Ground State Entropy of the Potts Antiferromagnet on Triangular Lattice Strips

Details Ground State Entropy of the Potts Antiferromagnet on Triangular Lattice Strips Ground State Entropy of the Potts Antiferromagnet on Triangular Lattice Strips

2000-04-10

arxiv.org/abs/cond-mat/0004129v2

Annals Phys. 290 (2001) 124-155

Physics

Condensed Matter

Statistical Mechanics

50 pages, latex, 6 encapsulated postscript figures

Scientific paper

10.1006/aphy.2001.6143

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.

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