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:2006.13061 · Physical review. D/Physical review. D.

Asymptotic low-temperature behavior of two-dimensional RPN−1 models

Claudio Bonati, Alessio Mauro Franchi, Andrea Pelissetto C, Ettore Vicari

Published Aug 28, 2020• 11 citations
arXivHTMLPDF

Abstract

We investigate the low-temperature behavior of two-dimensional (2D) ${\mathrm{RP}}^{N\ensuremath{-}1}$ models, characterized by a global $\mathrm{O}(N)$ symmetry and a local ${\mathbb{Z}}_{2}$ symmetry. For $N=3$ we perform large-scale simulations of four different 2D lattice models: two standard lattice models and two different constrained models. We also consider a constrained mixed $\mathrm{O}(3)\text{\ensuremath{-}}{\mathrm{RP}}^{2}$ model for values of the parameters such that vector correlations are always disordered. We find that all these models show the same finite-size scaling (FSS) behavior, and therefore belong to the same universality class. However, these FSS curves differ from those computed in the 2D O(3) $\ensuremath{\sigma}$ model, suggesting the existence of a distinct 2D ${\mathrm{RP}}^{2}$ universality class. We also performed simulations for $N=4$, and the corresponding FSS results also support the existence of an ${\mathrm{RP}}^{3}$ universality class, different from the O(4) one.

DOI
10.1103/physrevd.102.034513

Related papers

Finite-size scaling in two-dimensional Ising spin-glass models

Francesco Parisen Toldin, Andrea Pelissetto C, Ettore Vicari

We study the finite-size behavior of two-dimensional spin-glass models. We consider the $\ifmmode\pm\else\textpm\fi{}J$ model for two different values of the probability of the antiferromagnetic bonds and the model with Gaussian distributed couplings. The analysis of renormalization-group invariant quantities, the overlap susceptibility, and the two-point correlation function confirms that they belong to the same universality class. We analyze in detail the standard finite-size scaling limit in terms of $T{L}^{1/\ensuremath{\nu}}$ in the $\ifmmode\pm\else\textpm\fi{}J$ model. We find that it holds asymptotically. This result is consistent with the low-temperature crossover scenario in which the crossover temperature, which separates the universal high-temperature region from the discrete low-temperature regime, scales as ${T}_{c}(L)\ensuremath{\sim}{L}^{\ensuremath{-}{\ensuremath{\theta}}_{S}}$ with ${\ensuremath{\theta}}_{S}\ensuremath{\approx}0.5$.

arXiv:1106.5720•Physical Review E•Nov 16, 2011•21 citations

Three-dimensional Z 2 -gauge N -vector models

Claudio Bonati, Andrea Pelissetto C, Ettore Vicari

We study the phase diagram and critical behaviors of three-dimensional lattice ${\mathbb{Z}}_{2}$-gauge $N$-vector models, in which an $N$-component real field is minimally coupled with ${\mathbb{Z}}_{2}$-gauge link variables. These models are invariant under global $\mathrm{O}(N$) and local ${\mathbb{Z}}_{2}$ transformations. They present three phases characterized by the spontaneous breaking of the global $\mathrm{O}(N$) symmetry and by the different topological properties of the ${\mathbb{Z}}_{2}$-gauge correlations. We address the nature of the three transition lines separating the three phases. The theoretical predictions are supported by numerical finite-size scaling analyses of Monte Carlo data for the $N=2$ model. In this case, continuous transitions can be observed along both transition lines where the $N$-component spins order, in the regimes of small and large inverse gauge coupling $K$. Even though these continuous transitions belong to the same $XY$ universality class, their critical modes turn out to be different. When the gauge variables are disordered (small $K$), the relevant order-parameter field is a gauge-invariant bilinear combination of the vector field. On the other hand, when the gauge variables are ordered (large $K$), the order-parameter field is the gauge-dependent $N$-vector field, whose critical behavior can only be probed by using a stochastic gauge fixing that reduces the gauge freedom.

Physical review. B./Physical review. B•Jun 10, 2024•6 citations

Finite size scaling in the two-dimensional XY model and generalized universality

Guillermo Palma, T. Meyer, Raúl Labbé

In recent works [S. T. Bramwell, P. C. W. Holdsworth, and J.-F. Pinton, Nature (London) 396, 552 (1998); S. T. Bramwell et al., Phys. Rev. Lett. 84, 3744 (2000)], a generalized universality has been proposed, linking phenomena as dissimilar as two-dimensional (2D) magnetism and turbulence. To test these ideas, we performed Monte Carlo simulations of the 2D XY model. We found that the shape of the probability distribution function for the magnetization M is non-Gaussian and independent of the system size-in the range of the lattice sizes studied-below the Kosterlitz-Thoules temperature. However, our results suggest that in the full 2D XY model the shape of these distributions has a slight dependence on temperature-for finite volume-below the lattice-shifted critical temperature T*(L). This behavior can be explained by using renormalization group arguments and an extended finite-size scaling analysis, and by the existence of bounds for M.

arXiv:cond-mat/0007289•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Aug 16, 2002•15 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

Three-dimensional ${\mathbb Z}_2$-gauge $N$-vector models

Claudio Bonati, Andrea Pelissetto C, Ettore Vicari

We study the phase diagram and critical behaviors of three-dimensional lattice ${\mathbb Z}_2$-gauge $N$-vector models, in which an $N$-component real field is minimally coupled with a ${\mathbb Z}_2$-gauge link variables. These models are invariant under global O($N$) and local ${\mathbb Z}_2$ transformations. They present three phases characterized by the spontaneous breaking of the global O($N$) symmetry and by the different topological properties of the ${\mathbb Z}_2$-gauge correlations. We address the nature of the three transition lines separating the three phases. The theoretical predictions are supported by numerical finite-size scaling analyses of Monte Carlo data for the $N=2$ model. In this case, continuous transitions can be observed along both transition lines where the spins order, in the regime of small and large inverse gauge coupling $K$. Even though these continuous transitions belong to the same $XY$ universality class, their critical modes turn out to be different. When the gauge variables are disordered (small $K$), the relevant order-parameter field is a gauge-invariant bilinear combination of the vector field. On the other hand, when the gauge variables are ordered (large $K$), the order-parameter field is the gauge-dependent $N$-vector field, whose critical behavior can only be probed by using a stochastic gauge fixing that reduces the gauge freedom.

arXiv:2404.07050•Apr 10, 2024•0 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.

arXiv:2512.01956•Dec 1, 2025•0 citations
We investigate the low-temperature behavior of two-dimensional (2D) RPN\ensuremath−1{\mathrm{RP}}^{N\ensuremath{-}1}RPN\ensuremath−1 models, characterized by a global O(N)\mathrm{O}(N)O(N) symmetry and a local Z2{\mathbb{Z}}_{2}Z2​ symmetry. For N=3N=3N=3 we perform large-scale simulations of four different 2D lattice models: two standard lattice models and two different constrained models. We also consider a constrained mixed O(3)\ensuremath-RP2\mathrm{O}(3)\text{\ensuremath{-}}{\mathrm{RP}}^{2}O(3)\ensuremath-RP2 model for values of the parameters such that vector correlations are always disordered. We find that all these models show the same finite-size scaling (FSS) behavior, and therefore belong to the same universality class. However, these FSS curves differ from those computed in the 2D O(3) \ensuremathσ\ensuremath{\sigma}\ensuremathσ model, suggesting the existence of a distinct 2D RP2{\mathrm{RP}}^{2}RP2 universality class. We also performed simulations for N=4N=4N=4, and the corresponding FSS results also support the existence of an RP3{\mathrm{RP}}^{3}RP3 universality class, different from the O(4) one.
We study the finite-size behavior of two-dimensional spin-glass models. We consider the $\ifmmode\pm\else\textpm\fi{}J$ model for two different values of the probability of the antiferromagnetic bonds and the model with Gaussian distributed couplings. The analysis of renormalization-group invariant quantities, the overlap susceptibility, and the two-point correlation function confirms that they belong to the same universality class. We analyze in detail the standard finite-size scaling limit in terms of $T{L}^{1/\ensuremath{\nu}}$ in the $\ifmmode\pm\else\textpm\fi{}J$ model. We find that it holds asymptotically. This result is consistent with the low-temperature crossover scenario in which the crossover temperature, which separates the universal high-temperature region from the discrete low-temperature regime, scales as ${T}_{c}(L)\ensuremath{\sim}{L}^{\ensuremath{-}{\ensuremath{\theta}}_{S}}$ with ${\ensuremath{\theta}}_{S}\ensuremath{\approx}0.5$.
We study the phase diagram and critical behaviors of three-dimensional lattice Z2{\mathbb{Z}}_{2}Z2​-gauge NNN-vector models, in which an NNN-component real field is minimally coupled with Z2{\mathbb{Z}}_{2}Z2​-gauge link variables. These models are invariant under global O(N\mathrm{O}(NO(N) and local Z2{\mathbb{Z}}_{2}Z2​ transformations. They present three phases characterized by the spontaneous breaking of the global O(N\mathrm{O}(NO(N) symmetry and by the different topological properties of the Z2{\mathbb{Z}}_{2}Z2​-gauge correlations. We address the nature of the three transition lines separating the three phases. The theoretical predictions are supported by numerical finite-size scaling analyses of Monte Carlo data for the N=2N=2N=2 model. In this case, continuous transitions can be observed along both transition lines where the NNN-component spins order, in the regimes of small and large inverse gauge coupling KKK. Even though these continuous transitions belong to the same XYXYXY universality class, their critical modes turn out to be different. When the gauge variables are disordered (small KKK), the relevant order-parameter field is a gauge-invariant bilinear combination of the vector field. On the other hand, when the gauge variables are ordered (large KKK), the order-parameter field is the gauge-dependent NNN-vector field, whose critical behavior can only be probed by using a stochastic gauge fixing that reduces the gauge freedom.
In recent works [S. T. Bramwell, P. C. W. Holdsworth, and J.-F. Pinton, Nature (London) 396, 552 (1998); S. T. Bramwell et al., Phys. Rev. Lett. 84, 3744 (2000)], a generalized universality has been proposed, linking phenomena as dissimilar as two-dimensional (2D) magnetism and turbulence. To test these ideas, we performed Monte Carlo simulations of the 2D XY model. We found that the shape of the probability distribution function for the magnetization M is non-Gaussian and independent of the system size-in the range of the lattice sizes studied-below the Kosterlitz-Thoules temperature. However, our results suggest that in the full 2D XY model the shape of these distributions has a slight dependence on temperature-for finite volume-below the lattice-shifted critical temperature T*(L). This behavior can be explained by using renormalization group arguments and an extended finite-size scaling analysis, and by the existence of bounds for M.
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 $\ensuremath{\rho}<4$.
Three-dimensional Z2{\mathbb Z}_2Z2​-gauge NNN-vector models
We study the phase diagram and critical behaviors of three-dimensional lattice Z2{\mathbb Z}_2Z2​-gauge NNN-vector models, in which an NNN-component real field is minimally coupled with a Z2{\mathbb Z}_2Z2​-gauge link variables. These models are invariant under global O(NNN) and local Z2{\mathbb Z}_2Z2​ transformations. They present three phases characterized by the spontaneous breaking of the global O(NNN) symmetry and by the different topological properties of the Z2{\mathbb Z}_2Z2​-gauge correlations. We address the nature of the three transition lines separating the three phases. The theoretical predictions are supported by numerical finite-size scaling analyses of Monte Carlo data for the N=2N=2N=2 model. In this case, continuous transitions can be observed along both transition lines where the spins order, in the regime of small and large inverse gauge coupling KKK. Even though these continuous transitions belong to the same XYXYXY universality class, their critical modes turn out to be different. When the gauge variables are disordered (small KKK), the relevant order-parameter field is a gauge-invariant bilinear combination of the vector field. On the other hand, when the gauge variables are ordered (large KKK), the order-parameter field is the gauge-dependent NNN-vector field, whose critical behavior can only be probed by using a stochastic gauge fixing that reduces the gauge freedom.
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.