In this research, the 2-dimensional Ising system was studied using a Monte-Carlo simulation based on the Metropolis Algorithm. A systemof 2500 spins in a square lattice was simulated at varying temperatures and magnetic fields to obtain the critical temperature. Furthermore, themagnetisation and energy per spin and specific heat and magnetic susceptibility were determined as function of temperature. Lastly, the effect ofa magnetic field on the magnetisation of the system was studied.
In this research, the 2-dimensional Ising system was studied using a Monte-Carlo simulation based on the Metropolis Algorithm. A systemof 2500 spins in a square lattice was simulated at varying temperatures and magnetic fields to obtain the critical temperature. Furthermore, themagnetisation and energy per spin and specific heat and magnetic susceptibility were determined as function of temperature. Lastly, the effect ofa magnetic field on the magnetisation of the system was studied.
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.
Using the Monte Carlo Simulation (MCS) method combined with the Metropolis algorithm, we explore the magnetic properties of a two-dimensional (2D) spin system with spins 1, 2 and 3. The adapted simulation to the Ising model aims to study the impact of the correction term a of the interaction with second neighbors and the crystal field D on the critical temperature [Formula: see text] of the 2D anisotropic square lattice for different integer values of spins. The obtained results demonstrate that the critical temperature [Formula: see text] is significantly affected by the competing interactions between the anisotropy parameter, second-neighbor interaction and the crystal field. The considered model exhibits a critical phase transition regardless of the spin S value where the critical temperature increases with the value of S. Note that this phase transition occurs under the condition [Formula: see text]. Moreover, the obtained results show that the complex interactions give rise to a variety of phase diagrams, thereby revealing the richness of possible magnetic behaviors in the studied system.
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.
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 have studied the thermodynamic and magnetic properties of the spin 1/2 XXZ Ising model on a two dimensional square lattice using Monte Carlo simulation, in the presence of transverse magnetic field for varying lattice sizes (N=8-40) and exchange interaction strengths. The variations in specific heat and susceptibility for different N values is almost negligible, except for N=8. Interestingly, the magnetization showed large variation with N. This could be attributed to the increase in the disorder of spin alignment with increasing N.
We simulate the Ising Model with the Monte Carlo method and use the algorithms of Metropolis to update the distribution of spins. We found that, in the specific case of the three-dimensional Ising Model, methods of Metropolis are efficient. Studying the system near the point of phase transition, we observe that the magnetization goes to zero. In our simulations we analyzed the behavior of the magnetization and magnetic susceptibility to verify the phase transition in a paramagnetic to ferromagnetic material. The behavior of the magnetization and of the magnetic susceptibility as a function of the temperature suggest a phase transition around KT/J ≈ 4.5 and was evidenced the problem of finite size of the lattice to work with large lattice.