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

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 ( ϵ t ) , x 2 ( ϵ t ) , … } , ϵ ≪ 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 ϵ . This adiabatic perturbation theory provides an efficient tool for calculating nonlinear response properties.