TY - JOUR

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

AU - Chang, Shu Chiuan

AU - Salas, Jesús

AU - Shrock, Robert

N1 - Funding Information:
The research of R.S. was supported in part by the NSF Grant PHY-9722101. The research of J.S. was partially supported by CICyT (Spain) Grant AEN99-0990. One of us (R.S.) wishes to acknowledge H. Kluepfel for related collaborative work.

PY - 2002

Y1 - 2002

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 square-lattice strip graphs G for a variety of transverse widths Lt and for arbitrarily great length Lℓ, with free longitudinal boundary conditions and free and periodic transverse boundary conditions. These have the form Z(G, q, v) = ∑j = 1NZ, G, λ CZ, G, j(λZ, G, j)Lℓ. We give general formulas for NZ, G, j and its specialization to v = -1 for arbitrary L t for both types of boundary conditions, as well as other general structural results on Z. 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 square 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 Lℓ → ∞, in the q plane for fixed v and in the v plane for fixed q.

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 square-lattice strip graphs G for a variety of transverse widths Lt and for arbitrarily great length Lℓ, with free longitudinal boundary conditions and free and periodic transverse boundary conditions. These have the form Z(G, q, v) = ∑j = 1NZ, G, λ CZ, G, j(λZ, G, j)Lℓ. We give general formulas for NZ, G, j and its specialization to v = -1 for arbitrary L t for both types of boundary conditions, as well as other general structural results on Z. 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 square 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 Lℓ → ∞, in the q plane for fixed v and in the v plane for fixed q.

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U2 - 10.1023/A:1015165926201

DO - 10.1023/A:1015165926201

M3 - Article

AN - SCOPUS:0036243727

VL - 107

SP - 1207

EP - 1253

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 5-6

ER -