Octupolar vortex crystal and toroidal moment in twisted bilayer MnPSe$_3$

Experimental detection of antiferromagnetic order in two-dimensional materials is a challenging task due to the absence of net dipole moments. Identifying multi-domain antiferromagnetic textures via the current techniques is even more difficult. In order to address this challenge, we investigate the higher order multipole moments in twisted bilayer MnPSe$_3$. While the monolayers of MnPSe$_3$ exhibit in-plane N\'eel antiferromagnetic order, our atomistic simulations indicate that the moir\'e superlattices display a two-domain phase on each layer. We show that the octupolar moments $M_{33}^+$ and $M_{33}^-$ are significant in this multi-domain phase at the domain walls. In addition, when $[M_{33}^+,M_{33}^-]$ are represented by the $x$ and $y$ components of a vector, the resultant pattern of these octupole moments winds around the antiferromagnetic domains and forms to vortex crystals which leads to octupolar toroidal moments, $T_{xyz}$ and $T_{z}^{\beta}$. $T_{xyz}$ and $T_{z}^{\beta}$ can give rise to a magnetoelectric effect and gyrotropic birefringence that may provide indirect ways of detecting multi-domain antiferromagnetic order. Our results highlight the importance of higher-order multipole moments for identification of complex spin textures in moir\'e magnets.