Exact Potts Model Partition Functions for Strips of the Triangular Lattice

Abstract

We present exact calculations of the Potts model partition function Z(G,q,v) for arbitrary q and temperature-like variable v on n-vertex strip graphs G of the triangular lattice for a variety of transverse widths equal to L vertices and for arbitrarily great length equal to m vertices, with free longitudinal boundary conditions and free and periodic transverse boundary conditions. These have the form Z(G,q,v)=\sum_{j=1}^{N_{Z,G,λ}} c_{Z,G,j}(λ_{Z,G,j})^{m-1}. We give general formulas for N_{Z,G,j} and its specialization to v=-1 for arbitrary L. The free energy is calculated exactly for the infinite-length limit of the graphs, and the thermodynamics is discussed. It is shown how the internal energy calculated for the case of cylindrical boundary conditions is connected with critical quantities for the Potts model on the infinite triangular lattice. Considering the full generalization to arbitrary complex q and v, we determine the singular locus {\cal B}, arising as the accumulation set of partition function zeros as m\to\infty, in the q plane for fixed v and in the v plane for fixed q.

59 pages, LaTeX2e. Self-unpacking archive containing the tex file, three sty files, 51 ps files, and a gziped Mathematica file (transfer_Tutte_tri.m.gz). Version published in journal

Cited by in corpus (25)

• Statistical Mechanics of Equilibrium and Nonequilibrium Phase Transitions: The Yang-Lee Formalism
• Spanning forests and the q-state Potts model in the limit q \to 0
• A Little Statistical Mechanics for the Graph Theorist
• Complex-temperature phase diagram of Potts and RSOS models
• Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. V. Further Results for the Square-Lattice Chromatic Polynomial
• Q-colourings of the triangular lattice: Exact exponents and conformal field theory
• 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
• Phase diagram of the triangular-lattice Potts antiferromagnet
• A generalized Beraha conjecture for non-planar graphs
• Potts model on recursive lattices: some new exact results
• Potts correlators and the static three-quark potential
• Cylinder partition function of the 6-vertex model from algebraic geometry
• 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
• Integrability vs non-integrability: Hard hexagons and hard squares compared
• Exact Potts Model Partition Functions for Strips of the Honeycomb Lattice
• Potts critical frontiers of inhomogeneous and asymmetric bow-tie lattices
• Partition Function Zeros of a Restricted Potts Model on Lattice Strips and Effects of Boundary Conditions
• Critical manifold of the Potts model: Exact results and homogeneity approximation
• Potts Partition Function Zeros and Ground State Entropy on Hanoi Graphs
• On Negative Mass, Partition Function and Entropy
• Exact Potts/Tutte Polynomials for Hammock Chain Graphs
• Chromatic Zeros on the Limit $G_{\infty }^{(p,\ell )}$ of the Family $G_m^{(p,\ell )}$ of Hierarchical Graphs