二维伊辛模型临界模拟——基于蒙特卡洛方法的计算物理研究

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

摘要: 为展示计算物理方法在现代统计物理研究中的应用，本文以二维正方格点伊辛模型为研究对象，对有限尺寸铁磁体系的临界行为进行数值模拟与分析。研究建立了包含最近邻相互作用的伊辛模型哈密顿量，并在周期边界条件下采用Metropolis蒙特卡洛方法进行抽样，计算能量、磁化强度、比热、磁化率和Binder累积量等物理量，以刻画体系由有序态向无序态转变的过程。模拟结果表明，在低温区域，体系自旋排列具有明显的长程有序特征，序参量保持在较高水平；随着温度升高，热涨落逐渐增强，磁化强度迅速衰减，体系逐步转变为无序状态。比热曲线在约T = 2.269处出现明显峰值，磁化率峰值受有限尺寸效应和温度离散步长影响而略向高温侧偏移，整体临界区域与二维正方格点伊辛模型的理论临界温度基本一致。自旋构型分析进一步显示，体系在临界附近存在显著的涨落增强和畴结构重组现象。同时，本文补充了热化判断标准，并针对L = 48临界区比热峰值进行了延长采样复核，以检验有限采样和热化不足对结果的影响。研究表明，基于蒙特卡洛方法的二维伊辛模型模拟能够较好再现连续相变、临界涨落和有限尺寸效应，可为理解复杂系统临界现象以及现代数据驱动物理方法提供基础案例。

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.