Solutions to the one-dimensional Heisenberg model

Abstract

In this thesis we take a look at the one-dimensional Heisenberg model. The Heisenberg model is the most used model to study the magnetic properties of materials and is of paramount importance for understanding magnets. First we sketch a brief derivation of the model and discuss its strengths and weaknesses. Afterwards we develop a transformation of the Heisenberg model to a model of spinless fermions, which allows for an easy solution of the Jz = 0 model. After solving the model for Jz = 0, we solve the model for general Jz. The results we find are very complicated and it is hard to calculate anything explicitly. So we move to to the thermodynamic limit to calculate the ground state energy for Jz = 1. After that we have found a solution for big system sizes, we use exact diagonalisation to solve the Heisenberg model. We do this for system sizes up too 12 particles. With our results we check how accurate some expansions for smaller systems sizes are. We find some accurate expansions for small ∆ and ∆ = 1. We also look at a energy gap of the system for ∆ > 1. We find how this gap behaves as a function of ∆.