Magnetic field induced directional dichroism of spin waves in multiferroic BiFeO$_3$ at THz frequencies*

Using far infrared spectroscopy in high magnetic fields we show that spin excitations in BiFeO$_3$ simultaneously interact with the electric and magnetic field components of light resulting in directional dichroism (DD) of absorption. DD in BiFeO$_3$ arises because an applied static magnetic field induces a toroidal moment in the cycloidal spin structure. Strong DD is observed even in the room-temperature state of the material. The results are explained on the microscopic level as an interplay of five different interactions: isotropic exchange couplings between nearest and next nearest neighbors, an easy-axis anisotropy along the ferroelectric polarization, Dzyaloshinskii-Moriya (DM) interaction that creates the cycloid and DM interaction that causes spin canting.