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Critical properties of a geometrically frustrated generalized $XY$ model withantiferromagnetic (AFM) and third-order antinematic (AN3) couplings on atriangular lattice are studied by Monte Carlo simulation. It is found that sucha generalization leads to a phase diagram consisting of three differentquasi-long-range ordered (QLRO) phases. Compared to the model with thesecond-order antinematic (AN2) coupling, besides the AFM and AN3 phases whichappear in the limits of relatively strong AFM and AN3 interactions,respectively, it includes an additional complex canted antiferromagnetic (CAFM)phase. It emerges at lower temperatures, wedged between the AFM and AN3 phases,as a result of the competition between the AFM and AN3 couplings, which isabsent in the model with the AN2 coupling. The AFM-CAFM and AN3-CAFM phasetransitions are concluded to belong to the weak Ising and weak three-statePotts universality classes, respectively. Additionally, all three QLRO phasesalso feature true LRO of the standard and generalized chiralities, which bothvanish simultaneously at second-order phase transitions with non-Ising criticalexponents and the critical temperatures slightly higher than the magnetic andnematic order-disorder transition temperatures.

New ordered phase in geometrically frustrated generalized $XY$ model