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Abstract

High-order harmonic generation (HHG), which has been widely studied in atomic gas, has recently been expanded to solids to study the highly nonlinear electronic response in condensed matter and produce coherent high-frequency radiation. Recently, attention has turned to topological materials and the use of HHG to characterize topological bands and invariants. However, the theoretical interpretation of the nonlinear electronic response in topological materials presents many challenges. In particular, the Bloch wavefunction phase of topological materials has undefined points in the Brillouin zone. This leads to singularities in the calculation of the interband and intraband transition dipole matrix elements of the semiconductor Bloch equations (SBEs). Here, we use the laser-electromagnetic velocity gauge p⋅A(t) to numerically integrate the SBEs and treat the singularity in the production of the electrical currents and HHG spectra with better numerical efficiency and more straightforward implementation. We used a prototype of Chern insulators (CIs), the Haldane model, to demonstrate our approach. The validity of the velocity gauge approach is demonstrated in the following way: for topologically trivial materials such as MoS₂, qualitative agreement is achieved with the results of the length gauge approach and the time-dependent density functional theory. For the application of the velocity gauge approach to topological materials, Chern insulator is taken, using the two-band Haldane model. We found a good qualitative agreement between the velocity gauge and the length gauge approach in view of (i) the selection rules, (ii) the linear cutoff law scaling, and (iii) anomalous circular dichroism. We conclude that the velocity-gauge approach for HHG provides a theoretical tool to investigate topological materials.