Orbital moir\'e and quadrupolar triple-q physics in a triangular lattice

We numerically study orders of planer type $(xy,x^2-y^2)$ quadrupoles on a triangular lattice with nearest-neighbor isotropic $J$ and anisotropic $K$ interactions. This type of quadrupoles 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 moir\'e 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.