Studies of Model Systems in Statistical Mechanic and QuantumField Theory

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Research will be conducted on solvable statistical mechanical and quantum field theoretical models. Particular emphasis will be placed on the chiral Potts model. The chiral Potts model has recently been solved for its eigenvalues and displays a number of novel effects. The model exhibits a commensurate-incommensurate phase, the magnetic analog of optical activity and a Pokrovsky- Talapov transition. Further studies will be undertaken of this class of exact level crossing transitions which may impact strongly interacting fermion systems. Order parameters and correlation functions in the incommensurate phase will be computed. The results of this solvable level crossing system will be used to study more general non-integrable models. More general integrable models will be examined. Connections with the mathematics of monopoles, algebraic geometry and topological field theory will be explored. %%% The study of complex, interacting, many-particle systems lies at the foundation of condensed matter physics and materials physics. Because of the complexity of these systems, radical approximations frequently need to be made in order to obtain solutions. However, occasionally a physical model, such as the Ising model, is able to be solved exactly. These exact solutions provide invaluable guidance to understanding the underlying physics and the mathematical structure of the exact model and related models. The chiral Potts model has recently been solved exactly and the proposed research will investigate the application of this model to various classes of physical phemonena and further explore the nature of the solutions.