TY - JOUR

T1 - Exact Potts model partition functions for strips of the triangular lattice

AU - Chang, Shu Chiuan

AU - Jacobsen, Jesper Lykke

AU - Salas, Jesús

AU - Shrock, Robert

PY - 2004/2

Y1 - 2004/2

N2 - 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 partition functions have the form Z(G, q, v) = ∑j = 1 N Z,G,λ cZ,G,j(λZ,G,j) m-1. We give general formulas for NZ,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 ℬ, arising as the accumulation set of partition function zeros as m → ∞, in the q plane for fixed v and in the v plane for fixed q. Explicit results for partition functions are given in the text for L = 3 (free) and L = 3, 4 (cylindrical), and plots of partition function zeros and their asymptotic accumulation sets are given for L up to 5. A new estimate for the phase transition temperature of the q = 3 Potts antiferromagnet on the 2D triangular lattice is given.

AB - 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 partition functions have the form Z(G, q, v) = ∑j = 1 N Z,G,λ cZ,G,j(λZ,G,j) m-1. We give general formulas for NZ,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 ℬ, arising as the accumulation set of partition function zeros as m → ∞, in the q plane for fixed v and in the v plane for fixed q. Explicit results for partition functions are given in the text for L = 3 (free) and L = 3, 4 (cylindrical), and plots of partition function zeros and their asymptotic accumulation sets are given for L up to 5. A new estimate for the phase transition temperature of the q = 3 Potts antiferromagnet on the 2D triangular lattice is given.

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U2 - 10.1023/b:joss.0000012508.58718.83

DO - 10.1023/b:joss.0000012508.58718.83

M3 - Article

AN - SCOPUS:3543067018

SN - 0022-4715

VL - 114

SP - 763

EP - 823

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

IS - 3-4

ER -