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

Abstract

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.

90 pages, LaTeX2e. Self-unpacking archive containing the tex file, three sty files, 39 ps files, and a Mathematica file (transfer1.m). Many small changes to improve clarity and new Section 7.3 added. To be published in J. Stat. Phys

References in corpus (7)

• Chromatic roots are dense in the whole complex plane
• Exact Potts Model Partition Functions on Ladder Graphs
• Exact Potts Model Partition Function on Strips of the Triangular Lattice
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• Ground State Entropy of the Potts Antiferromagnet on Strips of the Square Lattice
• Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models II. Extended Results for Square-Lattice Chromatic Polynomial
• Unusual corrections to scaling in the 3-state Potts antiferromagnet on a square lattice

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