Quantum geometry and nonlinear optical responses in rhombohedral trilayer graphene

The nonlinear optical response of a rhombohedral trilayer graphene system is theoretically studied to identify signatures of quantum geometry. Applying a perpendicular electric field known as a displacement field breaks inversion symmetry and introduces a gap in the material's band structure, allowing us to calculate the second-order DC response, known as the shift current, in a low-frequency regime. We then examine the shift current and various quantities describing the quantum geometry of the system to investigate whether the shift current exhibits any features associated with the quantum geometric properties of the system. In comparing rhombohedral trilayer graphene to Bernal-stacked bilayer graphene, we observe that the shift current conductivities of the two materials display similar features at certain frequencies and distinct differences at others. Further analysis of the conductivities of the two systems reveals that a sign change in the shift current of rhombohedral trilayer graphene, which does not occur in bilayer graphene, corresponds to a band inversion within the conduction or valence bands and also correlates with significant changes in the Berry curvature, quantum metric, Hermitian metric, and eigenvector projections of the system. However, a clear correlation between the features observed in the shift current and those in the quantum geometry has yet to be fully established.