Transfer Matrices for the Partition Function of the Potts Model on Cyclic and Mobius Lattice Strips

Abstract

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 M\"obius strip graphs $G$ of the square (sq), triangular (tri), and honeycomb (hc) lattices of width $L_y$ vertices and of arbitrarily great length $L_x$ vertices. For the cyclic case we express the partition function as $Z(\Lambda,L_y \times L_x,q,v)=\sum_{d=0}^{L_y} c^{(d)} Tr[(T_{Z,\Lambda,L_y,d})^m]$, where $\Lambda$ denotes lattice type, $c^{(d)}$ are specified polynomials of degree $d$ in $q$, $T_{Z,\Lambda,L_y,d}$ is the transfer matrix in the degree-$d$ subspace, and $m=L_x$ ($L_x/2$) for $\Lambda=sq, tri (hc)$, respectively. An analogous formula is given for M\"obius strips. We exhibit a method for calculating $T_{Z,\Lambda,L_y,d}$ for arbitrary $L_y$. Explicit results for arbitrary $L_y$ are given for $T_{Z,\Lambda,L_y,d}$ with $d=L_y$ and $d=L_y-1$. In particular, we find very simple formulas the determinant $det(T_{Z,\Lambda,L_y,d})$, and trace $Tr(T_{Z,\Lambda,L_y})$. 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.