Magnetodielectric detection of magnetic quadrupole order in   Ba(TiO)Cu$_4$(PO$_4$)$_4$ with Cu$_4$O$_{12}$ square cupolas
  In vortex-like spin arrangements, multiple spins can combine into emergent multipole moments. Such multipole moments have broken space-inversion and time-reversal symmetries, and can therefore exhibit linear magnetoelectric (ME) activity. Three types of such multipole moments are known: toroidal, monopole, and quadrupole moments. So far, however, the ME-activity of these multipole moments has only been established experimentally for the toroidal moment. Here, we propose a magnetic square cupola cluster, in which four corner-sharing square-coordinated metal-ligand fragments form a noncoplanar buckled structure, as a promising structural unit that carries an ME-active multipole moment. We substantiate this idea by observing clear magnetodielectric signals associated with an antiferroic ME-active magnetic quadrupole order in the real material Ba(TiO)Cu$_4$(PO$_4$)$_4$. The present result serves as a useful guide for exploring and designing new ME-active materials based on vortex-like spin arrangements. 
K. Kimura, P. Babkevich, M. Sera, M. Toyoda, K. Yamauchi, G. S.   Tucker, J. Martius, T. Fennell, P. Manuel, D. D. Khalyavin...
cond-mat.str-el