Limitations of mean-field approximations in describing shift-current and injection-current in materials

We theoretically investigate bulk photovoltaic effects, with a specific focus on shift-current and injection-current. Initially, we perform a numerical analysis of the direct current (dc) induced by a laser pulse with a one-dimensional model, utilizing mean-field theories such as time-dependent Hartree--Fock and time-dependent Hartree methods. Our numerical results, obtained with mean-field theories, reveal that the dc component of the current exists even after irradiation with linearly polarized light as a second-order nonlinear effect, indicating the generation of injection current. Conversely, when we employ the independent particle approximation, no injection current is generated by linearly polarized light. To develop the microscopic understanding of injection current within the mean-field approximation, we further analyze the dc component of the current with the perturbation theory, employing the mean-field approximations, the independent-particle approximation, and the exact solution of the many-body Schr\"odinger equation. The perturbation analysis clarifies that the injection current induced by linearly polarized light under the mean-field approximations is an artifact caused by population imbalance, created through quantum interference from unphysical self-excitation pathways. Therefore, investigation of many-body effects on the bulk photovoltaic effects have to be carefully conducted in mean-field schemes due to potential contamination by unphysical dc current. Additionally, we perform the first-principles electron dynamics calculation for BaTiO$_3$ based on the time-dependent density functional theory, and we confirm that the above findings from the one-dimensional model calculation and the perturbation analysis apply to realistic systems.