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arXiv:2512.01956

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

Tianning Xiao, Dingyun Yao, Polle, Lode, Zhijie Fan, Youjin Deng

Published Dec 1, 2025• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2512.01956

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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.
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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 $\ensuremath{\sim}1/{r}^{2+\ensuremath{\sigma}}$ 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.
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