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:0804.2972 · Physical Review B

Monte Carlo simulations of the directional-ordering transition in the two-dimensional classical and quantum compass model

S. C. Wenzel, Wolfhard Janke E

Published Aug 5, 2008• 70 citations
arXivHTMLPDF

Abstract

A comprehensive study of the two-dimensional (2D) compass model on the square lattice is performed for classical and quantum spin degrees of freedom using Monte Carlo and quantum Monte Carlo methods. We employ state-of-the-art implementations using Metropolis, stochastic series expansion, and parallel tempering techniques to obtain the critical ordering temperatures and critical exponents. In a preinvestigation we reconsider the classical compass model where we study and contrast the finite-size scaling behavior of ordinary periodic boundary conditions against annealed boundary conditions. It is shown that periodic boundary conditions suffer from extreme finite-size effects which might be caused by closed-loop excitations on the torus. These excitations also appear to have severe effects on the Binder parameter. On this footing we report on a systematic Monte Carlo study of the quantum compass model. Our numerical results are at odds with recent literature on the subject which we trace back to neglecting the strong finite-size effects on periodic lattices. The critical temperatures are obtained as ${T}_{c}=0.1464(2)J$ and ${T}_{c}=0.055(1)J$ for the classical and quantum versions, respectively, and our data support a transition in the 2D Ising universality class for both cases.

DOI
10.1103/physrevb.78.064402

Related papers

Re-examining the directional-ordering transition in the compass model with screw-periodic boundary conditions

S. C. Wenzel, Wolfhard Janke E, Andreas Martin Läuchli

We study the directional-ordering transition in the two-dimensional classical and quantum compass models on the square lattice by means of Monte Carlo simulations. An improved algorithm is presented which builds on the Wolff cluster algorithm in one-dimensional subspaces of the configuration space. This improvement allows us to study classical systems up to L=512. Based on this algorithm, we give evidence for the presence of strongly anomalous scaling for periodic boundary conditions which is much worse than anticipated before. We propose and study alternative boundary conditions for the compass model which do not make use of extended configuration spaces and show that they completely remove the problem with finite-size scaling. In the last part, we apply these boundary conditions to the quantum problem and present a considerably improved estimate for the critical temperature which should be of interest for future studies on the compass model. Our investigation identifies a strong one-dimensional magnetic ordering tendency with a large correlation length as the cause of the unusual scaling and moreover allows for a precise quantification of the anomalous length scale involved.

arXiv:1002.3508•Physical Review E•Jun 9, 2010•35 citations

Phase diagram and Ashkin-Teller universality in the classical square-lattice Heisenberg-compass model

Yuchen Fan

We determine the finite-temperature phase diagram and critical behavior of the classical square-lattice Heisenberg-compass model using large-scale Monte Carlo simulations and finite-size scaling. Six symmetry distinct ordered phases are identified. The four phases that simultaneously break the spin-lattice $C_4$ and in-plane spin-inversion symmetries undergo continuous transitions in the Ashkin-Teller universality class, with the associated critical lines terminating at four-state Potts points, beyond which the transitions become first order. In contrast, the two $z$-polarized phases display conventional two-dimensional Ising criticality. Our results reveal how the interplay between Heisenberg exchange and compass anisotropy organizes these distinct critical regimes, thereby completing the characterization of the model's thermal phase transitions.

arXiv:2603.09469•Physical review. B./Physical review. B•Feb 18, 2026•0 citations

Critical Behavior Simulation of the Two-Dimensional Ising Model—A Computational Physics Study Based on the Monte Carlo Method

佳钢 李

To demonstrate the application of computational physics in modern statistical physics, this paper investigates the critical behavior of a finite-size ferromagnetic system based on the two-dimensional square-lattice Ising model.A nearest-neighbor Ising Hamiltonian is constructed, and periodic boundary conditions are adopted to reduce finite-boundary effects.The Metropolis Monte Carlo algorithm is used to sample spin configurations at different temperatures, while energy, magnetization, specific heat, magnetic susceptibility and Binder cumulant are calculated to describe the transition from an ordered state to a disordered state.The simulation results show that the system exhibits clear long-range order at low temperatures, where the order parameter remains high.As temperature increases, thermal fluctuations become stronger, magnetization decreases rapidly, and the system gradually evolves into a disordered state.The specific heat curve shows a pronounced peak at approximately (T = 2.269), while the susceptibility peak shifts slightly toward the high-temperature side due to finite-size effects and the discrete temperature grid.Overall, the simulated critical region is consistent with the theoretical critical temperature of the two-dimensional square-lattice Ising model.Spin-configuration analysis further reveals enhanced fluctuations and domain reconstruction near the critical region.In addition, a thermalization criterion and an extended sampling test for L = 48 near the critical region are added to examine the influence of finite sampling and equilibration.These results indicate that Monte Carlo simulation of the two-dimensional Ising model can effectively reproduce continuous phase transition, critical fluctuation and finite-size effects, providing a basic computational case for understanding critical phenomena in complex systems and data-driven physical analysis.

Modern Physics•Jan 1, 2026•0 citations

Monte Carlo analysis of the phase transitions in the two-dimensional J1−J2XY model

Damien Loison, Pascal Simon

We consider the two-dimensional (2D) ${J}_{1}\ensuremath{-}{J}_{2}$ classical $\mathrm{XY}$ model on a square lattice. In the frustrated phase corresponding to ${J}_{2}>{J}_{1}/2,$ an Ising-like order parameter emerges by an ``order due to disorder'' effect. This leads to a discrete ${Z}_{2}$ symmetry plus the $U(1)$ global one. Using a powerful algorithm we show that the system undergoes two transitions at different but still very close temperatures, one of Kosterlitz-Thouless (KT) type and another one which does not belong to the expected Ising universality class. A new analysis of the KT transition has been developed in order to avoid the use of the nonuniversal helicity jump and to allow the computation of the exponents without a precise determination of the critical temperature. Moreover, our huge number of data enables us to exhibit the existence of large finite size effects explaining the dispersed results found in the literature concerning the more studied frustrated 2D, $\mathrm{XY}$ models.

arXiv:cond-mat/9912438•Physical review. B, Condensed matter•Mar 1, 2000•41 citations

Quantum-critical properties of the one- and two-dimensional random transverse-field Ising model from large-scale quantum Monte Carlo simulations

Calvin Krämer, Jan Alexander Koziol, Anja Langheld +2 more

We study the ferromagnetic transverse-field Ising model with quenched disorder at T = 0 T = 0 in one and two dimensions by means of stochastic series expansion quantum Monte Carlo simulations using a rigorous zero-temperature scheme. Using a sample-replication method and averaged Binder ratios, we determine the critical shift and width exponents \nu_\mathrm{s} ν s and \nu_\mathrm{w} ν w as well as unbiased critical points by finite-size scaling. Further, scaling of the disorder-averaged magnetisation at the critical point is used to determine the order-parameter critical exponent \beta β and the critical exponent \nu_{\mathrm{av}} ν a v of the average correlation length. The dynamic scaling in the Griffiths phase is investigated by measuring the local susceptibility in the disordered phase and the dynamic exponent z' z ′ is extracted. By applying various finite-size scaling protocols, we provide an extensive and comprehensive comparison between the different approaches on equal footing. The emphasis on effective zero-temperature simulations resolves several inconsistencies in existing literature.

arXiv:2403.05223•SciPost Physics•Aug 23, 2024•5 citations

On Critical Temperature and Finite Size Scaling of Continuous Spin $2d$ Ising Model

S. Mahapatra, Rudra Majhi, Jahangir Mohammed +3 more

In this paper, we have studied the critical temperature $T_c$ of continuous spin $2d$ square-lattice Ising model using Monte-Carlo simulation. We have considered spins $s$ in a bounded interval, where $s \in [-1,+1]$ in square-lattice configuration with periodic boundary condition. We have observed that the critical temperature $T_c$ is approximately $0.925$, showing a clear second order phase transition. Considering finite size scaling, we have also obtained the critical exponents associated with susceptibility, specific heat, magnetization and we find that these values are in good agreement with the corresponding values obtained for the standard $2d$ Ising universality class.

arXiv:2512.21748•Dec 25, 2025•0 citations
A comprehensive study of the two-dimensional (2D) compass model on the square lattice is performed for classical and quantum spin degrees of freedom using Monte Carlo and quantum Monte Carlo methods. We employ state-of-the-art implementations using Metropolis, stochastic series expansion, and parallel tempering techniques to obtain the critical ordering temperatures and critical exponents. In a preinvestigation we reconsider the classical compass model where we study and contrast the finite-size scaling behavior of ordinary periodic boundary conditions against annealed boundary conditions. It is shown that periodic boundary conditions suffer from extreme finite-size effects which might be caused by closed-loop excitations on the torus. These excitations also appear to have severe effects on the Binder parameter. On this footing we report on a systematic Monte Carlo study of the quantum compass model. Our numerical results are at odds with recent literature on the subject which we trace back to neglecting the strong finite-size effects on periodic lattices. The critical temperatures are obtained as Tc=0.1464(2)J{T}_{c}=0.1464(2)JTc​=0.1464(2)J and Tc=0.055(1)J{T}_{c}=0.055(1)JTc​=0.055(1)J for the classical and quantum versions, respectively, and our data support a transition in the 2D Ising universality class for both cases.
We study the directional-ordering transition in the two-dimensional classical and quantum compass models on the square lattice by means of Monte Carlo simulations. An improved algorithm is presented which builds on the Wolff cluster algorithm in one-dimensional subspaces of the configuration space. This improvement allows us to study classical systems up to L=512. Based on this algorithm, we give evidence for the presence of strongly anomalous scaling for periodic boundary conditions which is much worse than anticipated before. We propose and study alternative boundary conditions for the compass model which do not make use of extended configuration spaces and show that they completely remove the problem with finite-size scaling. In the last part, we apply these boundary conditions to the quantum problem and present a considerably improved estimate for the critical temperature which should be of interest for future studies on the compass model. Our investigation identifies a strong one-dimensional magnetic ordering tendency with a large correlation length as the cause of the unusual scaling and moreover allows for a precise quantification of the anomalous length scale involved.
We determine the finite-temperature phase diagram and critical behavior of the classical square-lattice Heisenberg-compass model using large-scale Monte Carlo simulations and finite-size scaling. Six symmetry distinct ordered phases are identified. The four phases that simultaneously break the spin-lattice C4C_4C4​ and in-plane spin-inversion symmetries undergo continuous transitions in the Ashkin-Teller universality class, with the associated critical lines terminating at four-state Potts points, beyond which the transitions become first order. In contrast, the two zzz-polarized phases display conventional two-dimensional Ising criticality. Our results reveal how the interplay between Heisenberg exchange and compass anisotropy organizes these distinct critical regimes, thereby completing the characterization of the model's thermal phase transitions.
To demonstrate the application of computational physics in modern statistical physics, this paper investigates the critical behavior of a finite-size ferromagnetic system based on the two-dimensional square-lattice Ising model.A nearest-neighbor Ising Hamiltonian is constructed, and periodic boundary conditions are adopted to reduce finite-boundary effects.The Metropolis Monte Carlo algorithm is used to sample spin configurations at different temperatures, while energy, magnetization, specific heat, magnetic susceptibility and Binder cumulant are calculated to describe the transition from an ordered state to a disordered state.The simulation results show that the system exhibits clear long-range order at low temperatures, where the order parameter remains high.As temperature increases, thermal fluctuations become stronger, magnetization decreases rapidly, and the system gradually evolves into a disordered state.The specific heat curve shows a pronounced peak at approximately (T = 2.269), while the susceptibility peak shifts slightly toward the high-temperature side due to finite-size effects and the discrete temperature grid.Overall, the simulated critical region is consistent with the theoretical critical temperature of the two-dimensional square-lattice Ising model.Spin-configuration analysis further reveals enhanced fluctuations and domain reconstruction near the critical region.In addition, a thermalization criterion and an extended sampling test for L = 48 near the critical region are added to examine the influence of finite sampling and equilibration.These results indicate that Monte Carlo simulation of the two-dimensional Ising model can effectively reproduce continuous phase transition, critical fluctuation and finite-size effects, providing a basic computational case for understanding critical phenomena in complex systems and data-driven physical analysis.
We consider the two-dimensional (2D) ${J}_{1}\ensuremath{-}{J}_{2}$ classical XY\mathrm{XY}XY model on a square lattice. In the frustrated phase corresponding to J2>J1/2,{J}_{2}>{J}_{1}/2,J2​>J1​/2, an Ising-like order parameter emerges by an ``order due to disorder'' effect. This leads to a discrete Z2{Z}_{2}Z2​ symmetry plus the U(1)U(1)U(1) global one. Using a powerful algorithm we show that the system undergoes two transitions at different but still very close temperatures, one of Kosterlitz-Thouless (KT) type and another one which does not belong to the expected Ising universality class. A new analysis of the KT transition has been developed in order to avoid the use of the nonuniversal helicity jump and to allow the computation of the exponents without a precise determination of the critical temperature. Moreover, our huge number of data enables us to exhibit the existence of large finite size effects explaining the dispersed results found in the literature concerning the more studied frustrated 2D, XY\mathrm{XY}XY models.
We study the ferromagnetic transverse-field Ising model with quenched disorder at T = 0 T = 0 in one and two dimensions by means of stochastic series expansion quantum Monte Carlo simulations using a rigorous zero-temperature scheme. Using a sample-replication method and averaged Binder ratios, we determine the critical shift and width exponents \nu_\mathrm{s} ν s and \nu_\mathrm{w} ν w as well as unbiased critical points by finite-size scaling. Further, scaling of the disorder-averaged magnetisation at the critical point is used to determine the order-parameter critical exponent \beta β and the critical exponent \nu_{\mathrm{av}} ν a v of the average correlation length. The dynamic scaling in the Griffiths phase is investigated by measuring the local susceptibility in the disordered phase and the dynamic exponent z' z ′ is extracted. By applying various finite-size scaling protocols, we provide an extensive and comprehensive comparison between the different approaches on equal footing. The emphasis on effective zero-temperature simulations resolves several inconsistencies in existing literature.
On Critical Temperature and Finite Size Scaling of Continuous Spin 2d2d2d Ising Model
In this paper, we have studied the critical temperature TcT_cTc​ of continuous spin 2d2d2d square-lattice Ising model using Monte-Carlo simulation. We have considered spins sss in a bounded interval, where s∈[−1,+1]s \in [-1,+1]s∈[−1,+1] in square-lattice configuration with periodic boundary condition. We have observed that the critical temperature TcT_cTc​ is approximately 0.9250.9250.925, showing a clear second order phase transition. Considering finite size scaling, we have also obtained the critical exponents associated with susceptibility, specific heat, magnetization and we find that these values are in good agreement with the corresponding values obtained for the standard 2d2d2d Ising universality class.