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

We study the chromatic polynomials (= zero-temperature antiferromagnetic Po tts-model partition functions) P-G(q) for mxn rectangular subsets of the sq uare lattice, with m less than or equal to 8 (free or periodic transverse b oundary conditions) and n arbitrary (free longitudinal boundary conditions) , using a transfer matrix in the Fortuin Kasteleyn representation. In parti cular, we extract the limiting curves of partition-function zeros when n -- > infinity, which arise from the crossing in modulus of dominant eigenvalue s (Beraha-Kahane-Weiss theorem). We also provide evidence that the Beraha n umbers B-2, B-3, B-4, B-5 are limiting points of partition-function zeros a s n --> infinity, whenever the strip width m is greater than or equal to 7 (periodic transverse b.c,) or greater than or equal to 8 (free transverse b .c.). Along the way, we prove that a noninteger Beraha number (except perha ps B-10) cannot be a chromatic root of any graph.