Date of Award

2005

Level of Access

Campus-Only Thesis

Degree Name

Master of Science (MS)

Department

Physics

Advisor

Susan R. McKay

Second Committee Member

David P. Feldman

Third Committee Member

David Hiebeler

Abstract

This study uses an information theoretic approach implemented with Monte Carlo simulations to investigate the nature of local ordering in the two-dimensional Ising spin glass on a square lattice, a system that has remained controversial and serves as a prototypical frustrated model system. Bimodal and Gaussian spin glasses, both included in this investigation, are similar in that they both possess competing interactions, but different in that the bimodal spin glass has a highly degenerate ground state while the Gaussian does not. Neither exhibits a finite temperature phase transition in the two-dimensional case, but both do in three dimensions. This study focuses on their similarities and differences in local structure as they are cooled through intermediate temperatures. To determine optimal simulation parameters, our study initially focused on equilibration times and autocorrelation functions, and also calculated traditional thermodynamic quantities. These quantities include the magnetization, internal energy, and Edwards-Anderson order parameter, along with their respective response functions, the magnetic susceptibility, the specific heat, and the spin-glass susceptibility. These calculations provide direct comparisons between the macroscopic properties and the local ordering obtained through the information theoretic approach as a function of temperature. Prior to applying this new approach to the spin glass, we tested the method on the two-dimensional Ising ferromagnet, since its exact entropy is known. Our comparisons between the exact Ising entropy and that calculated using information theoretic methods show excellent agreement. In the spin-glass systems studied, where an exact solution is not available, comparisons between the information theoretic results and those obtained using integral methods from thermodynamics show excellent agreement and indicate the equivalence of the information theoretic and the thermodynamic entropies when the method is applied to quenched random systems such as spin glasses. To apply the information theoretic approach in a system with quenched randomness, small shapes, typically consisting of eleven spins in this study, are placed in the lattice and each shape yields a local probability distribution of configurations. These probability distributions provide the basis for the calculation of a local contribution to the entropy. These probability distributions can also be used to calculate the excess entropy, a measure of structure and correlation in the system. Further, they provide a new way to probe local ordering in these glassy systems as a function of temperature or bond distribution. The information theoretic tools illustrate the differences in spin glasses with Gaussian and bimodal distributions of interactions and show how the entropy is not spread evenly over the lattice. In fact, regions of high or low entropy emerge at relatively high temperature and persist and grow as the system is cooled. These regions are most pronounced in the bimodal case. Thus, the information theoretic method provides a valuable new tool for the study of spin glasses. It yields both visual and quantitative indicators of the local ordering process in these intricate systems and has potential value for application to other quenched random systems.

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