Phase invariance of the semiconductor Bloch equations

We investigate the gauge invariance of the semiconductor Bloch equations (SBEs) in solid high-order harmonic generation (HHG). It is found that the gauge dependence of the SBEs can be attributed to the absence of Berry connection terms in the SBEs in previous studies. When the Berry connection terms are considered, the gauge invariance of the SBEs can be naturally preserved under an arbitrary global phase transformation. To satisfy the demand of the continuity of transition dipole matrix elements in one-dimensional calculations, we propose a gauge that is easy to implement numerically. Finally, the saddle point analysis shows that Berry connections can influence many properties of HHG, such as ellipticity dependence and even harmonic generation. This work uncovers the importance of the Berry connection terms that have been neglected for a long time in simulations of solid HHG.