Potts Model and Generalizations: Exact Results and Statistical Physics

Abstract

The q-state Potts model is a spin model that has been of longstanding interest as a many body system in statistical mechanics. Via a cluster expansion, the Potts model partition function Z(G,q,v), defined on a graph G=(V,E), where V is the set of vertices (sites) and E is the set of edges (bonds), is expressed as a polynomial in terms of q and a temperature-dependent Boltzmann variable v. An important special case (v=-1) is the zero-temperature Potts antiferromagnet, for which Z(G,q,-1)=P(G,q), where P(G,q) is the chromatic polynomial, counting the number of ways of assigning q colors to the vertices of graph G such that no two adjacent vertices have the same color. A natural generalization is to consider this model in a generalized external field that favors or disfavors spin values in a subset I_s={1, . . . ,s} of the total set of q-state spin values. In this dissertation, we calculate the exact partition functions of the generalized Potts model Z(G,q,s,v,w), where w is a field-dependent Boltzmann variable, for certain families of graphs. We also investigate its special case, viz. Z(G,q,s,-1,w)=Ph(G,q,s,w), which describes a weighted-set graph coloring problem. Nonzero ground-state entropy (per lattice site), S₀>0, is an important subject in statistical physics, as an exception to the third law of thermodynamics and a phenomenon involving large disorder even at zero temperature. The q-state Potts antiferromagnet is a model exhibiting ground-state entropy for sufficiently large q on a given lattice graph. Another part of the dissertation is devoted to the study of ground-state entropy, for which lower bounds on slabs of the simple cubic lattice and exact results on homeomorphic expansions of kagom&eacute lattice strips are presented. Next, we focus on the structure of chromatic polynomials for a particular class of graphs, viz. planar triangulations, and discuss implications for chromatic zeros and some asymptotic limiting quantities.