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Análisis de transiciones de fase en un ferromagneto ideal aplicando el método de Monte Carlo a los modelos de Ising y Heisenberg en 2 y 3 dimensiones

Zegarra Borrero, Antonio Federico

Published Dec 1, 2013• 0 citations
OpenAlex

Abstract

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.

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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.
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.
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.
Using both recently developed cluster-algorithm and histogram methods, we have carried out a high-resolution Monte Carlo study of static critical properties of classical ferromagnetic Heisenberg models. Extensive Monte Carlo simulations were performed at several temperatures in the critical region, using an improved cluster-updating scheme, on L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L simple-cubic and body-centered-cubic systems with L\ensuremath{\le}40. Thermodynamic quantities as a function of temperature in the vicinity of the critical point were obtained by an optimized multiple-histogram method, and the critical temperature and static critical exponents were extracted using finite-size scaling. Our best estimates for the inverse critical temperatures are 0.693 035(37) for the simple-cubic system and 0.486 798(12) for the body-centered-cubic system. Estimated static critical exponents for both systems agree with each other within their respective error bars, and the mean estimates \ensuremath{\nu}=0.7048(30) and \ensuremath{\gamma}=1.3873(85) are also in excellent agreement with field-theoretic predictions 0.705(3) and 1.386(4).
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.
The critical behavior of 2D magnetic easy-plane systems has mainly been studied by the classical XY model. However, the z components of the spins have to be considered in order to describe real systems, and their fluctuations cannot positively be neglected when quantum effects are to be included, quantum spins being intrinsically three-component objects. Therefore, Monte Carlo simulations are performed for the Heisenberg ferromagnet with easy-plane anisotropy (XXZ model) on a two-dimensional square lattice with a twofold aim: first, to obtain accurate quantitative results about the critical behavior of the classical model, showing the relevant role played by the out-of-plane fluctuations; second, to open the way for approaching the quantum thermodynamics by means of the effective Hamiltonian method that reduces the quantum thermodynamics of the XXZ ferromagnet to the investigation of an effective classical model with temperature-dependent renormalized interaction parameters. Specific heat, magnetic susceptibility, and correlation length are calculated in the critical region for lattice sizes up to 128×128. These quantities preserve the Kosterlitz–Thouless behavior of the XY model. For the transition temperature of the classical XX0 model we obtain the estimate kBTc/(JS̃2)=0.69±0.01.
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.