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

publication date

  • September 2011

start page

  • 1028

end page

  • 1122


  • 5


  • 144

International Standard Serial Number (ISSN)

  • 0022-4715

Electronic International Standard Serial Number (EISSN)

  • 1572-9613


  • 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.


  • chromatic polynomial; chromatic roots; tutte polynomial; potts model; transfer matrix; beraha-kahane-weiss theorem; planar graph; square lattice; extra-vertex boundary conditions