Magnetic Susceptibility in the Ising Model

Abstract

Ferromagnetism emerges when a collection of particle spins align such that their associated magnetic moments all point in the same direction, yielding a macroscopic sized net magnetic moment. The simplest theoretical description of ferromagnetism was given by the Ising model. This model was invented by Wilhelm Lenz in 1920 and it is named after his student Ernst Ising who introduced it in his PhD thesis in 1925. Ising has introduced his model by supposing nearest-neighbor interactions between spins on a lattice and an interaction with an external magnetic field H. Using statistical physics methods, the model has been solved exactly in one dimension (for any H) and in two dimensions (for H=0) and the corresponding free energy and spontaneous magnetization have been obtained in terms of the temperature. In two dimensions, it can be seen that this model can describe and explain phase transition phenomena. Another quantity of physical interest for this model and which has been widely investigated is the magnetic susceptibility. It has still no clear analytical status, however, we know that it can be expressed as an infinite sum of spin correlation functions. The aim of our work in this thesis is to present an algorithm, using quadratic recursion relations, which allows the calculations of the analytical expressions of these correlation functions C(M,N) in terms of the temperature for all values of M and N(which represent the position of the spn in a lattice) in the isotopic case (J= J') when the magnetic field is zero (H = 0). Key Words: Correlation function, Ising model, ferromagnetism, Toeplitz