Ground State Entropy of the Potts Antiferromagnet with Next-Nearest-Neighbor Spin-Spin Couplings on Strips of the Square Lattice

We present exact calculations of the zero-temperature partition function (chromatic polynomial) and $W(q)$, the exponent of the ground-state entropy, for the $q$-state Potts antiferromagnet with next-nearest-neighbor spin-spin couplings on square lattice strips, of width $L_y=3$ and $L_y=4$ vertices and arbitrarily great length $L_x$ vertices, with both free and periodic boundary conditions. The resultant values of $W$ for a range of physical $q$ values are compared with each other and with the values for the full 2D lattice. These results give insight into the effect of such non-nearest neighbor couplings on the ground state entropy. We show that the $q=2$ (Ising) and $q=4$ Potts antiferromagnets have zero-temperature critical points on the $L_x\to \mathrm{\infty }$ limits of the strips that we study. With the generalization of $q$ from ${\mathbb{Z}}_{+}$ to $\mathbb{C}$, we determine the analytic structure of $W(q)$ in the $q$ plane for the various cases.