Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models I. General Theory and Square-Lattice Chromatic Polynomial

Details Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models I. General Theory and Square-Lattice Chromatic Polynomial Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models I. General Theory and Square-Lattice Chromatic Polynomial

2000-04-19

arxiv.org/abs/cond-mat/0004330v2

J.Statist.Phys. 104 (2001) 609-699

Physics

Condensed Matter

Statistical Mechanics

90 pages, LaTeX2e. Self-unpacking archive containing the tex file, three sty files, 39 ps files, and a Mathematica file (trans

Scientific paper

We study the chromatic polynomials (= zero-temperature antiferromagnetic Potts-model partition functions) P_G(q) for m \times n rectangular subsets of the square lattice, with m \le 8 (free or periodic transverse boundary conditions) and n arbitrary (free longitudinal boundary conditions), using a transfer matrix in the Fortuin-Kasteleyn representation. In particular, we extract the limiting curves of partition-function zeros when n \to\infty, which arise from the crossing in modulus of dominant eigenvalues (Beraha-Kahane-Weiss theorem). We also provide evidence that the Beraha numbers B_2,B_3,B_4,B_5 are limiting points of partition-function zeros as n \to\infty whenever the strip width m is \ge 7 (periodic transverse b.c.) or \ge 8 (free transverse b.c.). Along the way, we prove that a noninteger Beraha number (except perhaps B_{10}) cannot be a chromatic root of any graph.

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