Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2609.39267

Logarithmic Quasi-Long-Range Order and Novel Continuous Phase Transition in the Two-Dimensional $XY$ Model with $1/r^4$ Interaction

Xianzhi Pan, Zhijie Fan, Youjin Deng

Published Sep 30, 2026• 0 citations
arXivHTMLPDF

Abstract

We investigate the phase diagram and critical properties of the two-dimensional classical XY model with interactions decaying as $1/r^{2+σ}$. At the marginal case $σ=2$, 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, $C(r)\sim(\ln r)^{-η_\ell}$ with $η_\ell\propto T$, 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 $1/r^4$ interaction logarithmically modifies the spin-wave stiffness kernel as $γ(k)\simeqκk^2\ln(1/k)$, which suppresses vortex proliferation and gives rise to a nonuniversal correlation-length exponent $ν\propto1/\sqrt{2πκ}$. Crucially, our simulations reveal that the short-range algebraic QLRO phase is unstable against a weak long-range perturbation for $σ\le2$. These results establish $σ=2$ as the crossover boundary between the long-range and short-range universality classes and provide a critical evaluation of recent theoretical proposals.

DOI
10.48550/arxiv.2609.39267

Related papers

Two-Dimensional XY Ferromagnet Induced by Long-Range Interaction

Tianning Xiao, Dingyun Yao, Chao Zhang +2 more

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.

arXiv:2404.08498•Chinese Physics Letters•May 29, 2025•10 citations

Vortex screening and the fate of the BKT transition with long-range couplings

Guido Giachetti

The stability of the Berezinskii--Kosterlitz--Thouless phase against long-range interactions remains an open problem. We address this question in the two-dimensional XY model with couplings decaying as $r^{-2-σ}$, by developing a real-space renormalization-group treatment near the Gaussian manifold of quasi-long-range ordered (QLRO) states. We retain the full many-body vortex interaction and show that, at leading order in the fugacity, screening by small vortex--antivortex pairs preserves its Gaussian-average representation without an expansion in the long-range coupling. Including spin waves perturbatively then yields a local stability criterion: a stable QLRO interval exists only for $σ>7/4$. We also investigate a Wilson--Fisher-like fixed point at finite coupling for $σ<7/4$, for which the retained flow predicts $ν^{-1}\sim\sqrt{7/4-σ}$ as it merges with the Gaussian line.

arXiv:2610.10171•Oct 7, 2026•0 citations

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

Dingyun Yao, Tianning Xiao, Chao Zhang +2 more

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.

arXiv:2411.01811•Physical review. B./Physical review. B•Sep 24, 2025•5 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 $\ensuremath{\rho}<4$.

arXiv:1905.06688•Physical review. B./Physical review. B•Aug 22, 2019•14 citations

Long-Range Order in a Strictly Short-Range Quasi-2D XY Model: When Critical Fluctuations Matter

Minghui Hu, Chao Zhang, Da‐jun Zhang +3 more

The phase of spins in the quasi-two-dimensional (q2D) XY model has emerged as a topic of significant interest across multiple subfields of physics. Conventional wisdom, rooted in the Mermin-Wagner theorem and supported by existing paradigms, asserts that true long-range (LR) order is prohibited in q2D systems with continuous symmetries and short-range (SR) interactions. In this Letter, we propose a strictly SR q2D XY model defined on a plane perpendicularly intersected by a group of parallel planes, where each plane consists of XY spins coupled via nearest-neighbor interactions. Through large-scale Monte Carlo simulations complemented by finite-size scaling analysis, we establish the complete phase diagram of the setup. A LR ordered phase emerges in the q2D model when the spins on the parallel planes develop a Berezinskii-Kosterlitz-Thouless critical phase. The LR ordered phase is anisotropic: true LR correlations develop exclusively along the direction of the intersection lines, while the perpendicular direction exhibits quasi-long-range order. Furthermore, the LR order exhibits Goldstone-mode physics. Our findings reveal a mechanism for stabilizing LR order in low-dimensional systems with continuous symmetries, thereby establishing a new platform for studying exotic superfluidity.

arXiv:2506.19637•Physical Review Letters•Apr 29, 2026•0 citations

Multicritical behaviour in the fully frustratedXYmodel and related systems

Martin Hasenbusch B, Andrea Pelissetto C, Ettore Vicari

We study the phase diagram and critical behaviour of the two-dimensional square-lattice fully frustrated XY model (FFXY) and of two related models, a lattice discretization of the Landau–Ginzburg–Wilson Hamiltonian for the critical modes of the FFXY model, and a coupled Ising XY model. We present a finite-size-scaling analysis of the results of high-precision Monte Carlo simulations on L × L square lattices, up to L = O (10 3 ). In the FFXY model and in the other models, when the transitions are continuous, there are two very close but separate transitions. There is an Ising chiral transition characterized by the onset of chiral long-range order while spins remain paramagnetic. Then, as temperature decreases, the systems undergo a Kosterlitz–Thouless spin transition to a phase with quasi-long-range order. The FFXY model and the other models, in a rather large parameter region, show a crossover behaviour at the chiral and spin transitions that is universal to some extent. We conjecture that this universal behaviour is due to a multicritical point. The numerical data suggest that the relevant multicritical point is a zero-temperature transition. A possible candidate is the O(4) point that controls the low-temperature behaviour of the 4-vector model.

arXiv:cond-mat/0509682•Journal of Statistical Mechanics Theory and Experiment•Dec 5, 2005•89 citations
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.
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 stability of the Berezinskii--Kosterlitz--Thouless phase against long-range interactions remains an open problem. We address this question in the two-dimensional XY model with couplings decaying as r−2−σr^{-2-σ}r−2−σ, by developing a real-space renormalization-group treatment near the Gaussian manifold of quasi-long-range ordered (QLRO) states. We retain the full many-body vortex interaction and show that, at leading order in the fugacity, screening by small vortex--antivortex pairs preserves its Gaussian-average representation without an expansion in the long-range coupling. Including spin waves perturbatively then yields a local stability criterion: a stable QLRO interval exists only for σ>7/4σ>7/4σ>7/4. We also investigate a Wilson--Fisher-like fixed point at finite coupling for σ<7/4σ<7/4σ<7/4, for which the retained flow predicts ν−1∼7/4−σν^{-1}\sim\sqrt{7/4-σ}ν−1∼7/4−σ​ as it merges with the Gaussian line.
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.
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 phase of spins in the quasi-two-dimensional (q2D) XY model has emerged as a topic of significant interest across multiple subfields of physics. Conventional wisdom, rooted in the Mermin-Wagner theorem and supported by existing paradigms, asserts that true long-range (LR) order is prohibited in q2D systems with continuous symmetries and short-range (SR) interactions. In this Letter, we propose a strictly SR q2D XY model defined on a plane perpendicularly intersected by a group of parallel planes, where each plane consists of XY spins coupled via nearest-neighbor interactions. Through large-scale Monte Carlo simulations complemented by finite-size scaling analysis, we establish the complete phase diagram of the setup. A LR ordered phase emerges in the q2D model when the spins on the parallel planes develop a Berezinskii-Kosterlitz-Thouless critical phase. The LR ordered phase is anisotropic: true LR correlations develop exclusively along the direction of the intersection lines, while the perpendicular direction exhibits quasi-long-range order. Furthermore, the LR order exhibits Goldstone-mode physics. Our findings reveal a mechanism for stabilizing LR order in low-dimensional systems with continuous symmetries, thereby establishing a new platform for studying exotic superfluidity.
We study the phase diagram and critical behaviour of the two-dimensional square-lattice fully frustrated XY model (FFXY) and of two related models, a lattice discretization of the Landau–Ginzburg–Wilson Hamiltonian for the critical modes of the FFXY model, and a coupled Ising XY model. We present a finite-size-scaling analysis of the results of high-precision Monte Carlo simulations on L × L square lattices, up to L = O (10 3 ). In the FFXY model and in the other models, when the transitions are continuous, there are two very close but separate transitions. There is an Ising chiral transition characterized by the onset of chiral long-range order while spins remain paramagnetic. Then, as temperature decreases, the systems undergo a Kosterlitz–Thouless spin transition to a phase with quasi-long-range order. The FFXY model and the other models, in a rather large parameter region, show a crossover behaviour at the chiral and spin transitions that is universal to some extent. We conjecture that this universal behaviour is due to a multicritical point. The numerical data suggest that the relevant multicritical point is a zero-temperature transition. A possible candidate is the O(4) point that controls the low-temperature behaviour of the 4-vector model.