Quantum Geometric Fluctuation-Dissipation Relation for Nonlinear Transport

The fluctuation-dissipation theorem connects equilibrium noise to linear response and forms a cornerstone of statistical and quantum physics, yet its extension to geometry-driven nonlinear transport remains largely unexplored. Here we establish a geometric fluctuation-dissipation relation linking dc current noise at linear order to second-order nonlinear responses-specifically shift and injection photocurrents-in the bulk photovoltaic effect of noncentrosymmetric gapped quantum materials. Using a microscopic density-matrix formalism, we show that linear dc current noise in the dc electric field arising from off-diagonal current correlations is universally governed by frequency-integrated nonlinear optical responses and is encoded in the quantum geometry of Bloch states. We further demonstrate that intrinsic and extrinsic noise contributions exhibit distinct symmetry properties and relaxation-time dependencies, corresponding respectively to shift and injection photocurrents. We derive analytical expressions in a generic two-band model and numerically verify them in the Haldane model. Our results establish dc current noise at linear order as a direct probe of quantum geometry and nonlinear optical response in gapped quantum materials, extending fluctuation-dissipation relations well beyond linear equilibrium transport.