Mathematical Sciences: Quantum Mechanical Classical Lattice Spin Systems

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In this project the principal investigator will study the phase transition between the Haldane phase and the dimerized phase in the spin-1 Heisenberg antiferromagnetic, one-dimensional quantum spin system. In particular, he will prove that this phase transition does indeed exist by showing that an order- parameter proposed for this transition is zero in one phase and nonzero in the other. In addition, the principal investigator will continue his rigorous study of the majority rule renormalization group transformation for Ising-type systems. Finally he will study the existence of continuous spectra for the infinite volume limit of the Hamiltonian of a quantum spin system. The behavior of matter at the microscopic scale is very often what determines the behavior of matter at the macroscopic or observable scales of ordinary life. For example, what we call the temperature of a liquid or gas is simply the macroscopic or cumulative effect of the motion of countless atoms and molecules on the microscopic scale of the material. In this project the principal investigator will study various mathematical models of the behavior of atoms at the microscopic or quantum level, in order to establish connections between microscopic and macroscopic properties of matter.