Phase Transition Phenomena in Quantum Spin Systems and in Polyampholyte Gels.

We use the Suzuki-Trotter (ST) transformation to map exactly fully quantum mechanical Hamiltonians in d-dimensions to a classical system in (d + 1)-dimensions. We study the two-dimensional classical Ising model that is equivalent, via the ST mapping, to the XXZ-Heisenberg quantum -spin chain. By imposing appropriate boundary conditions to the Ising model, the spin waves of the quantum model are studied. We reproduce the entire energy spectrum of the two-spin-wave states and derive bound-state energies of the three-spin-wave states. Next, I use the ST mapping to study the fully quantum mechanical XY model in two dimensions. In the equivalent classical model, the phase transition is intuitively described and new order parameters are invented. A Monte Carlo (MC) study confirms that this picture's transition takes the Kosterlitz-Thouless form. Two additional local symmetries which have, to date, been neglected in Quantum Monte Carlo simulations are revealed and used. Next we study phase behavior in gels. The newly developed Bond Fluctuation Method (BFM) allows cross-linked polymer networks to be studied via Monte Carlo simulation. I study the scaling behavior of gels, determining the scaling exponent nu in two and three dimensions. The distance between cross-links follows the scaling law for self-avoiding random walks, R_{L} ~ N^nu , which confirms a supposition of Flory. Tanaka and colleagues showed that ionic gels, which are composed of acidic monomer units, exist in expanded or collapsed phases. Two interactions--the quality of the solvent and the work done by a gas of counterions- -suffice to characterize the first-order phase transition in these BFM simulations in two dimensions. A technique is introduced which prevents local attractive interactions from hindering global relaxation. Recent experiments by Annaka and Tanaka have yielded multiple coexistence loops for gels with random positive and negative ionic groups, demonstrating the existence of up to seven distinct macroscopic phases distinguished by volume discontinuities. We introduce for this system a microscopic model in which the randomness translates into random fields resulting in competing quenched random interactions in a spin system. The many phases observed in this model are similar to the experimental results and are understood as randomly coordinated phases. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253 -1690.).