Extending the wave-packet ansatz for Bloch electrons to include interband contributions and applying the time-dependent variational principle yields leading-order nonadiabatic corrections to the Lagrangian, including an energy-gap-renormalized quantum metric that recasts dynamics as geodesic motion.
Luttinger, The effect of a magnetic field on electrons in a periodic potential, Phys
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Nonadiabatic Wave-Packet Dynamics: Nonadiabatic Metric, Quantum Geometry, and Gravitational Analogy
Extending the wave-packet ansatz for Bloch electrons to include interband contributions and applying the time-dependent variational principle yields leading-order nonadiabatic corrections to the Lagrangian, including an energy-gap-renormalized quantum metric that recasts dynamics as geodesic motion.