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

Jesús Salas, Alan D. Sokal

    Research output: Contribution to journalArticle

    Abstract

    We study the chromatic polynomials (= zero-temperature antiferromagnetic Potts-model partition functions) PG(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.

    Original languageEnglish (US)
    Pages (from-to)609-699
    Number of pages91
    JournalJournal of Statistical Physics
    Volume104
    Issue number3-4
    DOIs
    StatePublished - Aug 2001

    Keywords

    • Antiferromagnetic Potts model
    • Beraha numbers
    • Beraha-Kahane-Weiss theorem
    • Chromatic polynomial
    • Chromatic root
    • Fortuin-Kasteleyn representation
    • Square lattice
    • Temperley-Lieb algebra
    • Transfer matrix

    ASJC Scopus subject areas

    • Statistical and Nonlinear Physics
    • Mathematical Physics

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