In this work we present a study of the thermodynamical properties of the spin-1=2 two-dimensional Ising model with ferromagnetic interactions at zero external magnetic eld. We used Monte Carlo simulations, through Metropolis andWang-Landau algorithms. The critical exponents and the corresponding critical temperatures are obtained by both methods using the nite-size scaling hypothesis. A comparison between the exact values and the results obtained from both algorithms was made, not only quantitatively but also in terms of computational times.
In this work we present a study of the thermodynamical properties of the spin-1=2 two-dimensional Ising model with ferromagnetic interactions at zero external magnetic eld. We used Monte Carlo simulations, through Metropolis andWang-Landau algorithms. The critical exponents and the corresponding critical temperatures are obtained by both methods using the nite-size scaling hypothesis. A comparison between the exact values and the results obtained from both algorithms was made, not only quantitatively but also in terms of computational times.
In this work we study the magnetic properties of the ferromagnetic Ising model on a square lattice, in zero external magnetic eld, through Monte Carlo simulations. The initial part comprises the theoretical background and the simulations as the Monte Carlo code, the single histogram technique and the estimate of the error bars. For the particular four-spin cluster the exact results have been obtained and compared to the simulational ones. Such comparison granted a better understanding of the process of generating the computational data besides a way of conrming the correctness of the code. Together with the single histogram technique and the nite size scaling hypothesis. We have obtained the mean values of the magnetization, energy, specic heat, susceptibility and the Binder cumulant. It was possible to obtain the thermodynamic quantities as a function of temperature and the system size. It was evident the existence of a phase transition with its peculiar behavior. Accordingly, we have obtained the critical temperature as well as the critical exponents of the model. The values of the critical indices and the critical temperature agree well with the exact ones and those coming from other computational methods.
The present thesis is focused on the study of the models of Ising and Heisenberg (classical) for ferromagnetism, the theory of phase transitions and the Monte Carlo method as an efficient computational scheme for dealing with these kinds of strongly interacting systems. With the aid of this method, computational simulations were performed for the above mentioned models in 2 and 3 dimensions (2D and 3D) in order to compare these results with the predictions of some theoretical models. The method followed to model these systems is indicated as well as the process for constructing phase diagrams for the main thermodynamic quantities of interest (mean magnetization, mean energy, specific heat, magnetic susceptibility) and their behavior in a vicinity of the critical point is analyzed. Furthermore, a quantitative analysis is carried out to calculate the corresponding critical exponents (_; _; and _) and to prove the phenomenon of universality they fulfill for different types of crystalline structure (3D case) and for the 2 models previously cited (Ising and Heisenberg). The programs written down in C++ that were constructed with the aim of performing the respective simulations are also included. \nKeywords: models of Ising and Heisenberg, ferromagnetism, phase transitions, Monte Carlo method, critical point, critical exponents, universality.
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.
This discussion serves as an introduction to the use of Monte Carlo simulations as a useful way to evaluate the observables of a ferromagnet. Key background is given about the relevance and effectiveness of this stochastic approach and in particular the applicability of the Metropolis-Hastings algorithm. Importantly the potentially devastating effects of spontaneous magnetization are highlighted and a means to avert this is examined. An Ising model is introduced and used to investigate the properties of a two dimensional ferromagnet with respect to its magnetization and energy at varying temperatures. The observables are calculated and a phase transition at a critical temperature is also illustrated and evaluated. Lastly a finite size scaling analysis is undertaken to determine the critical exponents and the Curie temperature is calculated using a ratio of cumulants with differing lattice sizes. The results obtained from the simulation are compared to exact calculations to endorse the validity of this numerical process. A copy of the code used, written in C++, is enclosed and is freely available for use and modification under the General Public License.
In this paper, we theoretically study the critical properties of the classical spin-1 Ising model using two approaches: 1) the analytical low-temperature series expansion and 2) the numerical Metropolis Monte Carlo technique. Within this analysis, we discuss the critical behavior of one-, two- and three-dimensional systems modeled by the first-neighbor spin-1 Ising model for different types of exchange interactions. The comparison of the results obtained according the Metropolis Monte Carlo simulations allows us to highlight the limits of the widely used mean-field theory approach. We also show, via a simple transformation, that for the special case where the bilinear and bicubic terms are set equal to zero in the Hamiltonian the partition function of the spin-1 Ising model can be reduced to that of the spin-1/2 Ising model with temperature dependent external field and temperature independent exchange interaction times an exponential factor depending on the other terms of the Hamiltonian and confirm this result numerically by using the Metropolis Monte Carlo simulation. Finally, we investigate the dependence of the critical temperature on the strength of long-range interactions included in the Ising Hamiltonian comparing it with that of the first-neighbor spin-1/2 Ising model.
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.