Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models VI. Square Lattice with Extra-Vertex Boundary Conditions

Jesús Salas, Alan D. Sokal

    Research output: Contribution to journalArticle

    Abstract

    We study, using transfer-matrix methods, the partition-function zeros of the square-lattice q-state Potts antiferromagnet at zero temperature (= square-lattice chromatic polynomial) for the boundary conditions that are obtained from an m×n grid with free boundary conditions by adjoining one new vertex adjacent to all the sites in the leftmost column and a second new vertex adjacent to all the sites in the rightmost column. We provide numerical evidence that the partition-function zeros are becoming dense everywhere in the complex q-plane outside the limiting curve B∞(sq) for this model with ordinary (e. g. free or cylindrical) boundary conditions. Despite this, the infinite-volume free energy is perfectly analytic in this region.

    Original languageEnglish (US)
    Pages (from-to)1028-1122
    Number of pages95
    JournalJournal of Statistical Physics
    Volume144
    Issue number5
    DOIs
    StatePublished - Sep 2011

    Fingerprint

    Matrix Function
    Potts Model
    Transfer Matrix
    Square Lattice
    Partition Function
    Transfer Function
    partitions
    apexes
    boundary conditions
    Boundary conditions
    Zero
    Adjacent
    Vertex of a graph
    Chromatic Polynomial
    Antiferromagnet
    Transfer Matrix Method
    free boundaries
    Free Boundary
    matrix methods
    Free Energy

    Keywords

    • Beraha-Kahane-Weiss theorem
    • Chromatic polynomial
    • Chromatic roots
    • Extra-vertex boundary conditions
    • Planar graph
    • Potts model
    • Square lattice
    • Transfer matrix
    • Tutte polynomial

    ASJC Scopus subject areas

    • Statistical and Nonlinear Physics
    • Mathematical Physics

    Cite this

    Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models VI. Square Lattice with Extra-Vertex Boundary Conditions. / Salas, Jesús; Sokal, Alan D.

    In: Journal of Statistical Physics, Vol. 144, No. 5, 09.2011, p. 1028-1122.

    Research output: Contribution to journalArticle

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