Ground State Entropy of the Potts Antiferromagnet on Triangular Lattice Strips

Abstract

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 $L_x \to \infty$ limit, the exponent of the ground-state entropy, $W=e^{S_0/k_B}$. The strips considered, with their boundary conditions ($BC$) are (a) $(FBC_y,PBC_x)=$ cyclic, (b) $(FBC_y,TPBC_x)=$ M\"obius, (c) $(PBC_y,PBC_x)=$ toroidal, and (d) $(PBC_y,TPBC_x)=$ 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) $(PBC_y,FBC_x)=$ cylindrical, $L_y=5,6$. Several interesting features are found, including the presence of terms in $P$ proportional to $\cos(2\pi L_x/3)$ for case (c). The continuous locus of points ${\cal B}$ where $W$ is nonanalytic in the $q$ plane is discussed for each case and a comparative discussion is given of the respective loci ${\cal B}$ 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.