We have investigated the phase transition of two-dimensional Ising model by Monte Carlo method.Our results reveal that a finite-temperature phase transition occurs in the Ising model.The calculation result agrees well with the theoretical one.The specific heat and the susceptibility peak at the critical temperature while they descend at both high and low temperature.The spontaneous magnetization approaches to unit when the temperature is below the critical temperature.
We have investigated the phase transition of two-dimensional Ising model by Monte Carlo method.Our results reveal that a finite-temperature phase transition occurs in the Ising model.The calculation result agrees well with the theoretical one.The specific heat and the susceptibility peak at the critical temperature while they descend at both high and low temperature.The spontaneous magnetization approaches to unit when the temperature is below the critical temperature.
Phase transitions are widespread physical phenomena in nature and the phase transition of ferromagnetic substances which can be described by Ising model is an important part of physics. This paper discusses the phase transition of the low-dimensional Ising model. For one-dimensional Ising model, the exact solution is given and no phase transition is found when there is no external magnetic field. For two-dimensional Ising model, Monte Carlo method along with importance sampling, careful balance conditions and the Metropolis algorithm is introduced to numerically simulate it. The trends of order parameters like magnetization, specific heat and magnetic susceptibility are analyzed and the mutation of some order parameters is found around critical temperature. The influences of external magnetic field and the scale of system are discussed and critical temperature of phase transition is explored by several methods, which is eventually found around 2.269 ( kBT/J). Lastly, compared with theoretical values, the disadvantages are pointed out and optimization suggestions are raised for improvement.
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 location of the phase transition in the two dimensional Ising model were determined using Monte Carlo simulation with importance sampling. The magnetization persite ,energy per site , magnetic susceptibility , specific heat of a Ferromagnetic materials were calculated as a function of temperature for spin lattice interaction in zero and nonzero magnetic field . There is thus a phase transition defined by the Curie temperature. The Monte Carlo method was used to check the results and to confirm the phase transition . The internal interaction results were found to be consistent with what was expected. As a magnetic field is applied, the spins tend to align with it for ˃ and its effect is not significant at a very high temperature because of the thermal agitation. For ˂ , the alignment of the spins is possible only if the amplitude of the field is big enough.
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.
In this project, the location of the phase transition in the two dimensional Ising model will be determined using Monte Carlo simulation with importance sampling. the magnetization per site [m ], energy per site [J ], magnetic susceptibility, specific heat of a Ferromagnetic materials are Calculated as a function of temperature T for 10´10, 20´ 20, 40´ 40 , 50´50 spin lattice interaction by using Monte Carlo Simulation of the 2D Ising Model for some experimental values of ferromagnetic materials such as Gadolinium Chloride ( ) 3 GdCl at Curie temperature c T B = 2.2 J k , and ferromagnetic thin film from Nickel ( ) i N growth on cooper (Cu)at Curie temperature c B T = 2.772 J k , in zero and nonzero magnetic field. It was noticed that above a certain temperature ( ) c T the material will be in a paramagnetic state, this will lead to that the average magnetization will be decrease and the average energy increase, while below that temperature ,it will be in a ferromagnetic state, and the average magnetization will increase and the average energy decrease. Moreover, above a certain temperature spontaneous magnetization will be zero
Abstract The two-dimensional anisotropic Ising model with competing interactions in the region of a transition from the ferromagnetic to paramagnetic phase has been studied by the Monte Carlo method using the Wang–Landau algorithm. State density distributions have been obtained and the order parameters corresponding to different energies have been calculated. The temperature dependences of the thermodynamic parameters including the internal energy, magnetization, free energy, entropy, specific heat at constant magnetization, and Binder cumulants have been calculated. Using the Binder cumulant method and histogram analysis, it has been shown that a single second-order phase transition from the ferromagnetic to paramagnetic phase occurs at two exchange coupling constants ratios |J1/J| = 0.1 and 0.2 in the two-dimensional anisotropic next-nearest neighbor Ising model. The phase transition temperatures have been calculated. The main critical exponents have been determined using the finite-size scaling theory.