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

TL;DR

This paper develops a comprehensive geometric framework for understanding adiabatic quantum mechanics, unifying geometric phases and exceptional points for finite-dimensional non-Hermitian Hamiltonians.

Contribution

It introduces a generalized bundle structure that captures geometric phases, eigenstate permutations, and non-Hermitian effects in a unified geometric formalism.

Findings

Generalizes holonomy interpretation to non-cyclic, non-Hermitian states

Defines a bundle of eigenrays with a natural connection for geometric phase

Recasts the framework as a principal bundle for unified phase and permutation analysis

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 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.