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
SN - 0022-4715
VL - 107
SP - 1207
EP - 1253
JO - Journal of Statistical Physics
JF - Journal of Statistical Physics
IS - 5-6
ER -