General Structural Results for Potts Model Partition Functions on Lattice Strips

Abstract

We present a set of general results on structural features of the $q$-state Potts model partition function $Z(G,q,v)$ for arbitrary $q$ and temperature Boltzmann variable $v$ for various lattice strips of arbitrarily great width $L_y$ vertices and length $L_x$ vertices, including (i) cyclic and Möbius strips of the square and triangular lattice, and (ii) self-dual cyclic strips of the square lattice. We also present an exact solution for the chromatic polynomial for the cyclic and Möbius strips of the square lattice with width $L_y=5$ (the greatest width for which an exact solution has been obtained so far for these families). In the $L_x\to \infty$ limit, we calculate the ground-state degeneracy per site, $W(q)$ and determine the boundary $\mathcal{B}$ across which $W(q)$ is singular in the complex $q$ plane.

49 pages, latex, four postscript figures

References in corpus (6)

• Exact Potts Model Partition Functions on Wider Arbitrary-Length Strips of the Square Lattice
• Exact Potts Model Partition Function for Strips of the Square Lattice
• 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
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• Exact T=0 Partition Functions for Potts Antiferromagnets on Sections of the Simple Cubic Lattice

Cited by in corpus (10)

• Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models III. Triangular-Lattice Chromatic Polynomial
• Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. IV. Chromatic polynomial with cyclic boundary conditions
• Transfer Matrices for the Partition Function of the Potts Model on Toroidal Lattice Strips
• Transfer Matrices for the Partition Function of the Potts Model on Cyclic and Mobius Lattice Strips
• Transfer Matrices for the Zero-Temperature Potts Antiferromagnet on Cyclic and Mobius Lattice Strips
• Lower Bounds on the Ground State Entropy of the Potts Antiferromagnet on Slabs of the Simple Cubic Lattice
• Ground State Entropy of the Potts Antiferromagnet on Homeomorphic Expansions of Kagome Lattice Strips
• Potts Partition Function Zeros and Ground State Entropy on Hanoi Graphs
• Improved Lower Bounds on the Ground-State Entropy of the Antiferromagnetic Potts Model
• Exact Potts/Tutte Polynomials for Hammock Chain Graphs