Potts transition in coupled XY models

Motivated by the long standing experimental mystery of the observation of a melting transition seemingly in the 3-state Potts universality class in thin films of liquid crystals, we revisit a model of coupled XY models. In these compounds, the interplay of the bond-orientational and herring-bone packing degrees of freedom, respectively invariant under rotation by π and by π/3, may lead to emergent discrete variables through the resulting anisotropic coupling. Using both large scale Monte-Carlo algorithms, with an adapted Wolff step and parallel tempering, and analytical methods, such as RG, we study the phase diagram of such a coupled XY model with global U(1) × Z₃ symmetry. The presence of the discrete order can highly frustrate the possible binding of vortices, and we investigate how this non-trivial interaction can lead to unconventional arrangements of the vortices. Our extensive numerical study focuses on the regime where both XY variables are equally present, and we discuss the possibility of a 3-state Potts transition at higher temperature than the BKT transition, as well as the nature of the meeting point between the BKT and Potts phase transition.