Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 003, 42 pages      arXiv:2107.02497      https://doi.org/10.3842/SIGMA.2022.003

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

Eric J. Pap ab, Daniël Boer b and Holger Waalkens a
a) Bernoulli Institute, University of Groningen, P.O. Box 407, 9700 AK Groningen, The Netherlands
b) Van Swinderen Institute, University of Groningen, 9747 AG Groningen, The Netherlands

Received July 23, 2021, in final form December 28, 2021; Published online January 13, 2022

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.

Key words: adiabatic quantum mechanics; geometric phase; exceptional point; quantum geometric tensor.

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