Emergent-Gravity Hall Effect from quantum geometry

We theoretically propose a Hall effect driven by effective gravitational fields that emerge from quantum geometry. In our work, we systematically investigate the role of non-adiabatic processes in the semiclassical equation of motion of an electron. Beyond the well-known anomalous velocity term induced by the Berry curvature, we identify additional contributions proportional to the Christoffel symbols constructed from a weighted quantum metric. These symbols, analogous to those in general relativity and cosmology, give rise to gravity-like effects on the motion of electrons. Our formalism treats various variables—such as real-space and momentum-space coordinates, external parameters, and time—on an equal footing. The resulting Christoffel symbols lead to novel Hall effects in phase spaces, which we term the "emergent-gravity Hall effect." We demonstrate these effects through model calculations and clarify the conditions under which a finite Hall response can arise. Our findings open a new avenue for exploring gravitational analogs in quantum systems.