TY - JOUR

T1 - Transfer matrices for the partition function of the Potts model on cyclic and Möbius lattice strips

AU - Chang, Shu Chiuan

AU - Shrock, Robert

N1 - Funding Information:
We thank J. Salas for discussions on transfer matrix methods during the work for Refs. [26,27] , A. Sokal for related discussions, and N. Biggs for discussion on sieve methods. 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 Nishina and Inoue Foundations, and he thanks Prof. M. Suzuki for further support. The NCTS Taipei address for S.C.C. applies after April 12, 2004.

PY - 2005

Y1 - 2005

N2 - We present a method for calculating transfer matrices for the q-state Potts model partition functions Z(G,q,ν), for arbitrary q and temperature variable ν, on cyclic and Möbius strip graphs G of the square (sq), triangular (tri), and honeycomb (hc) lattices of width Ly vertices and of arbitrarily great length Lx vertices. For the cyclic case we express the partition function as Z(λ,Ly × Lx,q,ν) = ∑d=0Lyc(d) Tr[(T Z,λ,Ly,d)m], where λ denotes lattice type, c(d) are specified polynomials of degree d in q, T Z,λ,Ly,d is the transfer matrix in the degree-d subspace, and m = Lx (Lx/2) for λ = sq, tri (hc), respectively. An analogous formula is given for Möbius strips. We exhibit a method for calculating TZ,λ,Ly,d for arbitrary Ly. Explicit results for arbitrary Ly are given for TZ,λ,Ly,d with d = Ly and Ly - 1. In particular, we find very simple formulas the determinant det(TZ,λ,Ly,d), and trace Tr(T Z,λ,Ly). Corresponding results are given for the equivalent Tutte polynomials for these lattice strips and illustrative examples are included. We also present formulas for self-dual cyclic strips of the square lattice.

AB - We present a method for calculating transfer matrices for the q-state Potts model partition functions Z(G,q,ν), for arbitrary q and temperature variable ν, on cyclic and Möbius strip graphs G of the square (sq), triangular (tri), and honeycomb (hc) lattices of width Ly vertices and of arbitrarily great length Lx vertices. For the cyclic case we express the partition function as Z(λ,Ly × Lx,q,ν) = ∑d=0Lyc(d) Tr[(T Z,λ,Ly,d)m], where λ denotes lattice type, c(d) are specified polynomials of degree d in q, T Z,λ,Ly,d is the transfer matrix in the degree-d subspace, and m = Lx (Lx/2) for λ = sq, tri (hc), respectively. An analogous formula is given for Möbius strips. We exhibit a method for calculating TZ,λ,Ly,d for arbitrary Ly. Explicit results for arbitrary Ly are given for TZ,λ,Ly,d with d = Ly and Ly - 1. In particular, we find very simple formulas the determinant det(TZ,λ,Ly,d), and trace Tr(T Z,λ,Ly). Corresponding results are given for the equivalent Tutte polynomials for these lattice strips and illustrative examples are included. We also present formulas for self-dual cyclic strips of the square lattice.

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U2 - 10.1016/j.physa.2004.08.023

DO - 10.1016/j.physa.2004.08.023

M3 - Article

AN - SCOPUS:10644262923

VL - 347

SP - 314

EP - 352

JO - Physica A: Statistical Mechanics and its Applications

JF - Physica A: Statistical Mechanics and its Applications

SN - 0378-4371

ER -