The 1/N Expansion in the Theory of Quantum Antiferromagnetism.

This thesis is concerned with the study of strongly -correlated electronic systems, specifically quantum antiferromagnets. The first chapter presents the introduction to the problem. The subject of the second chapter is the theory of spin correlations in constrained wave functions, which are variational ground states for quantum antiferromagnets. In order to cope with the constraint, the SU(2) symmetry group of usual spins is generalized to SU(N), and a large-N expansion is developed. This expansion is simpler than the familiar expansions of the quantum Heisenberg model, and thus more instructive. The diagrammatic rules of this expansion are used to prove certain identities to all orders in 1/N. The on-site spin fluctuations sum rule for arbitrary N is derived. The correlations of the one dimensional Valence Bonds Solid states and the Gutzwiller Projected Fermi Gas up to order 1/N are calculated. The comparison of the leading order terms to known results for N = 2 enhances our understanding of large-N approximations in general. The subject of the third chapter is the nature of the antiferromagnetic long-range order in Heisenberg model. A Luttinger-like theorem is proved, which states, that if there is long range order in mean-field theory (i.e. for N = infty ), then the spontaneous staggered magnetization does not vanish to all orders of the 1/N expansion. The proof uses a cancellation between self-energy diagrams and their tadpole counterparts, a feature special to the 1/N expansion.