Bulk photovoltaic effects and quantum geometry in the Haldane model

The bulk photovoltaic effect (BPVE) is the d.c. current generation in an inversion-symmetry broken material under light illumination. BPVE includes injection and shift currents, as a result of the change of velocity and displacement of wave packets during optical transitions, respectively. We derive the constraints on the nonvanishing components of 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 injection or shift current as constrained by the three-fold rotation symmetry. Additionally, due to the presence of $\mathcal{MT}$ symmetry, a separation of responses is expected in the Haldane model: one direction exhibits a time-reversal symmetry-allowed shift photocurrent, whereas another orthogonal direction manifests a $\mathcal{MT}$ allowed injection photocurrent. Furthermore, we demonstrate the microscopic quantum origin of the BPVE by calculating quantum metric and Hermitian connection. The vector field of the Hermitian connection in the Brillouin zone possesses vortices only in the topological phase, corresponding to nonzero Chern number. The addition of magnetic flux breaks the mirror symmetry, as captured by quantum metric, leading to nonvanishing injection current. Our findings establish the BPVE as an optical probe for quantum geometry and topological phase transitions.