Description of weakly periodic ground states for the Potts model with external field and a countable set of spin values on a Cayley tree

One of the main problems for the Hamiltonian of the Potts model is the description of all the limiting Gibbs measures corresponding to it. At low temperatures, one Gibbs measure corresponds to each ground state. Therefore, for the Potts model, the study of the set of ground states is also relevant. The work is devoted to the study of weakly periodic ground states for the Potts model with an external field and a countable set of spin values on the Cayley tree. It is known that weakly periodic ground states depend on the choice of the normal divisor of the group representation of the Cayley tree. It is also known that there is no normal divisor of an odd index, so in this paper considered the normal divisor of index two. In this paper, for the Potts model with an external field and a countable set of spin values on a Cayley tree of arbitrary order, sets of weakly periodic ground states corresponding to any normal divisors of the index two group representation of the Cayley tree are described. At the same time, it is proved that these sets include periodic ground states corresponding to normal divisors of index two, which were known previously. The sets of all weakly periodic (non-periodic) ground states are also found in the case of a normal divisor of index two, i.e. many new classes of ground states have been found.