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_xLa_3−xGa₅SiO₁₄, 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.

违反时间反转和空间反转对称性会对基本粒子和宇宙学产生深远的影响。这些对称性在相变处的自发破裂会在凝聚态系统中引起非常规的物理现象，例如磁螺旋，电磁子，光和自旋波的不可逆传播以及线性磁电（ME）效应所产生的铁电。极化与施加的磁场和电场感应的磁化强度成正比。在这里，我们报告了掺杂的镧铁矿Ho _x La _{3− x} Ga ₅ SiO ₁₄的实验研究。，显示出在无序顺磁状态下线性和高度非线性的ME反应令人费解的组合：其极化随磁场线性增长，但在磁场矢量旋转时振荡多次。我们提出了一个简单的现象学哈密顿量，描述了这种不寻常的行为，并使用稀土离子的磁性多极与电场的耦合从微观上导出了它。