Nonadiabatic Wave-Packet Dynamics: Nonadiabatic Metric, Quantum Geometry, and Gravitational Analogy

We develop a unified theory for the nonadiabatic wave-packet dynamics of Bloch electrons subject to slowly varying spatial and temporal perturbations. Extending the conventional wave-packet ansatz to include interband contributions, we derive equations for the interband coefficients using the time-dependent variational principle, referred to as the wave-packet coefficient equation. Solving these equations and integrating out interband contributions yields the leading-order nonadiabatic corrections to the wave-packet Lagrangian. These corrections appear in three forms: (i) a nonadiabatic metric in real and momentum space, which we identify with the energy-gap-renormalized quantum metric, (ii) modified Berry connections associated with the motion of the wave-packet center, and (iii) an energy correction arising from spatial and temporal variations of the Hamiltonian. This metric reformulates the wave-packet dynamics as geodesic motion in phase space, enabling an analogue-gravity perspective in condensed matter systems. As an application, we analyze one-dimensional Dirac electron systems under a slowly varying exchange field $\bm{m}$. Our results demonstrate that variations in the magnitude of $\bm{m}$ are important to nonadiabatic dynamics, in sharp contrast to the adiabatic regime where directional variations of $\bm{m}$ are crucial.