Adiabatic driving and parallel transport for parameter-dependent Hamiltonians

TL;DR

This paper applies perturbation theory to study the parallel transport of eigenvectors in parameter-dependent Hamiltonians, linking the resulting non-Abelian connection to Berry phase via geometric and gauge-theoretic concepts.

Contribution

It introduces a perturbative method to define a non-Abelian connection for eigenvector transport, connecting it to the Maurer-Cartan form and Berry phase.

Findings

The non-Abelian connection is an average of the Maurer-Cartan form.

Holonomy of the connection relates to Berry phase.

Perturbative approach simplifies analysis of eigenvector transport.

Abstract

We use the Van Vleck-Primas perturbation theory to study the problem of parallel transport of the eigenvectors of a parameter-dependent Hamiltonian. The perturbative approach allows us to define a non-Abelian connection $\mathcal{A}$ that generates parallel translation via unitary transformation of the eigenvectors. It is shown that the connection obtained via the perturbative approach is an average of the Maurer-Cartan 1-form of the one-parameter subgroup generated by the Hamiltonian. We use the Yang-Mills curvature and the non-Abelian Stokes' theorem to show that the holonomy of the connection $\mathcal{A}$ is related to the Berry phase.