  Violation of time reversal and spatial inversion symmetries has profound
consequences for elementary particles and cosmology. Spontaneous breaking of
these symmetries at phase transitions gives rise to unconventional physical
phenomena in condensed matter systems, such as ferroelectricity induced by
magnetic spirals, electromagnons, non-reciprocal propagation of light and spin
waves, and the linear magnetoelectric (ME) effect - the electric polarization
proportional to the applied magnetic field and the magnetization induced by the
electric field. Here, we report the experimental study of the holmium-doped
langasite, Ho$_{x}$La$_{3-x}$Ga$_5$SiO$_{14}$, showing a puzzling combination
of linear and highly non-linear ME responses in the disordered paramagnetic
state: its electric polarization grows linearly with the magnetic field but
oscillates many times upon rotation of the magnetic field vector. We propose a
simple phenomenological Hamiltonian describing this unusual behavior and derive
it microscopically using the coupling of magnetic multipoles of the rare-earth
ions to the electric field.
