Structure gauges and laser gauges for the semiconductor Bloch equations in high-order harmonic generation in solids

The semiconductor Bloch equations (SBEs) are routinely used for simulations of strong-field laser-matter interactions in condensed matter. In systems without inversion or time-reversal symmetries, the Berry connections and transition dipole phases (TDPs) must be included in the SBEs, which in turn requires the construction of a smooth and periodic structure gauge for the Bloch states. Here, we illustrate a general approach for such a structure-gauge construction for topologically trivial systems. Furthermore, we investigate the SBEs in the length and velocity gauges and discuss their respective advantages and shortcomings for the high-harmonic generation (HHG) process. We find that in cases where we require dephasing or separation of the currents into interband and intraband contributions, the length-gauge SBEs are computationally more efficient. In calculations without dephasing and where only the total current is needed, the velocity-gauge SBEs are structure-gauge independent and are computationally more efficient. We employ two systems as numerical examples to highlight our findings: a one-dimensional model of ZnO and the two-dimensional monolayer hexagonal boron nitride (hBN). The omittance of Berry connections or TDPs in the SBEs for hBN results in nonphysical HHG spectra. The structure- and laser-gauge considerations in the current work are not restricted to the HHG process and are applicable to all strong-field matter simulations with SBEs.