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 M₃₃⁺ and M₃₃^- are significant in this multidomain phase. In addition, when [M₃₃⁺,M₃₃^-] are represented by the x and y components of a vector, the resultant pattern of these octupole moments forms vortex crystals which leads to octupolar toroidal moments, T_xyz and T_z^β. T_xyz and T_z^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.