Representations and partition function zeros of the Potts model with and without boundaries
Seminar Room 1, Newton Institute
We consider the following four classes of models defined on the annulus, listed in order of increasing generality: 1. Restricted height models with face interactions 2. Fully-packed loop models based on the Temperley-Lieb algebra 3. TL loop models with one boundary generator 4. TL loop models with two boundary generators All of these are different formulations of the Potts model (Tutte polynomial), where, for classes 3 and 4, spins on the rims of the annulus are required to take a different number of states than bulk spins. We show how each class reduces to the one preceding it, provided that one of its parameters takes particular, magical values. In an appropriate part of the phase diagram, these reductions lead to partition function zeros in the more general model. In particular, the reduction from class 2 to 1 gives zeros of the chromatic polynomial at the Beraha numbers, provided one is inside the so-called Berker-Kadanoff phase.
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