Mathematical Sciences: Exact Analysis of Lattice Statistics Models and Applications

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In this project the principal investigators will continue their study of new, higher-genus solutions of the Yang-Baxter equation in the chiral Potts model. These solutions describe phase transitions in experimentally realizable systems and they serve as representatives of various universality classes. In addition, the principal investigators will look at spin-spin correlation functions of the Z-invariant inhomogeneous Ising model on an irregular lattice, in order to obtain the correlation functions of certain quasicrystals and some quasiperiodic Ising lattices. Models in statistical mechanics are an important vehicle for improving our understanding of ordered structures such as crystals. In addition, the study of models like the chiral Potts model involves a nice interplay between mathematics and physics that serves to enhance both fields. In this project the principal investigators will look for new solutions of the Yang- Baxter equation that describe quasicrystals and quasiperiodic lattices. This work touches upon other topics on the border between mathematics and physics such as quantum groups and knot theory, monopoles and string theory, phase transitions in condensed matter and biomembranes in biophysics.