Abstract
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.
Highlights
The two–dimensional (2D) q–state Potts model [1] is one of the most studied models in Statistical Mechanics
The ferromagnetic regime of the Potts model is the best understood case: exact results have been obtained for the ferromagnetic-paramagnetic phase transition temperature Tc(q), the order of the transition, the phase diagram, and the characterization in terms of conformal field theory (CFT) of the universality classes. (See e.g., Ref. [2].)
This paper is organized as follows: in Section 2 we show in detail how to obtain the chromatic polynomial of a strip with toroidal boundary conditions, and some structural properties: dimensionality of the transfer matrix and properties of the amplitudes
Summary
The two–dimensional (2D) q–state Potts model [1] is one of the most studied models in Statistical Mechanics. Despite many efforts over more than 50 years, its exact free energy and phase diagram are still unknown. The ferromagnetic regime of the Potts model is the best understood case: exact (albeit not always rigorous) results have been obtained for the ferromagnetic-paramagnetic phase transition temperature Tc(q) (at least for several regular lattices), the order of the transition (continuous for 0 ≤ q ≤ 4), the phase diagram, and the characterization in terms of conformal field theory (CFT) of the universality classes. We know the exact free energy along some curves of the phase diagram (q, T ) (where T is the temperature) for some regular lattices [2]; but in this regime, this (partial) solubility of the model does not imply criticality (as it is been observed for the ferromagnetic regime with 0 ≤ q ≤ 4).
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