Studies of Lattice Spin Models and Their Critical Behavior.

This dissertation considers how the global properties of a system affect its critical behavior. The first chapter studies the Ising model on Fractal lattices, i.e., lattices of noninteger dimensionality, with the end result that fractals violate universality. The second chapter demonstrates that topological excitations, known as vortex strings, are essential for the paramagnetic-ferromagnetic phase transition of the three dimensional O(2) model. Furthermore, for a system consisting of only a coupling to the vortex string density, there exist long range order in the presence of a ground state possessing finite disordering entropy. The generalized phase diagram for this model illustrates the result of competition between a continuously varying term in the Hamiltonian and a discreetly varying term. Finally, the third chapter examines how the order of a phase transition can change when the global properties of the spin manifold are altered. In this case, the generalized P^ {rm N} models in three dimensions are studied and compared to their S^{ rm N} counterparts. These two models have the same local properties but they differ in their global topology.