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:2409.02144 · Chinese Physics B

Note on Dirac monopole theory and Berry geometric phase

Li-Chen Zhao

Published Nov 14, 2025• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1674-1056/ae1f7f

Related papers

Anisotropic spin system, quantized Dirac monopole and the Berry phase

P. Bandyopadhyay

The Berry phase, acquired by an anisotropic spin system with spin J ≥1 when it is adiabatically rotated in a closed circuit, is considered to be associated with a non-quantized Dirac monopole, and has a geometrical as well as a topological component owing to the Dirac string. Here, it is argued that the Berry phase of a spin state with J ≥1 can be associated with a quantized Dirac monopole when the corresponding spin is taken to be an entangled state of a composite system of 1/2 spins. Evidently, this avoids the contribution of the Dirac string, and the Berry phase is purely geometrical in nature.

Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences•Aug 4, 2010•12 citations

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

Dirac monopole magnets in non-Hermitian systems

Hou–Yong Yu, Tao Jiang, Li-Chen Zhao

We theoretically establish that non-Hermitian perturbations induce a topological transformation of point-like Dirac monopoles into extended monopole distributions, characterized by distinct charge configurations emergent from three distinct Berry connection forms. Using piecewise adiabatic evolution, we confirm the validity of these configurations through observations of complex geometric phases. Most critically, we find a quantitative relation $Δϕ_d = Δϕ_g$, which quantifies how cumulative minute energy differences (\(Δϕ_d\)) manifest as geometric phase shifts (\(Δϕ_g\)) uniquely in non-Hermitian systems. We further propose a scheme leveraging soliton dynamics in dissipative two-component Bose-Einstein condensates, enabling direct measurement of these topological signatures. These results establish a milestone for understanding Dirac monopole charge distributions and measuring complex geometric phases in non-Hermitian systems, with far-reaching implications for topological quantum computing and non-Hermitian photonics.

arXiv:2508.15186•Aug 21, 2025•0 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

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

Geometric phase in PT -symmetric quantum mechanics

Jiangbin Gong, Qinghai Wang

Unitary evolution in $\mathcal{P}$$\mathcal{T}$-symmetric quantum mechanics (QM) with a time-dependent metric is found to yield an interesting class of adiabatic processes. As an explicit example, a Berry-like phase associated with a $\mathcal{P}$$\mathcal{T}$-symmetric two-level system is derived and is interpreted as the flux of a fictitious monopole with a tunable charge plus a singular string component with nontrivial phase contributions. To gain more insight, the Hermitian analog of our non-Hermitian problem is also analyzed, which results in an intriguing class of geometric-phase problems in conventional QM as well, where the Hamiltonian includes a perturbative term that is proportional to the rate of change in adiabatic parameters.

arXiv:1003.3076•Physical Review A•Jul 13, 2010•68 citations
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, acquired by an anisotropic spin system with spin J ≥1 when it is adiabatically rotated in a closed circuit, is considered to be associated with a non-quantized Dirac monopole, and has a geometrical as well as a topological component owing to the Dirac string. Here, it is argued that the Berry phase of a spin state with J ≥1 can be associated with a quantized Dirac monopole when the corresponding spin is taken to be an entangled state of a composite system of 1/2 spins. Evidently, this avoids the contribution of the Dirac string, and the Berry phase is purely geometrical in nature.
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.
We theoretically establish that non-Hermitian perturbations induce a topological transformation of point-like Dirac monopoles into extended monopole distributions, characterized by distinct charge configurations emergent from three distinct Berry connection forms. Using piecewise adiabatic evolution, we confirm the validity of these configurations through observations of complex geometric phases. Most critically, we find a quantitative relation Δϕd=ΔϕgΔϕ_d = Δϕ_gΔϕd​=Δϕg​, which quantifies how cumulative minute energy differences (ΔϕdΔϕ_dΔϕd​) manifest as geometric phase shifts (ΔϕgΔϕ_gΔϕg​) uniquely in non-Hermitian systems. We further propose a scheme leveraging soliton dynamics in dissipative two-component Bose-Einstein condensates, enabling direct measurement of these topological signatures. These results establish a milestone for understanding Dirac monopole charge distributions and measuring complex geometric phases in non-Hermitian systems, with far-reaching implications for topological quantum computing and non-Hermitian photonics.
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.
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.
Unitary evolution in P\mathcal{P}PT\mathcal{T}T-symmetric quantum mechanics (QM) with a time-dependent metric is found to yield an interesting class of adiabatic processes. As an explicit example, a Berry-like phase associated with a P\mathcal{P}PT\mathcal{T}T-symmetric two-level system is derived and is interpreted as the flux of a fictitious monopole with a tunable charge plus a singular string component with nontrivial phase contributions. To gain more insight, the Hermitian analog of our non-Hermitian problem is also analyzed, which results in an intriguing class of geometric-phase problems in conventional QM as well, where the Hamiltonian includes a perturbative term that is proportional to the rate of change in adiabatic parameters.