Emergent Curved space induced by adiabatic approximation

Berry curvature, serving as the imaginary part of quantum geometric tensor (QGT), gives rise to an effective Lorentz force to the dynamics of the adiabatic parameter. However, it is not clear whether the real part of QGT, the quantum metric, has any dynamical consequence as the Berry gauge force. We show in a general way that during an adiabatic process, the particle in a hybrid quantum system governed by an equation of motion second order in time is subject to an induced gravitational force. The adiabatic dynamics can be described by a geodesic equation as if the spacetime is curved by the quantum metric. As an example, we demonstrate the above result in a simple toy model.