Finite Lattice Methods in One and Two-Dimensional Quantum Spin Systems.

This is a study of finite lattice methods applied to quantum spin models on one and two dimensional lattices. The motivation for the use of these techniques is provided by placing them in the context of the wider field of approximation methods in the statistical mechanics of spin models. After reviewing previous finite lattice calculations on quantum spin models, the theory needed to efficiently generate and analyse finite lattice data is given. Finite lattice calculations on three quantum spin models are then presented. The first calculation provides an estimate of the transverse susceptibility of the one dimensional antiferromagnetic XY model, by extrapolation from calculations on rings of up to 10 sites. Good convergence is found for K(,B) T(, )>(, )0(.)4 and these results are compared with recent c - axis susceptibility measurements on the compound Cs(,2)CoCl(,4). Chapters 5 and 6 study the quantum hamiltonian analog of the two dimensional axial next nearest neighbour Ising model,. (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI). A perturbation analysis using non degenerate Rayleigh -Schrodinger perturbation theory is followed by a mass gap scaling analysis. These analyses provide accurate phase boundaries for, and ellucidate several interesting features of, the phase diagram of this model. In particular, the model is shown to exhibit a phase whose modulation varies continuously with a change in the interaction parameters in the hamiltonian. This "modulated" phase region lies between a normal ordered phase and a paramagnetic phase. Evidence is given that the transition from the paramagnetic to modulated phase is exponential in nature. The third model studied is a spin 1 biquadratic Heisenberg model with hamiltonian,. (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI). Accurate estimates of the ground state energy in both one and two dimensions are found by calculating this quantity on 4,6,8 and 10 site lattices. Comment is made about the ground state phase diagram, and about the existence of intermediate spin states and states with a complicated magnetic ordering. The thesis concludes by commenting on the performance of finite lattice methods in the three models studied and by outlining certain directions for future advance.