Bulk photovoltaic effects in the Haldane model

Abstract: The bulk photovoltaic effect (BPVE) refers to the direct current generation in a noncentrosymmetric material under illumination and can be applied to solar energy technology. BPVE includes injection and shift currents, led by the change of velocity and displacement of wave packet during optical transitions, respectively. We derive the constraints on the conductivity tensors imposed by mirror-time ($\mathcal{MT}$) symmetry for two-dimensional systems. For the Haldane model, we show that linearly polarized light can induce shift and injection currents, which vanish under circularly polarized light as constrained by the three-fold rotation and $\mathcal{MT}$ symmetry. Additionally, due to the presence of $\mathcal{MT}$ symmetry, a separation of responses is observed in the Haldane model: one direction exhibits a time-reversal symmetry-allowed response, whereas another manifests a parity-time symmetry-allowed response. Across the topological phase transition, the injection current does not change sign, whereas shift current shows a sign flip. The vector field of the Hermitian connection in the Brillouin zone possesses vortices in the topological phase, but not in the trivial phase. Furthermore, we calculate the related quantum geometry, including Berry curvature, quantum metric and Hermitian connection, and demonstrate the microscopic quantum origin of the BPVE.