Phase diagram of the Potts model with the number of spin states $q$ = 4 on a hexagonal lattice

We have carried out Monte Carlo investigations of the phase transitions, thermodynamic properties and ground-state magnetic structures in two-dimensional 4-state Potts model on a hexagonal lattice with the competing exchange interactions. Researches are carried out for the value of interaction of next-nearest neighbor in the range of 0.0 $\le r\le$ 1.0. Taking into account of antiferromagnetic interaction of next-nearest neighbor is shown to lead to the violation of the magnetic ordering. The phase diagram of dependence of critical temperature on the value of interaction of next-nearest neighbor is plotted. Of the orders of phase transitions has been carried out. It is established that the phase transition of the first order is observed in the ranges of 0.0 $\le r\le$ 0.2 and 0.7 $\le r\le$ 1.0. In the range of 0.3 $\le r\le$ 0.6 frustration are observed.