. We demonstrate that for \ensuremathσ\ensuremath≤2\ensuremath{\sigma}\ensuremath{\le}2\ensuremathσ\ensuremath≤2, the system undergoes a second-order phase transition into a ferromagnetic phase characterized by the emergence of long-range order. In the low-temperature phase, due to the presence of the Goldstone mode, the correlation function saturates to a nonzero constant in the form of a power law for \ensuremathσ<2\ensuremath{\sigma}<2\ensuremathσ<2, with decaying exponent 2\ensuremath−\ensuremathσ2\ensuremath{-}\ensuremath{\sigma}2\ensuremath−\ensuremathσ, and in the form of the inverse logarithm of distance for \ensuremathσ=2\ensuremath{\sigma}=2\ensuremathσ=2. Moreover, the critical points and exponents are also determined for various \ensuremathσ\ensuremath{\sigma}\ensuremathσ. We provide compelling evidence that the crossover between nonclassical and short-range regimes occurs at \ensuremathσ=2\ensuremath{\sigma}=2\ensuremathσ=2. This work presents a detailed account of the simulation methodology, extensive numerical data, and new insights into the physics of long-range interacting systems.
Abstract The crossover between short-range and long-range (LR) universal behaviors remains a central theme in the physics of LR interacting systems. The competition between LR coupling and the Berezinskii–Kosterlitz–Thouless mechanism makes the problem more subtle and less understood in the two-dimensional (2D) XY model, a cornerstone for investigating low-dimensional phenomena and their implications in quantum computation. We study the 2D XY model with algebraically decaying interaction ∼1/r 2+σ . Utilizing an advanced update strategy, we conduct LR Monte Carlo simulations of the model up to a linear size of L = 8192. Our results demonstrate continuous phase transitions into a ferromagnetic phase for σ < 2, which exhibit the simultaneous emergence of a long-ranged order and a power-law decaying correlation function due to the Goldstone mode. Furthermore, we find logarithmic scaling behaviors in the low-temperature phase at σ = 2. The observed scaling behaviors in the low-temperature phase for σ ≤ 2 agree with our theoretical analysis. Our findings request further theoretical understanding and can be of practical application in cutting-edge experiments like Rydberg atom arrays.
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 $\ensuremath{\rho}<4$.
Logarithmic Quasi-Long-Range Order and Novel Continuous Phase Transition in the Two-Dimensional XYXYXY Model with 1/r41/r^41/r4 Interaction
We investigate the phase diagram and critical properties of the two-dimensional classical XY model with interactions decaying as 1/r2+σ1/r^{2+σ}1/r2+σ. At the marginal case σ=2σ=2σ=2, we show that the low-temperature phase is characterized by logarithmic quasi-long-range order (log-QLRO), where spin correlations decay as a power of the logarithm of distance, C(r)∼(ln⁡r)−ηℓC(r)\sim(\ln r)^{-η_\ell}C(r)∼(lnr)−ηℓ​ with ηℓ∝Tη_\ell\propto Tηℓ​∝T, and that the transition into this phase is a continuous transition beyond both the Ginzburg-Landau-Wilson paradigm and the Berezinskii-Kosterlitz-Thouless mechanism. Our analysis combines a near-exact Gaussian spin-wave theory, an adiabatic renormalization-group analysis, and large-scale Monte Carlo simulations. We find that the 1/r41/r^41/r4 interaction logarithmically modifies the spin-wave stiffness kernel as γ(k)≃κk2ln⁡(1/k)γ(k)\simeqκk^2\ln(1/k)γ(k)≃κk2ln(1/k), which suppresses vortex proliferation and gives rise to a nonuniversal correlation-length exponent ν∝1/2πκν\propto1/\sqrt{2πκ}ν∝1/2πκ​. Crucially, our simulations reveal that the short-range algebraic QLRO phase is unstable against a weak long-range perturbation for σ≤2σ\le2σ≤2. These results establish σ=2σ=2σ=2 as the crossover boundary between the long-range and short-range universality classes and provide a critical evaluation of recent theoretical proposals.
Determining the threshold value σ∗σ_*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−ηSRσ_* = 2 - η_{\mathrm{SR}}σ∗​=2−ηSR​, 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/r2+σ1/r^{2+σ}1/r2+σ. Our previous simulations for the XY, Heisenberg, and percolation models consistently indicate a universal boundary at σ∗=2σ_* = 2σ∗​=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=8192L = 8192L=8192. By analyzing the Fortuin-Kasteleyn critical polynomial RpR_pRp​, the Binder ratio QmQ_mQm​, and the anomalous dimension ηηη, we obtain convergent and self-consistent evidence that the universality class already changes sharply at σ=2σ= 2σ=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∗=2σ_* = 2σ∗​=2.
Algebraically decaying interactions ∼1/rd+σ\sim 1/r^{d+σ}∼1/rd+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=8192L=8192L=8192. We show that the system enters a long-range-ordered phase through a single continuous phase transition for all σ≤2σ\leq 2σ≤2, including the marginal case σ=2σ=2σ=2. In contrast, for σ>2σ> 2σ>2 it recovers the SR asymptotically free behavior with no finite-temperature transition. This places the LR--SR crossover threshold at σ∗=2σ_* = 2σ∗​=2. To characterize the ordered phase, we introduce an LR simple random walk with a fixed total length L∼O(Ld)\mathcal{L} \sim\mathcal{O}(L^d)L∼O(Ld). This fixed-L\mathcal LL 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=2σ= 2σ=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 purpose of this article is to give an overview of results concerning ordering and critical properties of two-dimensional ferromagnets including the dipolar interaction.We investigate a two-dimensional xy-model extended by the dipolar interaction.Describing our system by a nonlinear σ-model and using renormalization group methods we predict a phase transition to an ordered state.This transition is due to the long-range dipolar interaction.The ferromagnetic phase is governed by a low temperature fixed-point with infinite dipolar coupling.In the critical regime we find exponential behavior for the correlation length and the order parameter in contrast to the usual power laws.The nature of the transition shows a striking similarity to the Kosterlitz-Thouless transition.We show that there is a whole class of longrange xy-models leading to such non-standard behavior.Parameterizing the divergencies in terms of the correlation length we are able to calculate the critical exponents.These exponents are correct in any loop order.