Orbital moiré and quadrupolar triple-q physics in a triangular lattice

We numerically study orders of planer type <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mo>(</a:mo><a:mi>x</a:mi><a:mi>y</a:mi><a:mo>,</a:mo><a:msup><a:mi>x</a:mi><a:mn>2</a:mn></a:msup><a:mo>−</a:mo><a:msup><a:mi>y</a:mi><a:mn>2</a:mn></a:msup><a:mo>)</a:mo></a:mrow></a:math> quadrupoles on a triangular lattice with nearest-neighbor isotropic <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>J</b:mi></b:math> and anisotropic <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>K</c:mi></c:math> 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-<d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mi mathvariant="bold-italic">q</d:mi></d:math> 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-<f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mi mathvariant="bold-italic">q</f:mi></f:math> orders, we find various orders including incommensurate triple-<h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mi mathvariant="bold-italic">q</h:mi></h:math> quasi-long-range orders with orbital moiré and a four-sublattice triple-<j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:mi mathvariant="bold-italic">q</j:mi></j:math> partial order. Our Monte Carlo simulations demonstrate that the phase transition to the latter triple-<l:math xmlns:l="http://www.w3.org/1998/Math/MathML"><l:mi mathvariant="bold-italic">q</l:mi></l:math> 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. Published by the American Physical Society 2024