Phase diagram of the chromatic polynomial on a torus

Jesper Lykke Jacobsen ^1, 2, Jesus Salas ³

Nuclear Physics B – Proceedings Supplements 783 (2007) 238-296

We study the zero-temperature partition function of the Potts antiferromagnet (i.e., the chromatic polynomial) on a torus using a transfer-matrix approach. We consider square- and triangular-lattice strips with fixed width L, arbitrary length N, and fully periodic boundary conditions. On the mathematical side, we obtain exact expressions for the chromatic polynomial of widths L=5,6,7 for the square and triangular lattices. On the physical side, we obtain the exact « phase diagrams » for these strips of width L and infinite length, and from these results we extract useful information about the infinite-volume phase diagram of this model: in particular, the number and position of the different phases.

• 1. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
  CNRS : UMR8626 – Université Paris XI – Paris Sud
• 2. Service de Physique Théorique (SPhT),
  CNRS : URA2306 – CEA : DSM/SPHT
• 3. Grupo de Modelizacion, Simulacion Numerica y Matematica Industrial,
  Universidad Carlos III de Madrid