Triple-$\mathcal{Q}$ partial magnetic orders induced by quadrupolar interactions: Triforce order scenario for UNi$_4$B

Cluster Toroidal Multipoles Formed by Electric-Quadrupole and Magnetic-Octupole Trimers: A Possible Scenario for Hidden Orders in Ca$_5$Ir$_3$O$_{12}$

“…Among them, the odd-parity toroidal multipoles, i.e., the odd-rank MT and even-rank ET multipoles, have been extensively studied, since they give rise to physical phenomena related to the spatial-parity breaking, such as the magnetoelectric effect under the MT dipole, [8][9][10][11][12][13][14][15][16][17] nonlinear (spin) transport [18][19][20][21][22][23][24][25] and nonreciprocal magnon excitations under the MT dipole and octupole, [26][27][28][29][30][31] and Edelstein effect and rotation-field induced electric polarization under the ET monopole and quadrupole. [32][33][34][35][36][37] These unconventional electronic orderings have been proposed and identified in metallic materials, such as the MT dipole orderings in UNi 4 B, 12,[38][39][40][41][42][43][44] Ce 3 TiBi 5 , [45][46][47][48] and CeCoSi 36,[49][50]…”