Ground State Entropy of the Potts Antiferromagnet on Triangular Lattice Strips

We present exact calculations of the zero-temperature partition function (chromatic polynomial) $P$ for the $q$-state Potts antiferromagnet on triangular lattice strips of arbitrarily great length $L_x$ vertices and of width $L_y=3$ vertices and, in the $L_x\to \mathrm{\infty }$ limit, the exponent of the ground-state entropy, $W=e^{S_0\mathrm{/}{k}_B}$. The strips considered, with their boundary conditions ($BC$) are (a) $(FB{C}_y,PB{C}_x)=$ cyclic, (b) $(FB{C}_y,TPB{C}_x)=$ M"obius, © $(PB{C}_y,PB{C}_x)=$ toroidal, and (d) $(PB{C}_y,TPB{C}_x)=$ Klein bottle, where $F$, $P$, and $TP$ denote free, periodic, and twisted periodic. Exact calculations of $P$ and $W$ are also given for wider strips, including (e) cyclic, $L_y=4$, and (f) $(PB{C}_y,FB{C}_x)=$ cylindrical, $L_y=5,6$. Several interesting features are found, including the presence of terms in $P$ proportional to $\cos (2\pi L_x\mathrm{/}3)$ for case ©. The continuous locus of points $\mathcal{B}$ where $W$ is nonanalytic in the $q$ plane is discussed for each case and a comparative discussion is given of the respective loci $\mathcal{B}$ for families with different boundary conditions. Numerical values of $W$ are given for infinite-length strips of various widths and are shown to approach values for the 2D lattice rapidly. A remark is also made concerning a zero-free region for chromatic zeros.