In this thesis we take several different analytic and numerical approaches to studying the classical J-K model. This model describes an interacting many-body system of spins with continuous symmetry which interact via 2-site nearest-neighbour exchange terms and 4-site ring-exchange terms. We begin by looking at the traditional solution of the XY model, in order to gain insight into the behaviour and general properties of the system. Classical Monte Carlo simulations will then be used to study the properties of the J-K model in different regimes of phase space. We will see that we can use properties from the theoretical solution of the XY model to study the Kosterlitz-Thouless phase transition numerically. We then extend our simulation to study the aspect ratio scaling of the superfluid density in the XY model. It will also be shown that there exists a finite temperature phase transition in the pure-K ring-exchange model. After this we will develop a mapping from the 1D quantum Bose-Hubbard model to the 2D J-K model and use this mapping to search for topological phases in classical Hamiltonians. However, we find that our mapping fails to reproduce the topological phase present in the quantum model. Finally we will look at the XY model using tools from information theory. A method for measuring mutual information in classical Monte Carlo simulations is developed. We then show that this measurement of mutual information can be used as a completely new way to identify the Kosterlitz-Thouless phase transition in Monte Carlo simulations.
We study the ground state of the two-dimensional (2D) disordered Hubbard model by means of the projector quantum Monte Carlo (PQMC) method. This approach allows us to investigate the ground state properties of this model for lattice sizes up to
10×10, at quarter filling, for a broad range of interaction and disorder strengths. Our results show that the ground state of this system of spin-1/2 fermions remains localised in the presence of the short-ranged Hubbard interaction.