Representations of the Lie algebra su(2) and the AKLT model

Abstract

In this thesis, we will provide an overview of the basics of Lie theory, starting with matrix Lie groups and moving on to the Lie algebras associated to those matrix Lie groups. We introduce representations of Lie groups and algebras, and we provide an extensive example on the representations of $\su(2)$. This can then be applied to understanding the coupling of spins. We define the Heisenberg model, and speak about the Haldane conjecture. We derive the VBS state, and use this to construct its parent Hamiltonian, the AKLT model. Then we show that the AKLT Hamiltonian satisfies the Haldane conjecture, providing evidence for its truth.