Quantum covariant derivative

TL;DR

This paper introduces a quantum covariant derivative that is gauge- and coordinate-invariant, enabling more efficient calculations of nonlinear adiabatic responses in quantum systems.

Contribution

It presents the first formulation of a quantum covariant derivative compatible with quantum geometric structures and uses it to develop an invariant adiabatic perturbation theory.

Findings

The quantum covariant derivative is covariant under gauge and coordinate transformations.

It is compatible with the quantum geometric tensor.

The new perturbation theory simplifies calculations of nonlinear adiabatic responses.

Abstract

The covariant derivative capable of differentiating and parallel transporting tangent vectors and other geometric objects induced by a parameter-dependent quantum state is introduced. It is proved to be covariant under gauge and coordinate transformations and compatible with the quantum geometric tensor. The quantum covariant derivative is used to derive a gauge- and coordinate-invariant adiabatic perturbation theory, providing an efficient tool for calculations of nonlinear adiabatic response properties.