Mathematical Sciences: Statistical Mechanics of Classical and Quantum Lattice Systems

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Abstract Kennedy 9623509 Three types of lattice systems will be studied - quantum spin systems, weakly self-avoiding walks and real space renormalization group transformations for classical spin systems. The primary quantum spin system that we will study is the Heisenberg antiferromagnet on a ladder (2 sites wide, infinitely many sites long). This particular model contains several interesting quantum spin models as limiting cases and should help us understand the relationship between the variety of exotic phases that occur in these models. Weakly self-avoiding walks are nearest neighbor walks on the lattice with the probability of a walk reduced according to the number of self intersections it contains. The goal here is to develop real space renormalization group transformations for the usual model and for a model in which the energy penalty for a self intersection decays as the length of the loop formed by the intersection raised to a power. This latter model has interesting behavior even in one dimension. For classical lattice spin systems like the Ising model, we will continue our study of real space renormalization group transformations like majority rule. When this transformation is applied to the one dimensional Ising model with a power law interaction, there is a nice simplication when the power is close to one. This simplification should be the start of a rigorous definition of the transformation. We will also work on establishing the existence of the transformation in various two dimensional models. This project is devoted to the mathematical study of several systems that originate in physics - quantum spin systems, self avoiding walks and classical spin systems. Quantum and classical spin systems are models for the interactions of the spins of electrons in crystals. These systems contain a huge number of spins, but typically each spin only interacts with nearby spins. Nonetheless these microscopic local interactions can produce macroscopic effects, e.g., magnetism and super conductivity. Self avoiding walks are models for very long polymers. Again, there are a large number of basic units, but the interaction is local - different sections of the polymer cannot occupy the same location. This local interaction affects how the size of the polymer grows as the number of basic units making up the polymer increases. A common motivation for studying all of these models is to understand how simple local interactions in a mathematical model with a very large number of degrees of freedom produce global effects. One of the most remarkable discoveries in physics is that certain properties of these global effects do not depend on the details of the microscopic interactions. There are several quantifiable properties of the macroscopic behavior know as critical exponents which are believed to be exactly the same for a wide variety of microscopic interactions. One of the goals of this project is to further our mathematical understanding of this "universality."