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arXiv · 2107.02497

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

Eric J. Pap·Daniël Boer·Holger Waalkens
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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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Eric J. Pap, Daniël Boer, Holger Waalkens. 2021-07-06. A Unified View on Geometric Phases and Exceptional Points in Adiabatic Quantum Mechanics. https://doi.org/10.3842/sigma.2022.003

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