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Modern Physics

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

佳钢 李

Published Jan 1, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.12677/mp.2026.165016

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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.
This study presents a comprehensive Monte Carlo simulation of the two-dimensional (2-D) Ising model using the Metropolis algorithm to investigate critical phenomena and thermodynamic behavior in spin-lattice systems. The model, implemented in MATLAB with periodic boundary conditions, explores equilibrium properties such as magnetization, internal energy, specific heat, and susceptibility across a range of temperatures. By employing various initial spin configurations—ordered and random—the simulations demonstrate the system's ergodicity and convergence to thermal equilibrium. Key results include a sharp decline in magnetization and a pronounced peak in specific heat near the critical temperature, consistent with second-order phase transition behavior. The simulation captures microscopic domain evolution, highlighting the transition from ferromagnetic to paramagnetic phases as thermal fluctuations increase. The study further evaluates algorithmic efficiency, discusses the impact of lattice size on statistical accuracy, and proposes improvements using advanced cluster algorithms and parallel computing frameworks. The findings validate theoretical predictions from Onsager's solution and underscore the versatility of Monte Carlo techniques in modeling collective behavior in magnetic systems. The simulation framework offers a robust foundation for analyzing critical dynamics and extends its relevance to broader applications in material science, computational physics, and complex systems modeling.
Many problems in statistical physics resist exact solution once the number of interacting particles becomes large, and the two-dimensional Ising model is a classic case where numerical simulation fills that gap. This paper reports a Monte Carlo study of the two-dimensional Ising model using the Metropolis algorithm, undertaken to examine how magnetization, internal energy, specific heat, and magnetic susceptibility change with temperature. Spin configurations were generated on a square lattice under periodic boundary conditions, with each proposed spin flip accepted or rejected according to the Boltzmann weight of the resulting energy change. The simulation reproduces the expected transition from an ordered ferromagnetic state at low temperature to a disordered paramagnetic state at high temperature, with magnetization falling sharply and both specific heat and susceptibility peaking near a critical temperature of Tc ≈ 2.3–2.4 (in units of J/kB), consistent with the exact value obtained analytically for this system. These results confirm that the Metropolis algorithm, despite its simplicity, remains a dependable tool for locating second-order phase transitions and extracting thermodynamic quantities in lattice spin systems, and the same computational framework can be adapted to more elaborate models used in condensed matter and materials research.
The magnetic behavior of a two-dimensional nearest-neighbor Ising model with the presence of linear temperature variation in a thermal steady state was studied using the Wolff Monte Carlo simulation. The technique consists of fixing the temperatures of boundary spins, while the temperature field in the interior linearly varies with distance. It is found that with increasing the temperature difference between the two boundaries, the magnetization greatly reduces in magnitude while the susceptibility peaks tend to spread out over a temperature range. The detailed descriptions of these magnetization and susceptibility behaviors are elucidated from their spatial variation. The extraction of the ``critical temperatures'' is taken via the fourth-order cumulant of the magnetization. The critical temperatures are found to reduce slightly with increasing the temperature difference. This implies the vulnerability of the magnetization and susceptibility properties to the temperature variation in ferromagnetic materials, and to use such materials in temperature variation environments must be done with caution.
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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.
In this work we present a thorough analysis of the phase transitions that occur in a ferromagnetic two-dimensional Ising model, with only nearest-neighbors interactions, in the framework of the Tsallis nonextensive statistics. We performed Monte Carlo simulations on square lattices with linear sizes L ranging from 32 up to 512. The statistical weight of the Metropolis algorithm was changed according to the nonextensive statistics. Discontinuities in the m(T) curve are observed for $q\ensuremath{\le}0.5$. However, we have verified only one peak on the energy histograms at the critical temperatures, indicating the occurrence of continuous phase transitions. For the $0.5<q\ensuremath{\le}1.0$ regime, we have found continuous phase transitions between the ordered and the disordered phases, and determined the critical exponents via finite-size scaling. We verified that the critical exponents $\ensuremath{\alpha}$, $\ensuremath{\beta}$, and $\ensuremath{\gamma}$ depend on the entropic index qqq in the range $0.5<q\ensuremath{\le}1.0$ in the form $\ensuremath{\alpha}(q)=(10{q}^{2}\ensuremath{-}33q+23)/20$, $\ensuremath{\beta}(q)=(2q\ensuremath{-}1)/8$, and $\ensuremath{\gamma}(q)=({q}^{2}\ensuremath{-}q+7)/4$. On the other hand, the critical exponent $\ensuremath{\nu}$ does not depend on qqq. This suggests a violation of the scaling relations $2\ensuremath{\beta}+\ensuremath{\gamma}=d\ensuremath{\nu}$ and $\ensuremath{\alpha}+2\ensuremath{\beta}+\ensuremath{\gamma}=2$ and a nonuniversality of the critical exponents along the ferro-paramagnetic frontier.