Abstract
The covariant derivative suitable for differentiating and parallel transporting tangent vectors and other geometric objects induced by a parameter-dependent adiabatic quantum eigenstate is introduced. It is proved to be covariant under gauge and coordinate transformations and compatible with the quantum geometric tensor. For a quantum system driven by a Hamiltonian depending on slowly-varying parameters , , the quantum covariant derivative is used to derive a recurrence relation that determines an asymptotic series for the wave function to all orders in . This adiabatic perturbation theory provides an efficient tool for calculating nonlinear response properties.
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