Bloch gauge symmetry of the semiconductor Bloch equations [Invited]

The semiconductor Bloch equations (SBEs) are a well-established model for optical interactions in condensed matter. In particular, the SBEs in the electromagnetic length gauge preserve the band picture of periodic crystals and thus provide an intuitive and numerically efficient model of high harmonic generation (HHG) in solids. For materials with broken inversion or time-reversal symmetry, the length gauge SBEs involve complex transition dipole moments (TDMs), which depend on the choice of Bloch gauge. The numerical and conceptual complications resulting from this gauge freedom have impeded interpretation and key applications of HHG, such as the tomographic reconstruction of crystal band structure. We derive gauge invariant SBEs (GI-SBEs) that contain only gauge invariant structural quantities: the absolute value of TDMs, the shift vector, and for more than two bands a triple product of TDM phases. The GI-SBEs provide insight into the physics of HHG in solids with broken inversion symmetry, which we demonstrate in gapped graphene.