Ground State Entropy of Potts Antiferromagnets on Homeomorphic Families of Strip Graphs

Abstract

We present exact calculations of the zero-temperature partition function, and ground-state degeneracy (per site), $W$, for the $q$-state Potts antiferromagnet on a variety of homeomorphic families of planar strip graphs $G=(Ch{)}_{k_1,k_2,\Sigma ,k,m}$, where $k_1$, $k_2$, $\Sigma$, and $k$ describe the homeomorphic structure, and $m$ denotes the length of the strip. Several different ways of taking the total number of vertices to infinity, by sending (i) $m\to \infty$ with $k_1$, $k_2$, and $k$ fixed; (ii) $k_1$ and/or $k_2\to \infty$ with $m$, and $k$ fixed; and (iii) $k\to \infty$ with $m$ and $p=k_1+k_2$ fixed are studied and the respective loci of points $\mathcal{B}$ where $W$ is nonanalytic in the complex $q$ plane are determined. The $\mathcal{B}$'s for limit (i) are comprised of arcs which do not enclose regions in the $q$ plane and, for many values of $p$ and $k$, include support for $Re(q)<0$. The $\mathcal{B}$ for limits (ii) and (iii) is the unit circle $|q-1|=1$.

42 pages, Latex, 11 encapsulated postscript figures, Physica A, in press

References in corpus (3)

• Chromatic Polynomials for Families of Strip Graphs and their Asymptotic Limits
• Families of Graphs with W_r({G},q) Functions That Are Nonanalytic at 1/q=0
• Families of Graphs With Chromatic Zeros Lying on Circles

Cited by in corpus (37)

• Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models I. General Theory and Square-Lattice Chromatic Polynomial
• Exact Potts Model Partition Functions on Ladder Graphs
• Thin Fisher Zeroes
• A Little Statistical Mechanics for the Graph Theorist
• Exact Potts Model Partition Functions on Wider Arbitrary-Length Strips of the Square Lattice
• Exact Potts Model Partition Function on Strips of the Triangular Lattice
• T=0 Partition Functions for Potts Antiferromagnets on Square Lattice Strips with (Twisted) Periodic Boundary Conditions
• Chromatic Polynomials, Potts Models and All That
• Chromatic Polynomials for $J(\prod H)I$ Strip Graphs and their Asymptotic Limits
• Ground State Entropy of Potts Antiferromagnets: Homeomorphic Classes with Noncompact W Boundaries
• Structural Properties of Potts Model Partition Functions and Chromatic Polynomials for Lattice Strips
• Ground State Entropy of Potts Antiferromagnets: Cases with Noncompact W Boundaries Having Multiple Points at 1/q = 0
• T=0 Partition Functions for Potts Antiferromagnets on Moebius Strips and Effects of Graph Topology
• Ground-State Degeneracy of Potts Antiferromagnets on Two-Dimensional Lattices: Approach Using Infinite Cyclic Strip Graphs
• Ground State Entropy of the Potts Antiferromagnet on Strips of the Square Lattice
• Exact Potts Model Partition Functions on Strips of the Honeycomb Lattice
• Ground State Entropy of the Potts Antiferromagnet on Cyclic Strip Graphs
• Ground State Entropy of Potts Antiferromagnets on Cyclic Polygon Chain Graphs
• Complex-Temperature Phase Diagrams for the q-State Potts Model on Self-Dual Families of Graphs and the Nature of the $q\to \infty$ Limit
• Zeros of Jones Polynomials for Families of Knots and Links
• Potts Model Partition Functions for Self-Dual Families of Strip Graphs
• Ground State Entropy of the Potts Antiferromagnet with Next-Nearest-Neighbor Spin-Spin Couplings on Strips of the Square Lattice
• T=0 Partition Functions for Potts Antiferromagnets on Lattice Strips with Fully Periodic Boundary Conditions
• Ground State Entropy of the Potts Antiferromagnet on Triangular Lattice Strips
• Reliability Polynomials and their Asymptotic Limits for Families of Graphs
• Exact Partition Function for the Potts Model with Next-Nearest Neighbor Couplings on Strips of the Square Lattice
• Exact T=0 Partition Functions for Potts Antiferromagnets on Sections of the Simple Cubic Lattice
• General Structural Results for Potts Model Partition Functions on Lattice Strips
• Potts model on recursive lattices: some new exact results
• Chromatic Polynomials for Lattice Strips with Cyclic Boundary Conditions
• Transfer Matrices for the Zero-Temperature Potts Antiferromagnet on Cyclic and Mobius Lattice Strips
• Exact Chromatic Polynomials for Toroidal Chains of Complete Graphs
• Asymptotic Behavior of Acyclic and Cyclic Orientations of Directed Lattice Graphs
• The Structure of Chromatic Polynomials of Planar Triangulation Graphs and Implications for Chromatic Zeros and Asymptotic Limiting Quantities
• End Graph Effects on Chromatic Polynomials for Strip Graphs of Lattices and their Asymptotic Limits
• Ground State Entropy of the Potts Antiferromagnet on Homeomorphic Expansions of Kagome Lattice Strips
• Exact Potts/Tutte Polynomials for Hammock Chain Graphs