Probing quantum geometry via second harmonic generation in ideal nodal-line semimetal CaAgP

The geometry of quantum states relates to the magnitude of optical responses in solids and the hybridization of electronic band states. This project investigates the quantum geometrical effect of the second harmonic generation (SHG) with a combination of first-principles calculation and spectroscopic measurements in an ideal nodal-line semimetal CaAgP. We introduce the torsion tensor into the SHG formalism by deriving the relation between the Hermitian connection and the quantum geometry tensor. Next, we study the relation between the SHG susceptibility spectrum, torsion tensor, and the Christoffel symbol for CaAgP. Firstly, our spectroscopic measurements agree well with the numerical calculations. Quantum geometry analysis shows a transition of dominance in the SHG spectrum, where the Christoffel symbol governs the low-frequency diverging part, and the torsion tensor dominates the visible light region where peaks originating from the joint density of states appear. We deduce that the Christoffel symbol enhances SHG through nodal points in the band structure, similar to that proposed for photocurrents. Furthermore, the dominance of the torsion tensor in the visible light region where trivial bands are considered shows that the torsion tensor can be regarded as a displacement field. Our study unveils the quantum geometrical effect in solids by proposing the physical representation of torsion in quantum states and showing the enhancement of SHG via the torsion-less part of the connection tensor.