Linear photogalvanic effect induced by quantum metric in PT -symmetric tilted nodal-line semimetals

Linear photogalvanic effect (LPGE), a second-order nonlinear optical effect under the irradiation by linearly polarized light, stems usually from shift current due to the contribution of Berry curvature. For tilted Nodal-line semimetals, the LPGE from the shift current is excluded by $PT$ symmetry. Interestingly, it is found that the remarkable LPGE arises from the injection current, which is instead induced by quantum metric, a real part of quantum geometry. By employing the combining Keldysh-Floquet method, we derive a formula to calculate the quantum metric induced LPGE. The dependence of the photogalvanic response is discussed in detail on dispersion tilt, chemical potential, and nodal-line radius. The linear photogalvanic conductivity exhibits rich characteristic frequencies, which are closely related to the Fermi surface geometry, determined by the relative size of the tilt and the chemical potential. These features may serve as a diagnostic tool to characterize the systemic parameters. We also check that the quantum metric induced LPGE has no the fingerprint of the parity anomaly, different from the Berry curvature induced nonlinear responses or linear response.