Abstract
We numerically study orders of planer type (xy,x2‑y2) quadrupoles on a triangular lattice with nearest-neighbor isotropic J and anisotropic K interactions. This type of quadrupole possesses unique single-ion anisotropy proportional to a third order of the quadrupole moments. This provides an unconventional mechanism of triple-q orders which does not exist for the degrees of freedom with odd parity under time-reversal operation such as magnetic dipoles. In addition to several single-q orders, we find various orders including incommensurate triple-q quasi-long-range orders with orbital moiré and a four-sublattice triple-q partial order. Our Monte Carlo simulations demonstrate that the phase transition to the latter triple-q state belongs to the universality class of the critical line of the Ashkin-Teller model in two dimensions close to the four-state Potts class. These results indicate a possibility of realizing unique quadrupole textures in simple triangular systems.