Bulk photovoltaic effects in the Haldane model

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. In contrast, circularly polarized light can not induce shift or injection currents, as constrained by the three-fold rotation symmetry. Additionally, due to the presence of $\mathcal{MT}$ symmetry, a separation of responses is shown in the Haldane model. Under linearly polarized light, shift current, allowed by time-reversal symmetry, flows perpendicularly to the injection current, allowed by $\mathcal{MT}$ symmetry. Across the topological phase transition, the injection current does not change sign since the group velocity's sign remains unchanged. On the contrary, shift current shows a sign flip, as a result of band inversion. Furthermore, we calculate quantum geometry, including quantum metric and symplectic connection, to demonstrate the microscopic quantum origin of the BPVE. We found that the vector field of symplectic connection in the Brillouin zone possesses vortices in the topological phase, but not in the trivial phase.