Multiple phase transitions in the $XY$ model with nematic-like couplings

Critical behavior of the two-dimensional generalized $XY$ model involving solely nematic-like terms of the second, third and fourth orders is studied by Monte Carlo method. We find that such a system can undergo three successive phase transitions. At higher temperatures there is a phase transition of the Berezinskii-Kosterlitz-Thouless type to the $q=4$ nematic-like phase, followed by two more transitions of the Ising type to the $q=2$ nematic-like and ferromagnetic phases, respectively. The $q$ nematic-like phases are characterized by spin alignments with angles $2k\pi \mathrm{/}q$, where $k\le q$ is an integer. The ferromagnetic phase appears at low temperatures even without the presence of magnetic interactions owing to a synergic effect of the nematic-like terms.