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, M?bius, 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 Z(L,L_y×L_x,q,v,w)=?_d=0^L_y[(c)tilde]^(d)Tr[(T_Z,L,Ly,d)^m]Z(Lambda,L_{y}times L_{x},q,v,w)=sum_{d=0}^{L_{y}}tilde{c}^{(d)}mathrm{Tr}[(T_{Z,Lambda,L_{y},d})^{m}] , where Λ denotes the lattice type, [(c)tilde]^(d)tilde{c}^{(d)} are specified polynomials of degree d in q, T_Z,L,Ly,dT_{Z,Lambda,L_{y},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 M?bius strips, while only T_Z,L,Ly,d=0T_{Z,Lambda,L_{y},d=0} appears for free strips. We exhibit a method for calculating T_Z,L,Ly,dT_{Z,Lambda,L_{y},d} for arbitrary L _y and give illustrative examples. Explicit results for arbitrary L _y are presented for T_Z,L,Ly,dT_{Z,Lambda,L_{y},d} with d=L _y and d=L _y −1. We find very simple formulas for the determinant det(T_Z,L,Ly,d)mathrm{det}(T_{Z,Lambda,L_{y},d}) . We also give results for self-dual cyclic strips of the square lattice.