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

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 L_y and arbitrarily great length L_x. 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, T_Z,Λ,Ly,d is the corresponding transfer matrix, and m=L_x (L_x/2) for Λ=sq,tri (hc), respectively. An analogous formula is given for Möbius strips, while only T_Z,Λ,Ly,d=0 appears for free strips. We exhibit a method for calculating T_Z,Λ,Ly,d for arbitrary L_y and give illustrative examples. Explicit results for arbitrary L_y are presented for T_Z,Λ,Ly,d with d=L_y and d=L_y-1. We find very simple formulas for the determinant det(T_Z,Λ,Ly,d). We also give results for self-dual cyclic strips of the square lattice.