#### Date of Award

2003

#### Level of Access Assigned by Author

Open-Access Thesis

#### Degree Name

Master of Science (MS)

#### Department

Physics

#### Advisor

Susan R. McKay

#### Second Committee Member

David P. Feldman

#### Third Committee Member

Charles W. Smith

#### Abstract

This investigation applies information theoretic techniques to study the ordering and structure of the Ising antiferromagnet with quenched disorder on a triangular lattice. The pure system shows no phase transition due to the high degree of frustration present. However, when quenched vacancies are introduced randomly into this geometrically frustrated system, a phase transition at finite temperature, to an ordered phase, can arise. If the vacancies are introduced on all three sublattices of the triangular lattice, then a two-dimensional spin-glass transition occurs. If dilution takes place on only one sublattice, then the other two sublattices develop magnetiza- tions below the critical temperature. These magnetizations are equal in magnitude but opposite in sign, producing a system that still exhibits no net magnetization. The diluted sublattice exhibits spin-glass ordering, but no net magnetization. Thus, this model exhibits two generic features that merit detailed study: a two- dimensional spin-glass transition and "order arising from disorder", in this case the introduction of randomly placed vacancies. To investigate these transitions and the ordering that occurs, we have used both traditional quantities from statistical me- chanics, such as the sublattice magnetizations, the Edwards-Anderson order param- eter, and the specific heat. In addition, we have calculated the Shannon entropy and the excess entropy, two quantities from the field of information theory which are frequently used to characterize the structure and complexity of dynamical systems. This study is the first to apply this type of Shannon entropy calculation to a tw* dimensional system with quenched randomness, so an initial part of our investigation included verifying that the Shannon entropy calculated this way in fact equals the thermodynamic entropy of the system. The Shannon entropy results show excellent agreement with entropies obtained by integrating the specific heat divided by the temperature. Our work is also the first study, to our knowledge, to use this approach for a triangular lattice and for a case in which individual members of the ensem- ble are not translationally invariant, although the ensemble as a whole does possess translational invariance. Thus we have established the broader applicability of this approach. The method that we have introduced to treat non-translationally invariant systems employs a set of shapes planted within the lattice. We calculate the entropy and excess entropy contributions of each shape and then average them to obtain the lattice properties. This procedure yields a spatial map of local properties. The distribution of these local properties provides a new way to characterize order in complex lattice systems. In the triangular antiferromagnet with quenched dilution, the distribution of local entropies shows a dramatic broadening at low temperatures, indicating that the total entropy of the system is not shared evenly across the lattice. The entropy contributions from some regions exhibit local reentrance as a function of temperature, even though the total entropy of the system decreases monotonically during cooling.

#### Recommended Citation

Robinson, Matthew D., "An Information Theoretic Study of the Ising Antiferromagnet with Quenched Dilution on a Triangular Lattice" (2003). *Electronic Theses and Dissertations*. 317.

https://digitalcommons.library.umaine.edu/etd/317