Introduction to Monte Carlo methods for an Ising Model of a Ferromagnet

Jacques Kotze

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arXiv · 0803.0217

Introduction to Monte Carlo methods for an Ising Model of a Ferromagnet

Jacques Kotze

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Abstract

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.

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Jacques Kotze. 2008-03-03. Introduction to Monte Carlo methods for an Ising Model of a Ferromagnet. https://arxiv.org/abs/0803.0217

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