Covariant formulation of the Berry connection in non-Hermitian systems

Abstract

Non-Hermitian systems exhibit spectral and topological phenomena absent in Hermitian physics; however, their geometric characterization remains subtle due to the intrinsic ambiguity of biorthogonal eigenspaces. Since left and right eigenvectors are not related by Hermitian conjugation, the associated Berry connection is generally nonunique, leading to complex geometric phases and ambiguously defined holonomies. Here we formulate a covariant geometric framework for non-Hermitian quantum systems based on the metric structure of the underlying Hilbert space. We show that, in the quantum regime with continuous state evolution, the conventional Berry connection and the associated Berry holonomy over closed parameter-space loops can be consistently defined only in the pseudo-Hermitian limit, where the spectrum is real. For generic non-Hermitian Hamiltonians with complex spectra, the relevant geometric object is instead the Aharonov--Anandan holonomy associated with cyclic evolution in projective Hilbert space. Within the pseudo-Hermitian regime, we construct a unique Hermitian Berry connection that is covariant under arbitrary $\mathrm{GL}(N,\mathbb{C})$ frame transformations and reduces to the standard Berry connection in the Hermitian limit. The resulting formalism separates the intrinsic geometry of the Hamiltonian eigenspace from contributions arising from the parameter dependence of the Hilbert-space metric, revealing that the conventional biorthogonal formulation generally mixes these distinct geometric effects. Consequently, geometric phases, synthetic gauge fields, and topological characteristics commonly attributed to non-Hermitian eigenspace geometry may, in part, originate from the underlying metric structure. Our framework therefore provides a consistent geometric foundation for Berry phases, non-Abelian holonomies, and topological invariants in non-Hermitian quantum systems.

13 pages, 1 figure

References in corpus (19)

• Topological Origin of Non-Hermitian Skin Effects
• PT-Symmetric Phonon Laser
• Dynamically encircling exceptional points: Exact evolution and polarization state conversion
• Thouless pumping and topology
• Chiral Coherent Perfect Absorption on Exceptional Surfaces
• Non-Hermitian engineering of terahertz light using exceptional points in electrically tuneable collective light-matter interactions
• Non-Dissipative Non-Hermitian Dynamics and Exceptional Points in Coupled Optical Parametric Oscillators
• Flattening the Curve with Einstein's Quantum Elevator: Hermitization of Non-Hermitian Hamiltonians via a Generalized Vielbein Formalism
• Homotopy, Symmetry, and Non-Hermitian Band Topology
• Restoring Adiabatic State Transfer in Time-Modulated Non-Hermitian Systems
• Emergent parallel transport and curvature in Hermitian and non-Hermitian quantum mechanics
• Creating and controlling exceptional points of non-Hermitian Hamiltonians via homodyne Lindbladian invariance
• Time-dependent C-operators as Lewis-Riesenfeld invariants in non-Hermitian theories
• The Stochastic-Quantum Correspondence
• Geometric contribution to adiabatic amplification in non-Hermitian systems
• Achieving the Quantum Fisher Information Bound in Pseudo-Hermitian Sensors
• Emerging (2+1)D electrodynamics and topological instanton in pseudo-Hermitian two-level systems
• Quantum geometrical effects in non-Hermitian systems
• Observation of Restored Adiabatic State Transfer in Time-Modulated Non-Hermitian Systems