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

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 show that the eigenrays form a bundle, a generalization of a principal bundle, which admits a natural connection yielding the (generalized) geometric phase. This bundle provides in addition a natural generalization of the quantum geometric tensor and derived tensors, and we show how it can incorporate the non-geometric dynamical phase as well. We finish by demonstrating how the bundle can be recast as a principal bundle, so that both the geometric phases and the permutations of eigenstates can be expressed simultaneously by means of standard holonomy theory.

Automated summary

This article presents a formal geometric framework for studying adiabatic quantum mechanics involving non-degenerate Hamiltonians in arbitrary finite-dimensional spaces. By examining the space of non-degenerate operators and the energy bands of Hamiltonian families, as well as the eigenrays forming a bundle, a generalized geometric phase is obtained. This framework also incorporates the non-geometric dynamical phase and can be recast as a principal bundle to simultaneously handle geometric phases and permutations of eigenstates.