Exact Partition Function for the Potts Model with Next-Nearest Neighbor Couplings on Strips of the Square Lattice

We present exact calculations of partition function $Z$ of the $q$-state Potts model with next-nearest-neighbor spin-spin couplings, both for the ferromagnetic and antiferromagnetic case, for arbitrary temperature, on $n$-vertex strip graphs of width $L_y=2$ of the square lattice with free, cyclic, and M"obius longitudinal boundary conditions. The free energy is calculated exactly for the infinite-length limit of these strip graphs and the thermodynamics is discussed. Considering the full generalization to arbitrary complex $q$ and temperature, we determine the singular locus $\mathcal{B}$ in the corresponding ${\mathbb{C}}^2$ space, arising as the accumulation set of partition function zeros as $n\to \mathrm{\infty }$.