Quantum Metric, Topology and Second Harmonic Generation

The quantum metric manifesting as the Riemannian metric in the parameter space of Bloch bands, characterizes the topology and geometry of quantum states. The second harmonic generation (SHG), as one of the fundamental nonlinear optical responses that links geometry of optical transitions to physical observables, despite being widely studied in various materials, its relation with quantum metric, especially in the dynamical regime, stays obscure. Here, we investigate the SHG within the Keldysh formalism and resolve the contributions from quantum metric. Using a Haldane model, we simulate the dynamic photocurrent, revealing a significant enhancement of SHG in the transparent region, i.e., for below-gap photon energies. Further, we show that such enhancement originates from the non-Hermitian nature of its complex band structure and quantum tunneling near the exceptional points. Such low-energy-photon SHG signals are highly sensitive to the topological phase transition, quantifying the quantum volume effect. Our work elucidates the physical origin of quantum metric contributed SHG and its relation with topology, providing an alternative route to probe the ultrafast topological phase transition in magnetic insulators.