Periodic Gibbs measures for the Potts model in translation-invariant and periodic external fields on the Cayley tree

We study the Potts model in translation-invariant and periodic external fields on the Cayley tree of order $${k\geq 2}$$ . For the Potts model in a translation-invariant external field for $$k\geq 2$$ , the nonuniqueness of the translation-invariant and periodic Gibbs measure is shown. It is proved that for the Potts model in an external field that is not translation-invariant, translation-invariant Gibbs measures do not exist on the Cayley tree of order $$k\geq 2$$ . Periodic Gibbs measures are also studied for the Potts model in a periodic external field. We prove that under certain conditions, the number of such measures can be at least three.

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