Structure of the Partition Function and Transfer Matrices for the Potts Model in a Magnetic Field on Lattice Strips

Abstract

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 $Z(\Lambda,L_{y}\times L_{x},q,v,w)=\sum_{d=0}^{L_{y}}\tilde{c}^{(d)}\mathrm{Tr}[(T_{Z,\Lambda,L_{y},d})^{m}]$ , where Λ denotes the lattice type, $\tilde{c}^{(d)}$ are specified polynomials of degree d in q, $T_{Z,\Lambda,L_{y},d}$ is the corresponding transfer matrix, and m=L _x (L _x /2) for Λ=sq,tri (hc), respectively. An analogous formula is given for Möbius strips, while only $T_{Z,\Lambda,L_{y},d=0}$ appears for free strips. We exhibit a method for calculating $T_{Z,\Lambda,L_{y},d}$ for arbitrary L _y and give illustrative examples. Explicit results for arbitrary L _y are presented for $T_{Z,\Lambda,L_{y},d}$ with d=L _y and d=L _y −1. We find very simple formulas for the determinant $\mathrm{det}(T_{Z,\Lambda,L_{y},d})$ . We also give results for self-dual cyclic strips of the square lattice.