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

Physical review. B, Solid state

Finite-size behavior of the Ising square lattice

David Paul Landau

Published Apr 1, 1976• 411 citations
Publisher

Abstract

An importance-sampling Monte Carlo method is used to study $N\ifmmode\times\else\texttimes\fi{}N$ Ising square lattices with nearest-neighbor interactions and either free edges or periodic boundary conditions. The internal energy, specific heat, order parameter, susceptibility, and near-neighbor spin-spin correlation functions of the finite lattices are determined as a function of $N$ and extrapolated to the corresponding infinite-system values. The effect of finite size is greater for free edges in all cases. The results agree well with predictions of finite size scaling theory and the shape functions as well as amplitudes of surface contribution terms are determined.

DOI
10.1103/physrevb.13.2997

Related papers

Finite size behavior of the Ising square lattice with free edges

David Paul Landau

Using an importance sampling Monte Carlo technique we have studied NxN Ising square lattices with free edges for 3≲N≲60. Both the shift in ’’ordering temperature’’ Tc(N) as well as the broadening of the transition due to finite N are greater than for identical lattices with periodic boundary conditions (p.b.c). We find that 1−Tc(N)/Tc(∝) =a/N with a=1.24±0.04 and that (C/R)max?0.33 ln N +0.2. For N≳30 the specific heat begins to approach the same asymptotic size dependence as found for much smaller lattices with p.b.c. Similarly, the finite size dependence of the order parameter is much more dramatic for free edges than p.b.c. and the limiting behavior is not reached until N≳30.

AIP conference proceedings•Jan 1, 1975•7 citations

Phase transitions in the Ising square lattice with next-nearest-neighbor interactions

David Paul Landau

An importance sampling Monte Carlo method has been used to study $L\ifmmode\times\else\texttimes\fi{}L$ Ising square lattices with periodic boundary conditions. The behavior of the thermal and magnetic properties was determined over a wide range of $R$, the ratio of next-nearest- to nearest-neighbor coupling. From these data we have extracted the $R$ dependence of the critical temperature and a range of other critical parameters. Within our experimental error the critical exponents are independent of $R$ but the critical amplitudes are not. The data do however suggest a change in $\ensuremath{\alpha}$ for $R$ slightly greater than \textonehalf{}. Variations in critical entropy and internal energy may be as large as those resulting from changes in lattice or spin dimensionalities. Our results are compared and contrasted with those obtained by a range of theoretical methods.

Physical review. B, Condensed matter•Feb 1, 1980•107 citations

Finite-size behavior of the simple-cubic Ising lattice

David Paul Landau

A Monte Carlo method is used to study $N\ifmmode\times\else\texttimes\fi{}N\ifmmode\times\else\texttimes\fi{}N$ simple-cubic Ising lattices with periodic boundary conditions and free edges. For both types of boundary conditions the position of the specific-heat maximum varies for large $N$ as $a{N}^{\ensuremath{-}\ensuremath{\lambda}}$, where $\ensuremath{\lambda}$ has the scaling value $\ensuremath{\lambda}={\ensuremath{\nu}}^{\ensuremath{-}1}$. Both the thermal and magnetic properties are shown to obey finite-size scaling. The free-edge data are shown to be consistent with a surface contribution described by the scaling exponents ${\ensuremath{\alpha}}_{s}=\ensuremath{\alpha}+\ensuremath{\nu}$, ${\ensuremath{\beta}}_{s}=\ensuremath{\beta}\ensuremath{-}\ensuremath{\nu}$, ${\ensuremath{\gamma}}_{s}=\ensuremath{\gamma}+\ensuremath{\nu}$. Using the free-edge data we also consider corrections to scaling in the infinite lattice and discuss "rounding" in real systems in terms of surface contributions from grains.

Physical review. B, Solid state•Jul 1, 1976•169 citations

Critical properties of an antiferromagnetic Ising model on a square lattice with interactions of the next-to-nearest neighbors

Akai Kurbanovich Murtazaev, Рамазанов Магомедшейх Курбанович, М. К. Бадиев

The critical properties of an antiferromagnetic Ising model on a square lattice with interactions of the next-to-nearest neighbors are investigated by a replica Monte-Carlo method. Using the finite-size scaling theory the static critical exponents of specific heat, ordering parameter, susceptibility, correlation radius as well as the Fisher exponent are calculated. An analysis of data is performed both with and without taking into account a correction to the finite-size scaling. It was found that in the model under consideration the second order phase transition is observed. It is shown that this model belongs to the new class of universality of critical behavior.

Low Temperature Physics•Dec 1, 2011•24 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

Ising Antiferromagnet with Nearest-Neighbor and Next-Nearest-Neighbor Interactions on a Square Lattice

Akai Kurbanovich Murtazaev, Рамазанов Магомедшейх Курбанович, М. К. Бадиев

The critical properties of two-dimensional antiferromagnetic Ising model in square lattice are investigated using the replica Monte-Carlo method with account of interactions of second nearest neighbors. The diagram of critical temperature dependence on an interaction value of second nearest neighbors is plotted. Static critical exponents of the heat capacity α, susceptibility γ, magnetization β, and correlation radius ν are calculated for this model using the finite-size scaling theory.

Diffusion and defect data, solid state data. Part B, Solid state phenomena/Solid state phenomena•Apr 1, 2014•0 citations
An importance-sampling Monte Carlo method is used to study N\ifmmode×\else\texttimes\fiNN\ifmmode\times\else\texttimes\fi{}NN\ifmmode×\else\texttimes\fiN Ising square lattices with nearest-neighbor interactions and either free edges or periodic boundary conditions. The internal energy, specific heat, order parameter, susceptibility, and near-neighbor spin-spin correlation functions of the finite lattices are determined as a function of NNN and extrapolated to the corresponding infinite-system values. The effect of finite size is greater for free edges in all cases. The results agree well with predictions of finite size scaling theory and the shape functions as well as amplitudes of surface contribution terms are determined.
Using an importance sampling Monte Carlo technique we have studied NxN Ising square lattices with free edges for 3≲N≲60. Both the shift in ’’ordering temperature’’ Tc(N) as well as the broadening of the transition due to finite N are greater than for identical lattices with periodic boundary conditions (p.b.c). We find that 1−Tc(N)/Tc(∝) =a/N with a=1.24±0.04 and that (C/R)max?0.33 ln N +0.2. For N≳30 the specific heat begins to approach the same asymptotic size dependence as found for much smaller lattices with p.b.c. Similarly, the finite size dependence of the order parameter is much more dramatic for free edges than p.b.c. and the limiting behavior is not reached until N≳30.
An importance sampling Monte Carlo method has been used to study $L\ifmmode\times\else\texttimes\fi{}L$ Ising square lattices with periodic boundary conditions. The behavior of the thermal and magnetic properties was determined over a wide range of RRR, the ratio of next-nearest- to nearest-neighbor coupling. From these data we have extracted the RRR dependence of the critical temperature and a range of other critical parameters. Within our experimental error the critical exponents are independent of RRR but the critical amplitudes are not. The data do however suggest a change in $\ensuremath{\alpha}$ for RRR slightly greater than \textonehalf{}. Variations in critical entropy and internal energy may be as large as those resulting from changes in lattice or spin dimensionalities. Our results are compared and contrasted with those obtained by a range of theoretical methods.
A Monte Carlo method is used to study $N\ifmmode\times\else\texttimes\fi{}N\ifmmode\times\else\texttimes\fi{}N$ simple-cubic Ising lattices with periodic boundary conditions and free edges. For both types of boundary conditions the position of the specific-heat maximum varies for large NNN as $a{N}^{\ensuremath{-}\ensuremath{\lambda}}$, where $\ensuremath{\lambda}$ has the scaling value $\ensuremath{\lambda}={\ensuremath{\nu}}^{\ensuremath{-}1}$. Both the thermal and magnetic properties are shown to obey finite-size scaling. The free-edge data are shown to be consistent with a surface contribution described by the scaling exponents ${\ensuremath{\alpha}}_{s}=\ensuremath{\alpha}+\ensuremath{\nu}$, ${\ensuremath{\beta}}_{s}=\ensuremath{\beta}\ensuremath{-}\ensuremath{\nu}$, ${\ensuremath{\gamma}}_{s}=\ensuremath{\gamma}+\ensuremath{\nu}$. Using the free-edge data we also consider corrections to scaling in the infinite lattice and discuss "rounding" in real systems in terms of surface contributions from grains.
The critical properties of an antiferromagnetic Ising model on a square lattice with interactions of the next-to-nearest neighbors are investigated by a replica Monte-Carlo method. Using the finite-size scaling theory the static critical exponents of specific heat, ordering parameter, susceptibility, correlation radius as well as the Fisher exponent are calculated. An analysis of data is performed both with and without taking into account a correction to the finite-size scaling. It was found that in the model under consideration the second order phase transition is observed. It is shown that this model belongs to the new class of universality of critical behavior.
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.