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:2508.15186

Dirac monopole magnets in non-Hermitian systems

Hou–Yong Yu, Tao Jiang, Li-Chen Zhao

Published Aug 21, 2025• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.2508.15186

Related papers

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

Spin-1 topological monopoles in the parameter space of ultracold atoms

Haiping Hu, Chuanwei Zhang

A magnetic monopole is a hypothetical elementary particle with an isolated magnetic pole. Its existence would directly lead to the quantization of electric charges. In recent years, analogs of magnetic monopoles, represented by topological defects in parameter spaces, have been studied in a wide range of physical systems. These works mainly focused on Abelian Dirac monopoles in spin-1/2 or non-Abelian Yang monopoles in spin-3/2 systems. Here we propose to realize three types of spin-1 topological monopoles and study their geometric properties using the parameter space formed by three hyperfine states of ultracold atoms coupled by radio-frequency fields. These spin-1 monopoles, characterized by different monopole charges, possess distinct Berry curvature fields and spin textures, which are directly measurable in experiments. The topological phase transitions between different monopoles are accompanied by the emergence of a spin ``vortex'' and can be intuitively visualized using Majorana's stellar representation. We show how to determine the Berry curvature, hence the geometric phase and monopole charge from dynamical effects. Our scheme provides a simple and highly tunable platform for observing and manipulating spin-1 topological monopoles, paving the way for exploring new topological quantum matter.

arXiv:1802.08222•Physical Review A•Jul 26, 2018•21 citations

Complex magnetic monopoles, geometric phases and quantum evolution in the vicinity of diabolic and exceptional points

Alexander I Nesterov, F Aceves de la Cruz

We consider the geometric phase and quantum tunneling in the vicinity of diabolic and exceptional points. We show that the geometric phase associated with the degeneracy points is defined by the flux of complex magnetic monopoles. In the limit of weak coupling, the leading contribution to the real part of the geometric phase is given by the flux of the Dirac monopole plus a quadrupole term, and the expansion of the imaginary part starts with a dipole-like field. For a two-level system governed by a generic non-Hermitian Hamiltonian, we derive a formula to compute the non-adiabatic, complex, geometric phase by integrating over the complex Bloch sphere. We apply our results to study a dissipative two-level system driven by a periodic electromagnetic field and show that, in the vicinity of the exceptional point, the complex geometric phase behaves like a step-function. Studying the tunneling process near and at the exceptional point, we find two different regimes: coherent and incoherent. The coherent regime is characterized by Rabi oscillations, with a one-sheeted hyperbolic monopole emerging in this region of the parameters. The two-sheeted hyperbolic monopole is associated with the incoherent regime. We show that the dissipation results in a series of pulses in the complex geometric phase which disappear when the dissipation dies out. Such a strong coupling effect of the environment is beyond the conventional adiabatic treatment of the Berry phase.

arXiv:0806.3720•Journal of Physics A Mathematical and Theoretical•Oct 22, 2008•34 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

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

Measuring the Adiabatic Non-Hermitian Berry Phase in Feedback-Coupled Oscillators

Yaashnaa Singhal, Enrico Martello, Agrawal, Shraddha +3 more

The geometrical Berry phase is key to understanding the behaviour of quantum states under cyclic adiabatic evolution. When generalised to non-Hermitian systems with gain and loss, the Berry phase can become complex, and should modify not only the phase but also the amplitude of the state. Here, we perform the first experimental measurements of the adiabatic non-Hermitian Berry phase, exploring a minimal two-site $\mathcal{PT}$-symmetric Hamiltonian that is inspired by the Hatano-Nelson model. We realise this non-Hermitian model experimentally by mapping its dynamics to that of a pair of classical oscillators coupled by real-time measurement-based feedback. As we verify experimentally, the adiabatic non-Hermitian Berry phase is a purely geometrical effect that leads to significant amplification and damping of the amplitude also for non-cyclical paths within the parameter space even when all eigenenergies are real. We further observe a non-Hermitian analog of the Aharonov--Bohm solenoid effect, observing amplification and attenuation when encircling a region of broken $\mathcal{PT}$ symmetry that serves as a source of imaginary flux. This experiment demonstrates the importance of geometrical effects that are unique to non-Hermitian systems and paves the way towards the further studies of non-Hermitian and topological physics in synthetic metamaterials.

arXiv:2205.02700•May 5, 2022•1 citation
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 ((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.
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.
A magnetic monopole is a hypothetical elementary particle with an isolated magnetic pole. Its existence would directly lead to the quantization of electric charges. In recent years, analogs of magnetic monopoles, represented by topological defects in parameter spaces, have been studied in a wide range of physical systems. These works mainly focused on Abelian Dirac monopoles in spin-1/2 or non-Abelian Yang monopoles in spin-3/2 systems. Here we propose to realize three types of spin-1 topological monopoles and study their geometric properties using the parameter space formed by three hyperfine states of ultracold atoms coupled by radio-frequency fields. These spin-1 monopoles, characterized by different monopole charges, possess distinct Berry curvature fields and spin textures, which are directly measurable in experiments. The topological phase transitions between different monopoles are accompanied by the emergence of a spin ``vortex'' and can be intuitively visualized using Majorana's stellar representation. We show how to determine the Berry curvature, hence the geometric phase and monopole charge from dynamical effects. Our scheme provides a simple and highly tunable platform for observing and manipulating spin-1 topological monopoles, paving the way for exploring new topological quantum matter.
We consider the geometric phase and quantum tunneling in the vicinity of diabolic and exceptional points. We show that the geometric phase associated with the degeneracy points is defined by the flux of complex magnetic monopoles. In the limit of weak coupling, the leading contribution to the real part of the geometric phase is given by the flux of the Dirac monopole plus a quadrupole term, and the expansion of the imaginary part starts with a dipole-like field. For a two-level system governed by a generic non-Hermitian Hamiltonian, we derive a formula to compute the non-adiabatic, complex, geometric phase by integrating over the complex Bloch sphere. We apply our results to study a dissipative two-level system driven by a periodic electromagnetic field and show that, in the vicinity of the exceptional point, the complex geometric phase behaves like a step-function. Studying the tunneling process near and at the exceptional point, we find two different regimes: coherent and incoherent. The coherent regime is characterized by Rabi oscillations, with a one-sheeted hyperbolic monopole emerging in this region of the parameters. The two-sheeted hyperbolic monopole is associated with the incoherent regime. We show that the dissipation results in a series of pulses in the complex geometric phase which disappear when the dissipation dies out. Such a strong coupling effect of the environment is beyond the conventional adiabatic treatment of the Berry phase.
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.
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.
The geometrical Berry phase is key to understanding the behaviour of quantum states under cyclic adiabatic evolution. When generalised to non-Hermitian systems with gain and loss, the Berry phase can become complex, and should modify not only the phase but also the amplitude of the state. Here, we perform the first experimental measurements of the adiabatic non-Hermitian Berry phase, exploring a minimal two-site PT\mathcal{PT}PT-symmetric Hamiltonian that is inspired by the Hatano-Nelson model. We realise this non-Hermitian model experimentally by mapping its dynamics to that of a pair of classical oscillators coupled by real-time measurement-based feedback. As we verify experimentally, the adiabatic non-Hermitian Berry phase is a purely geometrical effect that leads to significant amplification and damping of the amplitude also for non-cyclical paths within the parameter space even when all eigenenergies are real. We further observe a non-Hermitian analog of the Aharonov--Bohm solenoid effect, observing amplification and attenuation when encircling a region of broken PT\mathcal{PT}PT symmetry that serves as a source of imaginary flux. This experiment demonstrates the importance of geometrical effects that are unique to non-Hermitian systems and paves the way towards the further studies of non-Hermitian and topological physics in synthetic metamaterials.