Transfer matrices and partition-function zeros for antiferromagnetic potts models. I. General theory and square-lattice chromatic polynomial

We study the chromatic polynomials (= zero-temperature antiferromagnetic Potts-model partition functions) P_G(q) for m×n rectangular subsets of the square lattice, with m≤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→∞, which arise from the crossing in modulus of dominant eigenvalues (Beraha-Kahane-Weiss theorem). We also provide evidence that the Beraha numbers B₂, B₃, B₄, A₅ are limiting points of partition-function zeros as n→∞ whenever the strip width m is ≥7 (periodic transverse b.c.) or ≥8 (free transverse b.c.). Along the way, we prove that a noninteger Beraha number (except perhaps B₁₀) cannot be a chromatic root of any graph.