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

arXiv:2101.10541 · Physical review. D/Physical review. D.

Lensing of Dirac monopole in Berry’s phase

Kazuo Fujikawa, Koichiro Umetsu

Published May 7, 2021• 2 citations
arXivHTMLPDF

Abstract

Berry's phase, which is associated with the slow cyclic motion with a finite period, looks like a Dirac monopole when seen from far away but smoothly changes to a dipole near the level crossing point in the parameter space in an exactly solvable model. This topology change of Berry's phase is visualized as a result of the lensing effect; the monopole supposed to be located at the level crossing point appears at the displaced point when the variables of the model deviate from the precisely adiabatic movement. The effective magnetic field generated by Berry's phase is determined by a simple geometrical consideration of the magnetic flux coming from the displaced Dirac monopole.

DOI
10.1103/physrevd.103.096004

Related papers

Topology change from a monopole to a dipole in Berry’s phase

Shinichi Deguchi, Kazuo Fujikawa

The smooth topology change of Berry's phase from a Dirac monopolelike configuration to a dipole configuration, when one approaches the monopole position in the parameter space, is analyzed in an exactly solvable model. A novel aspect of Berry's connection ${\mathcal{A}}_{k}$ is that the geometrical center of the monopolelike configuration and the origin of the Dirac string are displaced in the parameter space. Gauss's theorem ${\ensuremath{\int}}_{S}(\ensuremath{\nabla}\ifmmode\times\else\texttimes\fi{}\mathcal{A})\ifmmode\cdot\else\textperiodcentered\fi{}d\stackrel{\ensuremath{\rightarrow}}{S}={\ensuremath{\int}}_{V}\ensuremath{\nabla}\ifmmode\cdot\else\textperiodcentered\fi{}(\ensuremath{\nabla}\ifmmode\times\else\texttimes\fi{}\mathcal{A})dV=0$ for a volume $V$ which is free of singularities shows that a combination of the monopolelike configuration and the Dirac string is effectively a dipole. The smooth topology change from a dipole to a monopole with a quantized magnetic charge ${e}_{M}=2\ensuremath{\pi}\ensuremath{\hbar}$ takes place when one regards the Dirac string as unobservable if it satisfies the Wu-Yang gauge invariance condition. In the transitional region from a dipole to a monopole, a half-monopole appears with an observable Dirac string, which is analogous to the Aharonov-Bohm phase of an electron for the magnetic flux generated by the Cooper pair condensation. The main topological features of an exactly solvable model are shown to be supported by a generic model of Berry's phase.

arXiv:1905.00698•Physical review. D/Physical review. D.•Jul 9, 2019•5 citations

Magnetic monopoles in quantum adiabatic dynamics and the immersion property of the control manifold

David Viennot

It is well known that the Berry phase of a cyclic adiabatic dynamical system appears formally as the flux of a magnetic field in the control parameter manifold. In this electromagnetic picture a level crossing appears as a Dirac magnetic monopole in this manifold. We make an extensive study of the magnetic monopole model of eigenvalue crossings. We show that the properties of the monopole magnetic field in the control manifold are determined by the immersion of the control manifold in a space given by the universal classifying theorem of fiber bundles. We give a detailed illustrative study of the simple but instructive case of a two level crossing of a system controlled by a two-dimensional manifold.

Journal of Mathematical Physics•Sep 1, 2006•5 citations

Resolution of monopole singularity in Berry's phase

Shinichi Deguchi, Kazuo Fujikawa

A monopole-like singularity generated by Berry's phase in momentum space and associated anomalous Poisson brackets have been discussed in various fields. In the applications, it is crucially important to understand precisely what is the monopole in momentum space implied by Berry's phase. Berry's phase is sensitive to how it is measured and thus very different from the genuine Dirac monopole; the monopole-like singularity in Berry's phase is a result of adiabatic approximation while the singularity disappears in non-adiabatic approximation. The resolution of monopole-like singularity in the non-adiabatic domain implies that no Dirac string appears close to the monopole position and the anomalous terms containing the monopole flux in quantum mechanical commutation relations are absent. The absence of modified commutators is also concluded if one understands Berry's phase in momentum space as an $O(\hbar)$ quantum mechanical effect with a quantized magnetic charge $e_{M}=2\pi\hbar$.

•Feb 27, 2019•1 citation

Note on Dirac monopole theory and Berry geometric phase

Li-Chen Zhao

Abstract This work revisit on the intrinsic connection between Dirac monopole theory and Berry geometric phases by extending Dirac's theory to the parameter space. Using the simplest two-mode Hamiltonian model, we explicitly visualize Dirac strings with endpoints in the parameter space, demonstrating that these endpoints correspond to accidental degeneracy points of energy eigenvalues in Hermitian systems. We show that non-integrable phase factors, induced by such Dirac strings, directly give rise to the well-known Berry connection and curvature, which can be re-derived by following the spirit of Dirac's monopole theory with the aid of the quantum adiabatic theorem. Our results indicate that the Berry geometric phase is essentially the non-integrable phase factor induced by Dirac strings with endpoints in the parameter space. These results further deepen our understanding on monopole and geometric phases, and would inspire more studies on them especially when Berry's framework is not applicable.

arXiv:2409.02144•Chinese Physics B•Nov 14, 2025•1 citation

Nonquantized Dirac Monopoles and Strings in the Berry Phase of Anisotropic Spin Systems

P. Bruno

The Berry phase of an anisotropic spin system that is adiabatically rotated along a closed circuit C is investigated. It is shown that the Berry phase consists of two contributions: (i) a geometric contribution which can be interpreted as the flux through C of a nonquantized Dirac monopole, and (ii) a topological contribution which can be interpreted as the flux through C of a Dirac string carrying a nonquantized flux, i.e., a spin analogue of the Aharonov-Bohm effect. Various experimental consequences of this novel effect are discussed.

arXiv:cond-mat/0404616•Physical Review Letters•Dec 9, 2004•25 citations

Aharonov-Bohm Effect and Geometric Phases — Exact and Approximate Topology

Kazuo Fujikawa

By analyzing an exactly solvable model in the second quantized formulation which allows a unified treatment of adiabatic and non-adiabatic geometric phases, it is shown that the topology of the adiabatic Berry's phase, which is characterized by the singularity associated with possible level crossing, is trivial in a precise sense. This topology of the geometric phase is quite different from the topology of the Aharonov-Bohm effect, where the topology is specified by the external local gauge field and it is exact for the slow as well as for the fast motion of the electron.

arXiv:1302.0456•WORLD SCIENTIFIC eBooks•Apr 1, 2014•0 citations
Berry's phase, which is associated with the slow cyclic motion with a finite period, looks like a Dirac monopole when seen from far away but smoothly changes to a dipole near the level crossing point in the parameter space in an exactly solvable model. This topology change of Berry's phase is visualized as a result of the lensing effect; the monopole supposed to be located at the level crossing point appears at the displaced point when the variables of the model deviate from the precisely adiabatic movement. The effective magnetic field generated by Berry's phase is determined by a simple geometrical consideration of the magnetic flux coming from the displaced Dirac monopole.
The smooth topology change of Berry's phase from a Dirac monopolelike configuration to a dipole configuration, when one approaches the monopole position in the parameter space, is analyzed in an exactly solvable model. A novel aspect of Berry's connection Ak{\mathcal{A}}_{k}Ak​ is that the geometrical center of the monopolelike configuration and the origin of the Dirac string are displaced in the parameter space. Gauss's theorem ${\ensuremath{\int}}_{S}(\ensuremath{\nabla}\ifmmode\times\else\texttimes\fi{}\mathcal{A})\ifmmode\cdot\else\textperiodcentered\fi{}d\stackrel{\ensuremath{\rightarrow}}{S}={\ensuremath{\int}}_{V}\ensuremath{\nabla}\ifmmode\cdot\else\textperiodcentered\fi{}(\ensuremath{\nabla}\ifmmode\times\else\texttimes\fi{}\mathcal{A})dV=0$ for a volume VVV which is free of singularities shows that a combination of the monopolelike configuration and the Dirac string is effectively a dipole. The smooth topology change from a dipole to a monopole with a quantized magnetic charge ${e}_{M}=2\ensuremath{\pi}\ensuremath{\hbar}$ takes place when one regards the Dirac string as unobservable if it satisfies the Wu-Yang gauge invariance condition. In the transitional region from a dipole to a monopole, a half-monopole appears with an observable Dirac string, which is analogous to the Aharonov-Bohm phase of an electron for the magnetic flux generated by the Cooper pair condensation. The main topological features of an exactly solvable model are shown to be supported by a generic model of Berry's phase.
It is well known that the Berry phase of a cyclic adiabatic dynamical system appears formally as the flux of a magnetic field in the control parameter manifold. In this electromagnetic picture a level crossing appears as a Dirac magnetic monopole in this manifold. We make an extensive study of the magnetic monopole model of eigenvalue crossings. We show that the properties of the monopole magnetic field in the control manifold are determined by the immersion of the control manifold in a space given by the universal classifying theorem of fiber bundles. We give a detailed illustrative study of the simple but instructive case of a two level crossing of a system controlled by a two-dimensional manifold.
A monopole-like singularity generated by Berry's phase in momentum space and associated anomalous Poisson brackets have been discussed in various fields. In the applications, it is crucially important to understand precisely what is the monopole in momentum space implied by Berry's phase. Berry's phase is sensitive to how it is measured and thus very different from the genuine Dirac monopole; the monopole-like singularity in Berry's phase is a result of adiabatic approximation while the singularity disappears in non-adiabatic approximation. The resolution of monopole-like singularity in the non-adiabatic domain implies that no Dirac string appears close to the monopole position and the anomalous terms containing the monopole flux in quantum mechanical commutation relations are absent. The absence of modified commutators is also concluded if one understands Berry's phase in momentum space as an O(ℏ)O(\hbar)O(ℏ) quantum mechanical effect with a quantized magnetic charge eM=2πℏe_{M}=2\pi\hbareM​=2πℏ.
Abstract This work revisit on the intrinsic connection between Dirac monopole theory and Berry geometric phases by extending Dirac's theory to the parameter space. Using the simplest two-mode Hamiltonian model, we explicitly visualize Dirac strings with endpoints in the parameter space, demonstrating that these endpoints correspond to accidental degeneracy points of energy eigenvalues in Hermitian systems. We show that non-integrable phase factors, induced by such Dirac strings, directly give rise to the well-known Berry connection and curvature, which can be re-derived by following the spirit of Dirac's monopole theory with the aid of the quantum adiabatic theorem. Our results indicate that the Berry geometric phase is essentially the non-integrable phase factor induced by Dirac strings with endpoints in the parameter space. These results further deepen our understanding on monopole and geometric phases, and would inspire more studies on them especially when Berry's framework is not applicable.
The Berry phase of an anisotropic spin system that is adiabatically rotated along a closed circuit C is investigated. It is shown that the Berry phase consists of two contributions: (i) a geometric contribution which can be interpreted as the flux through C of a nonquantized Dirac monopole, and (ii) a topological contribution which can be interpreted as the flux through C of a Dirac string carrying a nonquantized flux, i.e., a spin analogue of the Aharonov-Bohm effect. Various experimental consequences of this novel effect are discussed.
By analyzing an exactly solvable model in the second quantized formulation which allows a unified treatment of adiabatic and non-adiabatic geometric phases, it is shown that the topology of the adiabatic Berry's phase, which is characterized by the singularity associated with possible level crossing, is trivial in a precise sense. This topology of the geometric phase is quite different from the topology of the Aharonov-Bohm effect, where the topology is specified by the external local gauge field and it is exact for the slow as well as for the fast motion of the electron.