On weakly periodic Gibbs measures of the Potts model with a special external field on a Cayley tree

In the paper, we study the $q$-state (where $q=3,4,5,\dots$) Potts model with special external field on a Cayley tree of order $k\geq 2$. For antiferromagnetic Potts model with such an external field on the Cayley tree of order $k\geq 6$, the non-uniqueness of weakly periodic (non-periodic) Gibbs measures is proved. The weakly periodic Gibbs measures for the Potts model with zero external field are also studied. It is proved that under some conditions imposed on the parameters of the model there can be not less than $2^q-2$ such measures.