Adiabatic phases and group theory

Abstract

The construction of the adiabatic connection is studied in the case where the symmetry of a Hamiltonian is broken explicitly by a slowly varying perturbation. The type time variation of the perturbation corresponds to the one generated by the symmetry group of the unperturbed Hamiltonian. It is proven that the adiabatic connection for this type of system is completely determined by the group structure, up to a set of reduced matrix elements: systems with the same symmetries will have adiabatic connections differing at most in these reduced matrix elements. Several examples are detailed.