Mathematical Sciences: Classical and Quantum Mechanical Lattice Spin Systems

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This project is concerned with both classical and quantum mechanical spin systems on a lattice. The research will begin by developing techniques for rigorously studying systems which are in a phase in which there is a unique Gibbs state and exponential decay of the correlations. These techniques will be applied to various problems which can be cast in the form of showing that a given system is in such a phase. These problems include the majority rule renormalization group transformation, the predictions of Haldane for the spin 1 Heisenberg chain, and some classical statistical mechanics models that are believed to have a unique Gibbs state at all temperatures. Other problems to be studied include the existence of antiferromagnetic order in the two-dimensional spin 1/2 Heisberg antiferromagnet, Rayleigh- Schrodinger perturbation theory for quantum spin systems, and the generalization of Dobrushin and Schlosman's work on the uniqueness of the Gibbs state in classical systems to quantum systems.