Problems in Quantum and Classical Statistical Mechanics

University Of Arizona, Tucson AZ

Investigators

Abstract

--------------------------------- PI: Tom Kennedy, University of Arizona DMS 0201566 Abstract: The first part of the project concerns self-avoiding walks in two dimensions and their relation to Schramm's stochastic Loewner evolution process. Schramm's process is believed to describe the scaling limit of a variety of two-dimensional models, including the self-avoiding walk. The pivot algorithm provides a fast method for simulating self-avoiding walks. Recently, the principal investigator has found a new implementation of this algorithm that is as much as eighty times faster in two dimensions. This will be used to test both the conjectured conformal invariance of the self-avoiding walk and its equivalence with stochastic Loewner evolution. A version of the weakly self-avoiding walk in which the penalty for self-intersections decays with the length of the loop produced by the self-intersection will be investigated by simulations and perturbative means. The second part of the project concerns excited states in quantum spin systems. Recent work by the principal investigator has developed a method for studying the dispersion relation of one quasi-particle states and interface states in one dimension. The method is based on assuming a certain ansatz for the wave function of the states and then proving there is indeed an eigenstate of this form by a contraction mapping argument. This method will be used to study interface states in two dimensions and the excited states above these interface states. Self-avoiding walks are random walks which are not allowed to visit the same place more than once. They provide a model for linear polymers in a dilute solution. The interest in this model is, however, much broader since it is one of the simplest models that exhibits critical phenomena and in two dimensions conformal invariance. Recently there has been an explosion of conjectures relating the self-avoiding walk and other two-dimensional models to a new two dimensional stochastic process, called stochastic Loewner evolution, introduced by Schramm. Part of the project will study many of these conjectures for the self-avoiding walk by Monte Carlo simulations and by perturbative methods. The simulations will be done with a recent implementation of the pivot algorithm by the principal investigator that is much faster than previous implementations. This fast algorithm will also be used to study versions of the self-avoiding walk that are important in physical chemistry. Another part of the project is devoted to studying the low energy excitations in a variety of quantum spin systems. These are models of the behavior of the electron spins in crystals. At low temperatures the electron spins tend to align. However, several domains may form within which the spins are aligned, but between which the spins point in opposite directions. The interfaces between these domains will be studied by an approach which has proved very successful for studying one quasi-particle states. In particular, the nature of the excitations just above the interface states will be investigated.