Projector Method for Nonlinear Light-Matter Interactions and Quantum Geometry

We develop a systematic projector-based Feynman diagram framework that intrinsically encodes quantum geometry for nonlinear optical responses. By explicitly incorporating geometric quantities such as the quantum geometric tensor, quantum hermitian connection, and triple phase product, the method ensures component-wise gauge invariance and seamlessly extends to multiband systems, enabling accurate calculations of quantum geometry and nonlinear optical responses. We derive the projector formalism in Wannier function basis and implement the \textit{ab initio} calculations of shift current in GeS, demonstrating excellent agreement with the sum rule and Wilson loop approaches. This work extends projector-based representations within the Wannier functions basis, offering an efficient and reliable tool for investigating nonlinear light-matter interactions and quantum geometry in realistic materials.