Solitons in the O(3) nonlinear sigma model: Connections to planar antiferromagnetism and high-temperature superconductors

The motivation for this dissertation work is the unusual two-dimensionality of the new copper oxide high-temperature superconductors. Most of these compounds at zero doping exhibit planar antiferromagnetism with weak interlayer coupling. At very low doping, at least, the ground state of the compounds is thus a Neel state. An attempt is made in this thesis to extract the relevant behaviour of the added holes using a field theory formalism. The work begins with a brief review of Fermi liquids and how BCS instability arises in conventional superconductors. Experiments done in the normal state indicate that the new compounds exhibit strong electron correlations. A few of the experimental facts on the new compounds in the normal and the superconducting state are presented. With the hypothesis that the anti-ferromagnetism is the important agent in explaining the onset of superconductivity, the conventional approaches to the field theory description of the ordered Neel state is elaborated.; New formulations of the non-linear sigma model are made. The topological terms that arise impart novel spin statistics to the original spin waves and all other particles interacting with them. This term, the Chern-Simons term, whether it arises in the continuum limit or not is crucial to many scenarios of superconductivity. Using the BRST formalism for constrained systems, a consistent quantization of this theory is completed.; The non-linear sigma model has non-trivial soliton excitations in its spectrum only in two dimensions. It is tempting to propose that it is this unique feature that plays a significant role in an understanding of the new two-dimensional superconductors. The third chapter elaborates a description of the soliton excitations using the original language and the new mathematical maps. The catastrophe of the size instability of these solitons is described.; The last chapter begins with the description of holes moving in the antiferro-magnetic background. The chapter details the soliton-hole interactions which indicate the naturalness of s-wave cooper pairing. The experimental observation that the antiferro-magnetism is disordered at large doping could thus be viewed as a condensation of soliton-hole pairs. (Abstract shortened with permission of author.)...