Exact Potts Model Partition Functions on Strips of the Honeycomb Lattice

Abstract

We present exact calculations of the partition function of the $q$-state Potts model on (i) open, (ii) cyclic, and (iii) Möbius strips of the honeycomb (brick) lattice of width $L_y=2$ and arbitrarily great length. In the infinite-length limit the thermodynamic properties are discussed. The continuous locus of singularities of the free energy is determined in the $q$ plane for fixed temperature and in the complex temperature plane for fixed $q$ values. We also give exact calculations of the zero-temperature partition function (chromatic polynomial) and $W(q)$, the exponent of the ground-state entropy, for the Potts antiferromagnet for honeycomb strips of type (iv) $L_y=3$, cyclic, (v) $L_y=3$, Möbius, (vi) $L_y=4$, cylindrical, and (vii) $L_y=4$, open. In the infinite-length limit we calculate $W(q)$ and determine the continuous locus of points where it is nonanalytic. We show that our exact calculation of the entropy for the $L_y=4$ strip with cylindrical boundary conditions provides an extremely accurate approximation, to a few parts in $10^5$ for moderate $q$ values, to the entropy for the full 2D honeycomb lattice (where the latter is determined by Monte Carlo measurements since no exact analytic form is known).

48 pages, latex, with encapsulated postscript figures

References in corpus (7)

• Spanning Trees on Graphs and Lattices in d Dimensions
• Exact Potts Model Partition Functions on Ladder Graphs
• Exact Potts Model Partition Function on Strips of the Triangular Lattice
• Structural Properties of Potts Model Partition Functions and Chromatic Polynomials for Lattice Strips
• T=0 Partition Functions for Potts Antiferromagnets on Moebius Strips and Effects of Graph Topology
• Ground State Entropy of the Potts Antiferromagnet on Strips of the Square Lattice
• Ground State Entropy of the Potts Antiferromagnet with Next-Nearest-Neighbor Spin-Spin Couplings on Strips of the Square Lattice

Cited by in corpus (29)

• 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
• 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
• Potts Model Partition Functions for Self-Dual Families of Strip Graphs
• On Properties of the Ising Model for Complex Energy/Temperature and Magnetic Field
• Reliability Polynomials and their Asymptotic Limits for Families of Graphs
• General Structural Results for Potts Model Partition Functions on Lattice Strips
• Transfer Matrices for the Partition Function of the Potts Model on Cyclic and Mobius Lattice Strips
• Transfer Matrices for the Partition Function of the Potts Model on Toroidal Lattice Strips
• Potts model on recursive lattices: some new exact results
• Chromatic Polynomials for Lattice Strips with Cyclic Boundary Conditions
• Structure of the Partition Function and Transfer Matrices for the Potts Model in a Magnetic Field on Lattice Strips
• Exact Results for Average Cluster Numbers in Bond Percolation on Lattice Strips
• Transfer Matrices for the Zero-Temperature Potts Antiferromagnet on Cyclic and Mobius Lattice Strips
• Tutte polynomial of pseudofractal scale-free web
• Zeros of the Potts Model Partition Function in the Large-$q$ Limit
• Exact Chromatic Polynomials for Toroidal Chains of Complete Graphs
• Asymptotic Behavior of Acyclic and Cyclic Orientations of Directed Lattice Graphs
• Exact Potts Model Partition Functions for Strips of the Honeycomb Lattice
• Asymptotic Behavior of Spanning Forests and Connected Spanning Subgraphs on Two-Dimensional Lattices
• Partition Function Zeros of a Restricted Potts Model on Lattice Strips and Effects of Boundary Conditions
• Ground State Entropy of the Potts Antiferromagnet on Homeomorphic Expansions of Kagome Lattice Strips
• Partition Function Zeros of a Restricted Potts Model on Self-Dual Strips of the Square Lattice
• Potts Partition Function Zeros and Ground State Entropy on Hanoi Graphs
• Exponential Growth Constants for Spanning Forests on Archimedean Lattices: Values and Comparisons of Upper Bounds
• Improved Lower Bounds on the Ground-State Entropy of the Antiferromagnetic Potts Model
• Exact Potts/Tutte Polynomials for Hammock Chain Graphs
• Exact Results for Average Cluster Numbers in Bond Percolation on Infinite-Length Lattice Strips