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

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.