  Recently Second Harmonic Generation (SHG) for the yellow exciton series in
cuprous oxide has been demonstrated [J. Mund et al., Phys. Rev. B 98, 085203
(2018)]. Assuming perfect $O_{\mathrm{h}}$ symmetry, SHG is forbidden along
certain high-symmetry axes. Perturbations can break this symmetry and forbidden
transitions may become allowed. We investigate theoretically the effect of
external magnetic fields on the yellow exciton lines of cuprous oxide. We
identify two mechanisms by which an applied magnetic field can induce a second
harmonic signal in a forbidden direction. First of all, a magnetic field by
itself generally lifts the selection rules. In the Voigt configuration, an
additional magneto-Stark electric field appears. This also induces certain SHG
processes differing from those induced by the magnetic field alone.
Complementary to the manuscript by A. Farenbruch et al. [Phys. Rev. B,
submitted], we perform a full numerical diagonalization of the exciton
Hamiltonian including the complex valence band structure. Numerical results are
compared with experimental data.
