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

• URL: http://arxiv.org/abs/2107.02497v2
• Date: Thu, 13 Jan 2022 10:12:10 GMT
• Title: A Unified View on Geometric Phases and Exceptional Points in Adiabatic Quantum Mechanics
• Authors: Eric J. Pap, Dani\"el Boer and Holger Waalkens
• Abstract summary: 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.
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• License: http://creativecommons.org/licenses/by-sa/4.0/
• 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.

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