Exact Potts Model Partition Functions for Strips of the Triangular Lattice

Shu-Chiuan Chang 1, 2, Jesper-Lykke Jacobsen 3, Jesus Salas 4, 5, Robert Shrock 1

Journal of Statistical Physics 114 (2004) 763-823

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,\lambda}} c_{Z,G,j}(\lambda_{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.

  • 1. C. N. Yang Institute for Theoretical Physics,
    State University of New York at Stony Brook
  • 2. Department of Applied Physics, Faculty of Science,
    Tokyo University of Science
  • 3. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
    CNRS : UMR8626 – Université Paris XI - Paris Sud
  • 4. Departamento de Física Teórica, Facultad de Ciencias,
    Universidad de Zaragoza
  • 5. Dept. de Matemáticas,
    Universidad Carlos III de Madrid