A Unified View on Geometric Phases and Exceptional Points in Adiabatic Quantum Mechanics

Abstract

We present a formal geometric framework for the study of adiabatic quantum mechanics for arbitrary finite-dimensional non-degenerate Hamiltonians. This framework generalizes earlier holonomy interpretations of the geometric phase to non-cyclic states appearing for non-Hermitian Hamiltonians. We start with an investigation of the space of non-degenerate operators on a finite-dimensional state space. We then show how the energy bands of a Hamiltonian family form a covering space. Likewise, we demonstrate that the eigenrays form a bundle, a generalization of a principal bundle, which admits a natural connection that yields the (generalized) geometric phase. This bundle also provides a natural generalization of the quantum geometric tensor and derived tensors, and we demonstrate how it can incorporate the non-geometric dynamical phase as well. We conclude by demonstrating how the bundle can be recast as a principal bundle, allowing both the geometric phases and the permutations of eigenstates to be expressed simultaneously using standard holonomy theory.