Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Mathematical Physics

On the uniqueness of the Berry connection

Arno Böhm, Luis J. Boya, Brian K. Kendrick

Published Jul 1, 1992• 13 citations
Publisher

Abstract

Arguments are presented that make the choice of the connection, which gives rise to the Berry phase not only natural, but unique, both in the Abelian and the non-Abelian cases. Invariance is invoked under unitary transformations of the probability amplitudes in quantum systems to force the connection to be invariant under the unitary group. Because the action is not free, the horizontal subspace chosen by the connection has to be invariant under the little group, and that makes it orthogonal to the fiber’s direction, yielding the conventional Berry connection. The argument works just as well for the non-Abelian case, where the fibers are orthonormal frames (Stiefel manifolds), and the state space a Grassmanian, with a transitive but not free unitary action.

DOI
10.1063/1.529571

Related papers

VECTOR COHERENT STATES AND NON-ABELIAN GAUGE STRUCTURES IN QUANTUM MECHANICS

G. Giavarini, E. Onofri

We set the general formalism for calculating Berry's phase in quantum systems with Hamiltonian belonging to the algebra of a semisimple Lie group of any rank in the framework of generalized coherent states. Within this approach the geometric properties of Berry's connection are also studied, both in the Abelian and non-Abelian cases. In particular we call attention to the non-Abelian case where we make use of a vectorial generalization of coherent states. In this respect a thorough and self-contained exposition of the formalism of vector coherent states is given. The specific examples of the groups SU(3) and Sp(2) are worked out in detail.

International Journal of Modern Physics A•Nov 20, 1990•8 citations

Certain properties of Berry's phases in supersymmetric quantum mechanics. II

K M Cheng, P. C. W. Fung

For pt.I see ibid., vol.25, p.1745 (1992). In the first paper the authors studied certain crucial mathematical properties of Abelian Berry's phases in two quantum systems with supersymmetrically related Hamiltonians. They follow up such investigation by extending their analysis to nonAbelian Berry's phases in this present study. They present the derivation of an explicit expression for the difference in the relevant nonAbelian Berry's connections. Also they have derived an expression for a connection one form, which contain the non-Abelian Berry's connection, and which is invariant in the two supersymmetrically related quantum systems. To illustrate their findings mentioned above, they take the example of a Hamiltonian expressible in spin quadrupole. In this example, with a particular choice of supersymmetric partner, they show that the two Berry's connection one forms of the two supersymmetrically related systems may be related by gauge transformation.

Journal of Physics A Mathematical and General•Aug 21, 1992•1 citation

Invariant Berry connections

Luc Vinet

Berry connections are shown to be invariant under the action of symmetry groups on parameter space. This observation allows one to use the theory of invariant Yang-Mills potentials to evaluate these connections without computing the instantaneous energy eigenstates. Hamiltonians belonging to the algebra of a Lie group are examined in this light. They provide a wide class of systems admitting nontrivial adiabatic holonomy. The special case of the generalized harmonic oscillator is analyzed and the SO(2,1) invariance of the associated Berry connection is exhibited.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Apr 15, 1988•37 citations

Invariance of the generalised Berry phase under unitary transformations: application to the time-dependent generalised harmonic oscillator

Donald H. Kobe

A generalisation of the Berry phase, which reduces to the usual Berry phase in the adiabatic and cyclic case is derived in a gauge-invariant way. This generalised Berry phase, unlike the Berry phase, is invariant under unitary transformations. A time-dependent generalised harmonic oscillator with a cross term in the Hamiltonian is taken as an example. The Hamiltonian of this system is not in general the energy. An energy, the time derivative of which is the power, is obtained from the equation of motion. When the system is quantised, the generalised Berry phase is zero, and remains invariant under unitary transformations. On the other hand, if the energy is chosen to be the Hamiltonian, a non-zero generalised Berry phase is obtained, but the power is given incorrectly. In this case the total phase, which includes the dynamical phase and the generalised Berry phase, is equal in the adiabatic limit to the correct total phase.

Journal of Physics A Mathematical and General•Oct 7, 1990•26 citations

On geometric interpretation of Berry's phase and the Aharonov-Bohm effect

Mikhail Orionovich Katanaev

A geometric interpretation of Berry’s phase, its Wilczek–Zee non-Abelian generalization, and Aharonov–Bohm effect is given in terms of connections on principal fiber bundles. It is shown that a principal fiber bundle in all cases can be trivial while the connection and its holonomy group are nontrivial. Therefore the main role is played not by topological effects but geometrical ones. The Berry phase is also shown not to be connected to the adiabatic changes of the Hamiltonian parameters. 1

•Sep 2, 2009•0 citations

Covariant formulation of the Berry connection in non-Hermitian systems

Ievgen I. Arkhipov

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 ${\rm GL}(N,\mathbb C)$ 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.

quant-pharXiv:2601.19777•Physical Review Research•Jan 28, 2026•0 citations
Arguments are presented that make the choice of the connection, which gives rise to the Berry phase not only natural, but unique, both in the Abelian and the non-Abelian cases. Invariance is invoked under unitary transformations of the probability amplitudes in quantum systems to force the connection to be invariant under the unitary group. Because the action is not free, the horizontal subspace chosen by the connection has to be invariant under the little group, and that makes it orthogonal to the fiber’s direction, yielding the conventional Berry connection. The argument works just as well for the non-Abelian case, where the fibers are orthonormal frames (Stiefel manifolds), and the state space a Grassmanian, with a transitive but not free unitary action.
We set the general formalism for calculating Berry's phase in quantum systems with Hamiltonian belonging to the algebra of a semisimple Lie group of any rank in the framework of generalized coherent states. Within this approach the geometric properties of Berry's connection are also studied, both in the Abelian and non-Abelian cases. In particular we call attention to the non-Abelian case where we make use of a vectorial generalization of coherent states. In this respect a thorough and self-contained exposition of the formalism of vector coherent states is given. The specific examples of the groups SU(3) and Sp(2) are worked out in detail.
For pt.I see ibid., vol.25, p.1745 (1992). In the first paper the authors studied certain crucial mathematical properties of Abelian Berry's phases in two quantum systems with supersymmetrically related Hamiltonians. They follow up such investigation by extending their analysis to nonAbelian Berry's phases in this present study. They present the derivation of an explicit expression for the difference in the relevant nonAbelian Berry's connections. Also they have derived an expression for a connection one form, which contain the non-Abelian Berry's connection, and which is invariant in the two supersymmetrically related quantum systems. To illustrate their findings mentioned above, they take the example of a Hamiltonian expressible in spin quadrupole. In this example, with a particular choice of supersymmetric partner, they show that the two Berry's connection one forms of the two supersymmetrically related systems may be related by gauge transformation.
Berry connections are shown to be invariant under the action of symmetry groups on parameter space. This observation allows one to use the theory of invariant Yang-Mills potentials to evaluate these connections without computing the instantaneous energy eigenstates. Hamiltonians belonging to the algebra of a Lie group are examined in this light. They provide a wide class of systems admitting nontrivial adiabatic holonomy. The special case of the generalized harmonic oscillator is analyzed and the SO(2,1) invariance of the associated Berry connection is exhibited.
A generalisation of the Berry phase, which reduces to the usual Berry phase in the adiabatic and cyclic case is derived in a gauge-invariant way. This generalised Berry phase, unlike the Berry phase, is invariant under unitary transformations. A time-dependent generalised harmonic oscillator with a cross term in the Hamiltonian is taken as an example. The Hamiltonian of this system is not in general the energy. An energy, the time derivative of which is the power, is obtained from the equation of motion. When the system is quantised, the generalised Berry phase is zero, and remains invariant under unitary transformations. On the other hand, if the energy is chosen to be the Hamiltonian, a non-zero generalised Berry phase is obtained, but the power is given incorrectly. In this case the total phase, which includes the dynamical phase and the generalised Berry phase, is equal in the adiabatic limit to the correct total phase.
A geometric interpretation of Berry’s phase, its Wilczek–Zee non-Abelian generalization, and Aharonov–Bohm effect is given in terms of connections on principal fiber bundles. It is shown that a principal fiber bundle in all cases can be trivial while the connection and its holonomy group are nontrivial. Therefore the main role is played not by topological effects but geometrical ones. The Berry phase is also shown not to be connected to the adiabatic changes of the Hamiltonian parameters. 1
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 GL(N,C){\rm GL}(N,\mathbb C)GL(N,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.