Quantum covariant derivative: a tool for deriving adiabatic perturbation theory to all orders

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 $H=H(x)$ depending on slowly-varying parameters $x=\{x_1(\epsilon t),x_2(\epsilon t),\ldots\}$, $\epsilon\ll 1$, the quantum covariant derivative is used to derive a recurrence relation that determines an asymptotic series for the wave function to all orders in $\epsilon$. This adiabatic perturbation theory provides an efficient tool for calculating nonlinear response properties.