TY - JOUR

T1 - Structure of the partition function and transfer matrices for the potts model in a magnetic field on lattice strips

AU - Chang, Shu Chiuan

AU - Shrock, Robert

N1 - Funding Information:
Acknowledgements We thank F.Y. Wu for a valuable communication calling our attention to [15]. The research of R.S. was partially supported by the NSF grant PHY-00-98527. The research of S.C.C. was partially supported by the Taiwan NSC grant NSC-97-2112-M-006-007-MY3 and NSC-98-2119-M-002-001.

PY - 2009/11

Y1 - 2009/11

N2 - 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 Ly and arbitrarily great length Lx. For the cyclic case we prove that the partition function has the form, where Λ denotes the lattice type, c̃(d) are specified polynomials of degree d in q, TZ,Λ,Ly,d is the corresponding transfer matrix, and m=Lx (Lx/2) for Λ=sq,tri (hc), respectively. An analogous formula is given for Möbius strips, while only TZ,Λ,Ly,d=0 appears for free strips. We exhibit a method for calculating TZ,Λ,Ly,d for arbitrary Ly and give illustrative examples. Explicit results for arbitrary Ly are presented for TZ,Λ,Ly,d with d=Ly and d=Ly-1. We find very simple formulas for the determinant det(TZ,Λ,Ly,d). We also give results for self-dual cyclic strips of the square lattice.

AB - 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 Ly and arbitrarily great length Lx. For the cyclic case we prove that the partition function has the form, where Λ denotes the lattice type, c̃(d) are specified polynomials of degree d in q, TZ,Λ,Ly,d is the corresponding transfer matrix, and m=Lx (Lx/2) for Λ=sq,tri (hc), respectively. An analogous formula is given for Möbius strips, while only TZ,Λ,Ly,d=0 appears for free strips. We exhibit a method for calculating TZ,Λ,Ly,d for arbitrary Ly and give illustrative examples. Explicit results for arbitrary Ly are presented for TZ,Λ,Ly,d with d=Ly and d=Ly-1. We find very simple formulas for the determinant det(TZ,Λ,Ly,d). We also give results for self-dual cyclic strips of the square lattice.

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U2 - 10.1007/s10955-009-9868-0

DO - 10.1007/s10955-009-9868-0

M3 - Article

AN - SCOPUS:70549113174

SN - 0022-4715

VL - 137

SP - 667

EP - 699

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

IS - 4

ER -