Abstract
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
(
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.
Subject
General Physics and Astronomy,Mathematical Physics,Modeling and Simulation,Statistics and Probability,Statistical and Nonlinear Physics