Statistical physics of the 2D Ising model: simulations based on the Monte Carlo method

Abstract

The general objective of this undergraduate research project is to introduce the candidate to the application of numerical simulation techniques to problems in the context of Statistical Physics. Specifically, we are interested in addressing one of the most important models in statistical physics, the Ising model. In addition to representing a simplified mathematical representation for ferromagnetism, its 2D version on a square lattice is one of the simplest models to display a second-order phase transition. In this context, we plan to study the thermodynamic properties of the 2D Ising model on a square lattice by classical statistical mechanics using Monte Carlo simulation techniques.The project involves three key ingredients. The first concerns the basic study of the 2D Ising model, with the emphasis on theoretical statistical approach used for the description of the physical properties of this system. The second involves the study of the fundamentals of Monte Carlo simulation techniques used to compute statistical quantities in terms of ensemble averages. The third ingredient involves the implementation of Monte Carlo techniques to specifically study the Ising model in 2D. Among them are the single-spin flip as well as cluster algorithms and analyze their efficiencies upon approaching the thermodynamic conditions in which the phase transition occurs. In these efforts the results will also be compared to the exact solution that exists for the Ising model in 2D on a square lattice.