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

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 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,v)=∑d=0Lyc(d)Tr[(TZ,Λ,Ly,d)m], where Λ denotes lattice type, c(d) are specified polynomials of degree d in q, TZ,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(TZ,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.