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Structure of the Partition Function and Transfer Matrices for the Potts Model in a Magnetic Field on Lattice Strips

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Abstract

We determine the general structure of the partition function of the q-state Potts model in an external magnetic field, Z(G,q,v,w) for arbitrary q, temperature variable v, and magnetic field variable w, on cyclic, Möbius, and free strip graphs G of the square (sq), triangular (tri), and honeycomb (hc) lattices with width L y and arbitrarily great length L x . For the cyclic case we prove that the partition function has the form \(Z(\Lambda,L_{y}\times L_{x},q,v,w)=\sum_{d=0}^{L_{y}}\tilde{c}^{(d)}\mathrm{Tr}[(T_{Z,\Lambda,L_{y},d})^{m}]\) , where Λ denotes the lattice type, \(\tilde{c}^{(d)}\) are specified polynomials of degree d in q, \(T_{Z,\Lambda,L_{y},d}\) is the corresponding transfer matrix, and m=L x (L x /2) for Λ=sq,tri (hc), respectively. An analogous formula is given for Möbius strips, while only \(T_{Z,\Lambda,L_{y},d=0}\) appears for free strips. We exhibit a method for calculating \(T_{Z,\Lambda,L_{y},d}\) for arbitrary L y and give illustrative examples. Explicit results for arbitrary L y are presented for \(T_{Z,\Lambda,L_{y},d}\) with d=L y and d=L y −1. We find very simple formulas for the determinant \(\mathrm{det}(T_{Z,\Lambda,L_{y},d})\) . We also give results for self-dual cyclic strips of the square lattice.

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Authors and Affiliations

  1. Department of Physics, National Cheng Kung University, Tainan, 70101, Taiwan

    Shu-Chiuan Chang

  2. C.N. Yang Institute for Theoretical Physics, State University of New York, Stony Brook, NY, 11794, USA

    Robert Shrock

Authors
  1. Shu-Chiuan Chang
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  2. Robert Shrock
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Correspondence to Shu-Chiuan Chang.

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Chang, SC., Shrock, R. Structure of the Partition Function and Transfer Matrices for the Potts Model in a Magnetic Field on Lattice Strips. J Stat Phys 137, 667–699 (2009). https://doi.org/10.1007/s10955-009-9868-0

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