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

    Fingerprint

    Chromatic Polynomial
    Matrix Function
    Potts Model
    Transfer Matrix
    Square Lattice
    Partition Function
    Transfer Function
    partitions
    polynomials
    Transverse
    Zero
    Limiting
    Polynomial Zeros
    boundary conditions
    Boundary conditions
    set theory
    Strip
    strip
    Modulus
    eigenvalues

    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

    • Mathematical Physics
    • Physics and Astronomy(all)
    • Statistical and Nonlinear Physics

    Cite this

    Transfer matrices and partition-function zeros for antiferromagnetic potts models. I. General theory and square-lattice chromatic polynomial. / Salas, Jesús; Sokal, Alan D.

    In: Journal of Statistical Physics, Vol. 104, No. 3-4, 08.2001, p. 609-699.

    Research output: Contribution to journalArticle

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