Energy spectrum of the low-lying states in Sr$_2$FeSi$_2$O$_7$ and the nature of the magnetoelectric effect

A mechanism underlying the magnetoelectric effect is discussed. This mechanism is related to the combined action of an odd crystal field, spin-orbit coupling, and the interaction of the orbital angular momentum with an applied magnetic field. The effective operator describing the spin states of Fe$^{2+}$ ions is obtained. Such operator allows one to interpret the terahertz spectroscopy data and to calculate both the electric field effect on the magnetization and the magnetic field effect on the electric polarization of the sample. It is demonstrated that the magnetoelectric effect is enhanced with a decrease in the energy corresponding to the tetragonal distortion of ligand tetrahedra.