Study of Exactly Solvable Model Systems in Statistical Mechanics

Oklahoma State University, Stillwater OK

Investigators

Abstract

The main objective of the proposed study is to develop new mathematics and new numerical procedures to obtain exact expressions for correlation functions in statistical mechanical model systems together with procedures to evaluate them to high precision. This is of current interest as exactly solvable (or integrable) many-body systems appear in many areas of physics and mathematics. One can now construct a wide variety of experimentally accessible low-dimensional systems. The insight obtained from exact theoretical studies of such models is definitive. New exact results enhance the understanding of these experimental systems. For integrable models exactly solvable through solutions of the Yang?Baxter equations, one can usually find the free energy and order parameters. For various so-called Ising-type models within the Yang?Baxter class one now has a rather complete understanding of the pair correlations, in part due to recent work of the PIs. There has also been recent progress in calculating correlation functions for some of the other models of the Yang?Baxter class. But these results are still rather incomplete and the expressions derived so far are often unwieldy. The search for more accessible results is a major goal of the proposed research. Many potential applications depend on a good understanding of these correlations. In the proposed research the PIs will pay special attention to the chiral Potts model. Its integrable subcase remains a great mathematical challenge. Better understanding of this model could lead to theoretical breakthroughs in several areas of physics and mathematics. Recently Baxter succeeded in giving a proof of the sixteen year old conjecture of the order parameter. Combining his techniques with other methods developed by the PIs, some of the many challenging problems remaining will be investigated, such as the spin-spin and energy-energy correlations. Broader Impact: The proposed research relates to condensed matter physics, quantum computing, string theory and high-energy physics, knot theory, quantum groups, and various other areas of mathematics. The models to be studied should lead to a better understanding of a variety of phenomena studied in condensed matter physics. The models are also relevant for the study of entanglement in the theory of quantum computers, Bethe-Ansatz calculations in string theory, knot invariants, operator algebras, special functions, and could lead to new insights in biomembranes and DNA science. Some of the findings may be incorporated in tomorrow?s software packages and graduate and advanced undergraduate textbooks. The results may enable new materials and devices. Certain devices in nanotechnology, advances in coating of surfaces, high-temperature superconductivity, cell membranes, and potential applications in quantum computing are only a few of the many examples. Any progress made could have new important implications. The requested support includes funds for training a graduate student, some funds to get an undergraduate student involved part-time, funds for travel and consulting in order to continue and expand the intra- and interdisciplinary interactions with U.S. and international research partners, and funds to replace outdated computers and software.