TY - JOUR

T1 - Transfer matrices for the partition function of the Potts model on lattice strips with toroidal and Klein-bottle boundary conditions

AU - Chang, Shu Chiuan

AU - Shrock, Robert

N1 - Funding Information:
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 NSC grant NSC-93-2119-M-006-009. S.C.C would like to thank the support from National Center for Theoretical Sciences with grant NSC 94-2119-M-002-001.

PY - 2006/5/15

Y1 - 2006/5/15

N2 - We present a method for calculating transfer matrices for the q-state Potts model partition functions Z(G,q,v), for arbitrary q and temperature variable v, on strip graphs G of the square (sq), triangular (tri), and honeycomb (hc) lattices of width Ly vertices and of arbitrarily great length Lx vertices, subject to toroidal and Klein-bottle boundary conditions. For the toroidal case we express the partition function as Z(Λ,Ly×Lx,q,v)=∑d= 0Ly∑jbj(d)(λZ,Λ,Ly,d,j)m, where Λ denotes lattice type, bj(d) are specified polynomials of degree d in q, λZ,Λ,Ly,d,j are eigenvalues of the transfer matrix TZ,Λ,Ly,d in the degree-d subspace, and m=Lx (Lx/2) for Λ=sq,tri(hc), respectively. An analogous formula is given for Klein-bottle strips. We exhibit a method for calculating TZ,Λ,Ly,d for arbitrary Ly. In particular, we find some very simple formulas for the determinant det(TZ,Λ,Ly,d), and trace Tr(TZ,Λ,Ly). Illustrative examples of our general results are given, including new calculations of transfer matrices for Potts model partition functions on strips of the square, triangular, and honeycomb lattices with toroidal or Klein-bottle boundary conditions.

AB - We present a method for calculating transfer matrices for the q-state Potts model partition functions Z(G,q,v), for arbitrary q and temperature variable v, on strip graphs G of the square (sq), triangular (tri), and honeycomb (hc) lattices of width Ly vertices and of arbitrarily great length Lx vertices, subject to toroidal and Klein-bottle boundary conditions. For the toroidal case we express the partition function as Z(Λ,Ly×Lx,q,v)=∑d= 0Ly∑jbj(d)(λZ,Λ,Ly,d,j)m, where Λ denotes lattice type, bj(d) are specified polynomials of degree d in q, λZ,Λ,Ly,d,j are eigenvalues of the transfer matrix TZ,Λ,Ly,d in the degree-d subspace, and m=Lx (Lx/2) for Λ=sq,tri(hc), respectively. An analogous formula is given for Klein-bottle strips. We exhibit a method for calculating TZ,Λ,Ly,d for arbitrary Ly. In particular, we find some very simple formulas for the determinant det(TZ,Λ,Ly,d), and trace Tr(TZ,Λ,Ly). Illustrative examples of our general results are given, including new calculations of transfer matrices for Potts model partition functions on strips of the square, triangular, and honeycomb lattices with toroidal or Klein-bottle boundary conditions.

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U2 - 10.1016/j.physa.2005.08.076

DO - 10.1016/j.physa.2005.08.076

M3 - Article

AN - SCOPUS:33645161405

SN - 0378-4371

VL - 364

SP - 231

EP - 262

JO - Physica A: Statistical Mechanics and its Applications

JF - Physica A: Statistical Mechanics and its Applications

ER -