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Estudo computacional estático e dinâmico do modelo clássico de Heisenberg ferromagnético isotrópico

Alexandre de Aquino Soares

Published Mar 4, 2011• 0 citations

Abstract

We studied the energy, the magnetization and cumulants of the three-dimesional isotropic Heisenberg model in the phase transition region by using Metropolis algorithm. In order to study the critical temperature we implemented the vectorized Metropolis, hybrid Metropolis and Wolff (the latter one as prescribed by Chen et al (4)). We choseMetropolis:Overrelaxation 1:4 for its fastest independent measurements yield. We implemented Bereau and Swendsen's optimized multi- histogram technique (13) for this study. We obtained through finite size scaling the critical exponents = -0, 0709 +- 0,0099, = 0,3499 +- 0,0076, = 1,3880 +- 0,0060 and = 0,6903 +- 0,0034, the inverse criticaltemperature Kc = 0; 69314 +- 0,00032, and the universal parameter Binder cumulant for an infinite lattice at critical temperature U4B = 0, 62178 +- 0,00049 by means of a strategy involving jackknife to reduce bias. These exponents agree with experimental, theoretical and simulational determinations from literature. We also investigated energy and spin time autocorrelations in unidimensional lattices (an integrable L = 4 system and a non-integrable L = 18 system), by integration with fourth order Runge-Kutta, Predictor-Corrector, Suzuki-Trotter and optimized Forest-Ruth methods. We found that the use of quad precision, an integration step of approximately 10..6, N = 106 sampling, and longspin chains may be sucient to discriminate between an expected and an anomalous spin diffusion. Besides, each algorithm implementation may correspond to a dierent time operator for the non-integrable system.

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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.

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We have studied the critical properties of the three-dimensional random anisotropy Heisenberg model by means of numerical simulations using the Parallel Tempering method. We have simulated the model with two different disorder distributions, cubic and isotropic ones, with two different anisotropy strengths for each disorder class. For the case of the anisotropic disorder, we have found evidence of universality by finding critical exponents and universal dimensionless ratios independent of the strength of the disorder. In the case of isotropic disorder distribution the situation is very involved: we have found two phase transitions in the magnetization channel which are merging for larger lattices remaining a zero magnetization low-temperature phase. Studying this region using a spin-glass order parameter we have found evidence for a spin-glass phase transition. We have estimated effective critical exponents for the spin-glass phase transition for the different values of the strength of the isotropic disorder, discussing the crossover regime.