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arXiv:2609.12560

Persistence of BKT phase transition in the 2D nonanalytic XY model

Sihan Hu, Xianzhi Pan, Kun Chen, Yi Jiang, Youjin Deng

Published Sep 11, 2026• 0 citations
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Abstract

We study the two-dimensional XY model with the nonanalytic pair potential $2[(1-\cosδ)/2]^{p}$, whose small-angle law $\propto|δ|^{2p}$ carries a cusp for $p<1$ and a flat bottom for $p>1$, invalidating the harmonic spin-wave expansion. Two questions arise: the nature of the low-temperature ($T$) phase and of the phase transition. A naive energetic argument would predict genuine long-range order and an enhanced transition temperature for $p<1$, and no transition at all for $p>1$. Large-scale Monte Carlo simulations contradict both: for every $p>0$ the \dengrevxxi{low-$T$ phase} is quasi-long-range ordered, with anomalous dimension $η(T)\propto T^{1/p}$, and terminates at a Berezinskii--Kosterlitz--Thouless transition. Using a vortex-free noncompact lattice-field description and utilizing a duality transformation, we show that coarse-graining drives the height-difference distribution onto a single Gaussian fixed point, renormalizing the cusp and flatness into a finite harmonic stiffness that restores the spin-wave description and the BKT scenario for all $p>0$.

Related papers

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

Berezinskii-Kosterlitz-Thouless transition in the XY model on the honeycomb lattice: a comprehensive Monte Carlo analysis

Felipe E. F. de Andrade, L. N. Jorge, Cláudio José DaSilva

Abstract In this paper, we thoroughly examined the Berezinskii-Kosterlitz-Thouless (BKT) phase transition in the two-dimensional XY model on the honeycomb lattice. To address its thermodynamical behavior, we combined standard numerical Monte Carlo simulations with the simulated annealing (SA) protocol and entropic simulations based on the Wang-Landau (WL) algorithm. The transition temperature was determined using the second (ϒ) and fourth-order (ϒ4) helicity modulus as the order parameter. Our best finite-size scaling analysis suggests T BKT = 0.575(8) from SA and T BKT = 0.576(3) from WL. These values deviate significantly from the expected theoretical value of 1 / 2 . We believe that this discrepancy suggests that the theoretical assumptions regarding the analytical calculation may need to be revisited. Additionally, we calculated the vortex density and the formation energy of the vortex-antivortex pairs, where the obtained vortex formation energy is 2μ = 5.80(12). Upon comparison with the square lattice, our results support the notion of instability of the honeycomb lattice to support the spin long-range order and give additional backing to the critical behavior we found.

arXiv:2406.12076•Physica Scripta•May 8, 2025•3 citations

Persistence of the Berezinskii-Kosterlitz-Thouless Transition with Long-Range Couplings

Luis Walther, Josef Willsher, Johannes Knolle

The Berezinskii-Kosterlitz-Thouless (BKT) transition is an archetypal example of a topological phase transition, which is driven by the proliferation of vortices. In this Letter, we analyze the persistence of the BKT transition in the XY model under the influence of long-range algebraically decaying interactions of the form ∼1/r^{2+σ}. The model hosts a magnetized low temperature phase for sufficiently small σ. Crucially, in the presence of long-range interactions, spin waves renormalize the interaction between vortices, which stabilizes the BKT transition. As a result, we find that there is no direct transition from the magnetized to the disordered phase and that the BKT transition persists for arbitrary long-range exponents, which is distinct from previous results. The key methodological advance is the inclusion of coupling between spin wave and topological excitations. We do this by developing Landau-Peierls-type arguments and using higher-order renormalization group calculations, obtaining compatible results. We emphasize that Landau-Peierls-type arguments are a powerful tool for analyzing continuous spin models. We discuss the relevance of our findings for current Rydberg atom experiments, and highlight the importance of long-range couplings for other types of topological defects.

arXiv:2511.07305•Physical Review Letters•May 15, 2026•0 citations

Topological Lattice Actions for the 2d XY Model

Wolfgang Bietenholz A, Bögli, Michael, Niedermayer, Ferenc +3 more

We consider the 2d XY Model with topological lattice actions, which are invariant against small deformations of the field configuration. These actions constrain the angle between neighbouring spins by an upper bound, or they explicitly suppress vortices (and anti-vortices). Although topological actions do not have a classical limit, they still lead to the universal behaviour of the Berezinskii-Kosterlitz-Thouless (BKT) phase transition — at least up to moderate vortex suppression. In the massive phase, the analytically known Step Scaling Function (SSF) is reproduced in numerical simulations. However, deviations from the expected universal behaviour of the lattice artifacts are observed. In the massless phase, the BKT value of the critical exponent ηc is confirmed. Hence, even though for some topological actions vortices cost zero energy, they still drive the standard BKT transition. In addition we identify a vortex-free transition point, which deviates from the BKT behaviour.

•Jan 1, 2013•6 citations

BKT transitions of the XY and six-state clock models on the various two-dimensional lattices

Yutaka Okabe, Hiromi Otsuka

Abstract In a two-dimensional (2D) spin system, the XY model, characterized by planar rotational symmetry, exhibits a unique phenomenon known as the Berezinskii–Kosterlitz–Thouless (BKT) transition. In contrast, the clock model, which introduces discrete rotational symmetry, exhibits the BKT transition at two different temperatures due to this discreteness. In this study, we numerically investigate the BKT transition for XY and six-state clock models over various 2D lattices. We employ two primary methods: the Monte Carlo method, which analyzes the size dependence of the ratio of the correlation functions for two different distances, and a machine-learning approach to classify the different phases—namely, the low-temperature ordered phase, the intermediate BKT phase, and the high-temperature disordered phase. We identify the BKT transition temperatures for the XY and six-state models on honeycomb, kagome, and diced lattices. Combined with the previously calculated data for the triangular lattice, we then compare these values with the second-order phase transition temperatures of the 2D Ising model, for which exact solutions are known. Our results indicate that the ratio of the BKT transition temperatures for each lattice relative to the Ising model transition temperatures are close, although the values are not universal.

arXiv:2501.07388•Journal of Physics A Mathematical and Theoretical•Jan 17, 2025•7 citations

Quench dynamics of the 2dXYmodel

Asja Jelić, Leticia F. Cugliandolo

We investigate the out of equilibrium dynamics of the two-dimensional XY model when cooled across the Berezinskii–Kosterlitz–Thouless (BKT) phase transition using different protocols. We focus on the evolution of the growing correlation length and the density of topological defects (vortices). By using Monte Carlo simulations we first determine the time and temperature dependence of the growing correlation length after an infinitely rapid quench from above the transition temperature to the quasi-long-range order region. The functional form is consistent with a logarithmic correction to the diffusive law and it serves to validate dynamic scaling in this problem. This analysis clarifies the different dynamic roles played by bound and free vortices. We then revisit the Kibble–Zurek mechanism in thermal phase transitions in which the disordered state is plagued with topological defects. We provide a theory of quenching rate dependence in systems with the BKT-type transition that goes beyond the equilibrium scaling arguments. Finally, we discuss the implications of our results for a host of physical systems with vortex excitations, including planar ferromagnets and liquid crystals.

arXiv:1012.0417•Journal of Statistical Mechanics Theory and Experiment•Feb 18, 2011•89 citations
We study the two-dimensional XY model with the nonanalytic pair potential 2[(1−cos⁡δ)/2]p2[(1-\cosδ)/2]^{p}2[(1−cosδ)/2]p, whose small-angle law ∝∣δ∣2p\propto|δ|^{2p}∝∣δ∣2p carries a cusp for p<1p<1p<1 and a flat bottom for p>1p>1p>1, invalidating the harmonic spin-wave expansion. Two questions arise: the nature of the low-temperature (TTT) phase and of the phase transition. A naive energetic argument would predict genuine long-range order and an enhanced transition temperature for p<1p<1p<1, and no transition at all for p>1p>1p>1. Large-scale Monte Carlo simulations contradict both: for every p>0p>0p>0 the \dengrevxxi{low-TTT phase} is quasi-long-range ordered, with anomalous dimension η(T)∝T1/pη(T)\propto T^{1/p}η(T)∝T1/p, and terminates at a Berezinskii--Kosterlitz--Thouless transition. Using a vortex-free noncompact lattice-field description and utilizing a duality transformation, we show that coarse-graining drives the height-difference distribution onto a single Gaussian fixed point, renormalizing the cusp and flatness into a finite harmonic stiffness that restores the spin-wave description and the BKT scenario for all p>0p>0p>0.
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.
Abstract In this paper, we thoroughly examined the Berezinskii-Kosterlitz-Thouless (BKT) phase transition in the two-dimensional XY model on the honeycomb lattice. To address its thermodynamical behavior, we combined standard numerical Monte Carlo simulations with the simulated annealing (SA) protocol and entropic simulations based on the Wang-Landau (WL) algorithm. The transition temperature was determined using the second (ϒ) and fourth-order (ϒ4) helicity modulus as the order parameter. Our best finite-size scaling analysis suggests T BKT = 0.575(8) from SA and T BKT = 0.576(3) from WL. These values deviate significantly from the expected theoretical value of 1 / 2 . We believe that this discrepancy suggests that the theoretical assumptions regarding the analytical calculation may need to be revisited. Additionally, we calculated the vortex density and the formation energy of the vortex-antivortex pairs, where the obtained vortex formation energy is 2μ = 5.80(12). Upon comparison with the square lattice, our results support the notion of instability of the honeycomb lattice to support the spin long-range order and give additional backing to the critical behavior we found.
The Berezinskii-Kosterlitz-Thouless (BKT) transition is an archetypal example of a topological phase transition, which is driven by the proliferation of vortices. In this Letter, we analyze the persistence of the BKT transition in the XY model under the influence of long-range algebraically decaying interactions of the form ∼1/r^{2+σ}. The model hosts a magnetized low temperature phase for sufficiently small σ. Crucially, in the presence of long-range interactions, spin waves renormalize the interaction between vortices, which stabilizes the BKT transition. As a result, we find that there is no direct transition from the magnetized to the disordered phase and that the BKT transition persists for arbitrary long-range exponents, which is distinct from previous results. The key methodological advance is the inclusion of coupling between spin wave and topological excitations. We do this by developing Landau-Peierls-type arguments and using higher-order renormalization group calculations, obtaining compatible results. We emphasize that Landau-Peierls-type arguments are a powerful tool for analyzing continuous spin models. We discuss the relevance of our findings for current Rydberg atom experiments, and highlight the importance of long-range couplings for other types of topological defects.
We consider the 2d XY Model with topological lattice actions, which are invariant against small deformations of the field configuration. These actions constrain the angle between neighbouring spins by an upper bound, or they explicitly suppress vortices (and anti-vortices). Although topological actions do not have a classical limit, they still lead to the universal behaviour of the Berezinskii-Kosterlitz-Thouless (BKT) phase transition — at least up to moderate vortex suppression. In the massive phase, the analytically known Step Scaling Function (SSF) is reproduced in numerical simulations. However, deviations from the expected universal behaviour of the lattice artifacts are observed. In the massless phase, the BKT value of the critical exponent ηc is confirmed. Hence, even though for some topological actions vortices cost zero energy, they still drive the standard BKT transition. In addition we identify a vortex-free transition point, which deviates from the BKT behaviour.
Abstract In a two-dimensional (2D) spin system, the XY model, characterized by planar rotational symmetry, exhibits a unique phenomenon known as the Berezinskii–Kosterlitz–Thouless (BKT) transition. In contrast, the clock model, which introduces discrete rotational symmetry, exhibits the BKT transition at two different temperatures due to this discreteness. In this study, we numerically investigate the BKT transition for XY and six-state clock models over various 2D lattices. We employ two primary methods: the Monte Carlo method, which analyzes the size dependence of the ratio of the correlation functions for two different distances, and a machine-learning approach to classify the different phases—namely, the low-temperature ordered phase, the intermediate BKT phase, and the high-temperature disordered phase. We identify the BKT transition temperatures for the XY and six-state models on honeycomb, kagome, and diced lattices. Combined with the previously calculated data for the triangular lattice, we then compare these values with the second-order phase transition temperatures of the 2D Ising model, for which exact solutions are known. Our results indicate that the ratio of the BKT transition temperatures for each lattice relative to the Ising model transition temperatures are close, although the values are not universal.
We investigate the out of equilibrium dynamics of the two-dimensional XY model when cooled across the Berezinskii–Kosterlitz–Thouless (BKT) phase transition using different protocols. We focus on the evolution of the growing correlation length and the density of topological defects (vortices). By using Monte Carlo simulations we first determine the time and temperature dependence of the growing correlation length after an infinitely rapid quench from above the transition temperature to the quasi-long-range order region. The functional form is consistent with a logarithmic correction to the diffusive law and it serves to validate dynamic scaling in this problem. This analysis clarifies the different dynamic roles played by bound and free vortices. We then revisit the Kibble–Zurek mechanism in thermal phase transitions in which the disordered state is plagued with topological defects. We provide a theory of quenching rate dependence in systems with the BKT-type transition that goes beyond the equilibrium scaling arguments. Finally, we discuss the implications of our results for a host of physical systems with vortex excitations, including planar ferromagnets and liquid crystals.