Quantum Geometry as a Probe of Hidden Order in Quantum Materials

Electrons in crystals carry a quantum geometry, encoded in the Berry curvature, that shapes how they respond to applied fields. I will argue that this geometry can be used to probe order in materials, including symmetry breaking that conventional measurements miss. Second-order responses, such as the nonlinear Hall effect, detect broken inversion symmetry that linear transport cannot see. At KTaO₃ interfaces, where spin-orbit coupling is strong, we find large, reproducible fluctuations in this nonlinear response as the magnetic field is varied. They persist in devices hundreds of micrometers across and on every surface orientation studied, even where the linear conductivity looks perfectly symmetric. We attribute them to local polar distortions pinned to interfacial defects. The signal fades above about 10 K and its pattern is erased after warming to 40 K. Nonlinear transport thus becomes a new window on hidden structural order in quantum materials.