Geometric Phase for Non-Hermitian Hamiltonians and Its Holonomy Interpretation

Abstract

For an arbitrary possibly non-Hermitian matrix Hamiltonian H, that might involve exceptional points, we construct an appropriate parameter space M and the lines bundle L^n over M such that the adiabatic geometric phases associated with the eigenstates of the initial Hamiltonian coincide with the holonomies of L^n. We examine the case of 2 x 2 matrix Hamiltonians in detail and show that, contrary to claims made in some recent publications, geometric phases arising from encircling exceptional points are generally geometrical and not topological in nature.

22 pages, 1 figure, published version

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Cited by in corpus (47)

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• Time Dependent $\mathcal{P}\mathcal{T}$-Symmetric Quantum Mechanics
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• Topological Modes in a Laser Cavity through Exceptional State Transfer
• Resonance Phenomenon Related to Spectral Singularities, Complex Barrier Potential, and Resonating Waveguides
• Stochastic pump effect and geometric phases in dissipative and stochastic systems
• Topological quantum state control through exceptional-point proximity
• Quantum Geometric Tensor in $\mathcal{P}\mathcal{T}$-Symmetric Quantum Mechanics
• Geometric Phase in PT-Symmetric Quantum Mechanics
• Self-dual Spectral Singularities and Coherent Perfect Absorbing Lasers without PT-symmetry
• Transfer Matrices as Non-Unitary S-Matrices, Multimode Unidirectional Invisibility, and Perturbative Inverse Scattering
• Information Geometry of Complex Hamiltonians and Exceptional Points
• Floquet exceptional points and chirality in non-Hermitian Hamiltonians
• Partial topological Zak phase and dynamical confinement in non-Hermitian bipartite system
• Spectral Singularities of a General Point Interaction
• Scattering Experiments with Microwave Billiards at an Exceptional Point under Broken Time Reversal Invariance
• Non-abelian nature of systems with multiple exceptional points
• Application of Pseudo-Hermitian Quantum Mechanics to a Complex Scattering Potential with Point Interactions
• Non-Hermitian dynamics of slowly-varying Hamiltonians
• Non-Hermitian Hamiltonians with a Real Spectrum and Their Physical Applications
• An introduction to PT-symmetric quantum mechanics -- time-dependent systems
• Emergent parallel transport and curvature in Hermitian and non-Hermitian quantum mechanics
• A view on the open STIRAP problem
• Complex Berry phase and imperfect non-Hermitian phase transitions
• Non-Hermitian time-dependent perturbation theory: asymmetric transitions and transitionless interactions
• Adiabatic Approximation, Semiclassical Scattering, and Unidirectional Invisibility
• Discrete changes of current statistics in periodically driven stochastic systems
• Bloch vector, disclination and exotic quantum holonomy
• A Unified View on Geometric Phases and Exceptional Points in Adiabatic Quantum Mechanics
• Amplitude control of a quantum state in a non-Hermitian Rice-Mele model driven by an external field
• Consistency between adiabatic and nonadiabatic geometric phases for nonselfadjoint hamiltonians
• Almost quantum adiabatic dynamics and generalized time dependent wave operators
• Exceptional points and pseudo-Hermiticity in real potential scattering
• Non-hermitian time evolution: from static to parametric instability
• Adiabatic transport of qubits around a black hole
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• Topological adiabatic dynamics in classical mass-spring chains with clamps
• A mathematical formalism of non-Hermitian quantum mechanics and observable-geometric phases
• A unifying E2-quasi-exactly solvable model
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• Geometric criterion of topological phase transition for non-Hermitian systems