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

Journal of Applied Physics

The Ising model between one and two dimensions

M. A. Novotny

Published Apr 15, 1988• 2 citations
Publisher

Abstract

A method which continuously interpolates between one and two dimensions for any magnetic model on a square lattice is introduced. This interpolation scheme, which preserves the translational invariance of the lattice, can be performed for any Hamiltonian with pairwise interactions. The Potts model interpolated between one and two dimensions is studied using finite-size scaling of data obtained with a Monte Carlo method.

DOI
10.1063/1.340738

Related papers

Numerical Transfer Matrix Study of the Ising Model between One and Two Dimensions

M. A. Novotny

A method is presented which interpolates continuously between one and two dimensions for classical statistical-mechanical models. The interpolation scheme preserves the translational invariance of the system. A numerical transfer matrix study is made using the interpolation scheme on the Ising model. Results for the thermal and magnetic critical exponent as a function of dimension are obtained which are in good agreement with d = 1 + ε and d = 4 - ε expansions.

Europhysics Letters (EPL)•Jan 21, 1992•10 citations

Critical exponents for the Ising model between one and two dimensions

M. A. Novotny

Critical exponents are presented for the ferromagnetic Ising model in dimensions between one and two. The exponents are calculated by applying finite-size-scaling methods to numerical transfer-matrix data. The transfer matrix, which is translationally invariant, is written in such a fashion that interpolation to noninteger dimensions is possible. Results are also obtained for the first derivatives evaluated at d=2 for the critical exponents ${\mathit{y}}_{\mathit{T}}$ and ${\mathit{y}}_{\mathit{H}}$ as a function of dimension. All results are compared with series-expansion results in 1d\ensuremath{\le}2.

Physical review. B, Condensed matter•Aug 1, 1992•34 citations

Extrapolating Monte Carlo Simulations to Infinite Volume: Finite-Size Scaling at ξ / L ≫ 1

Sergio Caracciolo A, Robert G. Edwards, Sabino José Ferreira +2 more

We present a simple and powerful method for extrapolating finite-volume Monte Carlo data to infinite volume, based on finite-size-scaling theory. We discuss carefully its systematic and statistical errors, and we illustrate it using three examples: the two-dimensional three-state Potts antiferromagnet on the square lattice, and the two-dimensional $O(3)$ and $O(\ensuremath{\infty})$ \ensuremath{\sigma} models. In favorable cases it is possible to obtain reliable extrapolations (errors of a few percent) even when the correlation length is 1000 times larger than the lattice.

arXiv:hep-lat/9409004•Physical Review Letters•Apr 10, 1995•105 citations

Antiferromagnetic Potts models on the square lattice

Sabino José Ferreira, Alan D. Sokal

We study the antiferromagnetic q-state Potts model on the square lattice for q=3 and 4, using the Wang-Swendsen-Koteck\'y Monte Carlo algorithm and a finite-size-scaling extrapolation method. For q=3, we obtain good control up to correlation length \ensuremath{\xi}\ensuremath{\sim}5000; the data are consistent with \ensuremath{\xi}(\ensuremath{\beta})=${\mathit{Ae}}^{2\mathrm{\ensuremath{\beta}}}$\ensuremath{\beta}(1+${\mathit{a}}_{1}$${\mathit{e}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\beta}}}$+...) as \ensuremath{\beta}\ensuremath{\rightarrow}\ensuremath{\infty}. For q=4, the model is disordered even at zero temperature.

arXiv:hep-lat/9405015•Physical review. B, Condensed matter•Mar 1, 1995•26 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

NEW METHOD FOR THE EXTRAPOLATION OF FINITE-SIZE DATA TO INFINITE VOLUME

Caracciolo, Sergio, Edwards, R G, Ferreira, S J +2 more

We present a simple and powerful method for extrapolating finite-volume Monte Carlo data to infinite volume, based on finite-size-scaling theory. We discuss carefully its systematic and statistical errors, and we illustrate it using three examples: the two-dimensional three-state Potts antiferromagnet on the square lattice, and the two-dimensional O(3) and O(\\infty) \\sigma-models. In favorable cases it is possible to obtain reliable extrapolations (errors of a few percent) even when the correlation length is 1000 times larger than the lattice.

•Nov 28, 1994•2 citations
A method which continuously interpolates between one and two dimensions for any magnetic model on a square lattice is introduced. This interpolation scheme, which preserves the translational invariance of the lattice, can be performed for any Hamiltonian with pairwise interactions. The Potts model interpolated between one and two dimensions is studied using finite-size scaling of data obtained with a Monte Carlo method.
Critical exponents are presented for the ferromagnetic Ising model in dimensions between one and two. The exponents are calculated by applying finite-size-scaling methods to numerical transfer-matrix data. The transfer matrix, which is translationally invariant, is written in such a fashion that interpolation to noninteger dimensions is possible. Results are also obtained for the first derivatives evaluated at d=2 for the critical exponents yT{\mathit{y}}_{\mathit{T}}yT​ and yH{\mathit{y}}_{\mathit{H}}yH​ as a function of dimension. All results are compared with series-expansion results in 1d\ensuremath{\le}2.
We present a simple and powerful method for extrapolating finite-volume Monte Carlo data to infinite volume, based on finite-size-scaling theory. We discuss carefully its systematic and statistical errors, and we illustrate it using three examples: the two-dimensional three-state Potts antiferromagnet on the square lattice, and the two-dimensional O(3)O(3)O(3) and $O(\ensuremath{\infty})$ \ensuremath{\sigma} models. In favorable cases it is possible to obtain reliable extrapolations (errors of a few percent) even when the correlation length is 1000 times larger than the lattice.
We study the antiferromagnetic q-state Potts model on the square lattice for q=3 and 4, using the Wang-Swendsen-Koteck\'y Monte Carlo algorithm and a finite-size-scaling extrapolation method. For q=3, we obtain good control up to correlation length \ensuremath{\xi}\ensuremath{\sim}5000; the data are consistent with \ensuremath{\xi}(\ensuremath{\beta})=${\mathit{Ae}}^{2\mathrm{\ensuremath{\beta}}}$\ensuremath{\beta}(1+a1{\mathit{a}}_{1}a1​${\mathit{e}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\beta}}}$+...) as \ensuremath{\beta}\ensuremath{\rightarrow}\ensuremath{\infty}. For q=4, the model is disordered even at zero temperature.
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 present a simple and powerful method for extrapolating finite-volume Monte Carlo data to infinite volume, based on finite-size-scaling theory. We discuss carefully its systematic and statistical errors, and we illustrate it using three examples: the two-dimensional three-state Potts antiferromagnet on the square lattice, and the two-dimensional O(3) and O(\\infty) \\sigma-models. In favorable cases it is possible to obtain reliable extrapolations (errors of a few percent) even when the correlation length is 1000 times larger than the lattice.