Octupolar Vortex Crystal and Toroidal Moment in Twisted Bilayer MnPSe₃

Experimental detection of antiferromagnetic order in two-dimensional materials is a challenging task. Identifying multidomain antiferromagnetic textures via the current techniques is even more difficult. Therefore, we investigate the higher-order multipole moments in twisted bilayer MnPSe₃. While the monolayers of MnPSe₃ exhibit antiferromagnetic order, the moiré superlattices display a two-domain phase. We show that the octupolar moments <i>M</i>₃₃⁺ and <i>M</i>₃₃^- are significant in this multidomain phase. In addition, when [<i>M</i>₃₃⁺,<i>M</i>₃₃^-] are represented by the <i>x</i> and <i>y</i> components of a vector, the resultant pattern of these octupole moments forms vortex crystals which leads to octupolar toroidal moments, <i>T</i>_<i>xyz</i> and <i>T</i>_<i>z</i>^β. <i>T</i>_<i>xyz</i> and <i>T</i>_<i>z</i>^β can give rise to a magnetoelectric effect and gyrotropic birefringence that may provide indirect ways of detecting multidomain antiferromagnetic order. Our results highlight the importance of higher-order multipole moments for identification of complex spin textures in moiré magnets.