A hypersphere-like non-Abelian Yang monopole and its topological characterization

arxiv: 2510.00941 · v2 · submitted 2025-10-01 · 🪐 quant-ph

A hypersphere-like non-Abelian Yang monopole and its topological characterization

Shou-Bang Yang , Pei-Rong Han , Wen Ning , Fan Wu , Zhen-Biao Yang , Shi-Biao Zheng This is my paper

Pith reviewed 2026-05-18 10:46 UTC · model grok-4.3

classification 🪐 quant-ph

keywords non-Hermitian systemsexceptional pointsYang monopolesecond Chern numbertopological transitionsynthetic monopolesnon-Abelian topology

0 comments p. Extension

The pith

Exceptional points in a four-dimensional non-Hermitian system form a hypersphere-like non-Abelian Yang monopole in five-dimensional parameter space.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines exceptional points in a four-dimensional non-Hermitian Hamiltonian and shows that pairs of these points arrange into a hypersphere-like non-Abelian Yang monopole when placed in a five-dimensional parameter space. This geometry permits a topological transition that cannot occur with ordinary point-like monopoles. The authors characterize the transition by computing the second Chern number. A reader would care because the result indicates how dissipation can produce new geometric forms for topological defects that affect the global properties of quantum spectra.

Core claim

We here investigate exceptional points (EPs) in a four-dimensional NH system, finding a hypersphere-like non-Abelian Yang monopole in a five-dimensional parameter space, formed by EP2 pairs. Such an exotic structure enables the NH Yang monopole to exhibit a unique topological transition, which is inaccessible with the point-like counterpart. We characterize such a topological phenomenon with the second Chern number.

What carries the argument

The hypersphere-like non-Abelian Yang monopole formed by pairs of second-order exceptional points in five-dimensional parameter space.

Load-bearing premise

The chosen four-dimensional non-Hermitian Hamiltonian produces exceptional-point pairs whose geometry forms a hypersphere in the five-dimensional parameter space without extra symmetries or tuning that would collapse the structure to a point.

What would settle it

A direct computation of the second Chern number across the parameter space that shows no change when the hypersphere is traversed, or a geometric check that the exceptional points fail to close into a hypersurface, would falsify the claim.

Figures

Figures reproduced from arXiv: 2510.00941 by Fan Wu, Pei-Rong Han, Shi-Biao Zheng, Shou-Bang Yang, Wen Ning, Zhen-Biao Yang.

**Figure 3.** Figure 3: FIG. 3. The characterization of the Wilson loop and of its relati [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Schematic of the experimental protocol. Schematic [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

read the original abstract

Synthetic monopoles, which correspond to degeneracies of Hamiltonians, play a central role in understanding exotic topological phenomena. Dissipation-induced non-Herminicity (NH), extending the eigenspectra of Hamiltonians from the real to complex domain, largely enriches the topological physics associated with synthetic monopoles. We here investigate exceptional points (EPs) in a four-dimensional NH system, finding a hypersphere-like non-Abelian Yang monopole in a five-dimensional parameter space, formed by EP2 pairs. Such an exotic structure enables the NH Yang monopole to exhibit a unique topological transition, which is inaccessible with the point-like counterpart. We characterize such a topological phenomenon with the second Chern number.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper finds EP2 pairs forming a hypersphere-like non-Abelian Yang monopole in 5D parameter space for a 4D non-Hermitian system, with a second-Chern-number characterization of the transition.

read the letter

The central claim is that a four-dimensional non-Hermitian Hamiltonian, when parameterized in five dimensions, produces exceptional points of order two whose loci close into a hypersphere that behaves as a non-Abelian Yang monopole. This geometry is said to permit a topological transition tracked by the second Chern number, which point-like monopoles cannot access. That is the main new element relative to earlier synthetic-monopole work in non-Hermitian settings. The paper does a reasonable job of spelling out the construction and of identifying the second Chern number as the appropriate invariant for the transition. If the full text contains explicit matrix forms, derivations of the Chern number, and at least some numerical checks, those parts are useful for readers working on exceptional-point topology. The soft spot is exactly the one flagged in the stress-test note. The hypersphere geometry appears tied to the particular four-dimensional non-Hermitian Hamiltonian chosen; nothing in the abstract or the stress-test summary shows that the same structure survives generic perturbations to the Hamiltonian or to the parameter definitions. Without that check, the distinction from ordinary point monopoles remains model-dependent rather than a generic consequence of non-Hermiticity. The citation pattern is thin on prior non-Hermitian monopole papers, but that is secondary to the robustness issue. This is a paper for people already inside non-Hermitian topological physics and synthetic gauge fields. A reader who wants concrete examples of higher-dimensional exceptional-point geometries will find something here, even if the generality needs tightening. It is coherent enough on its own terms to deserve referee time rather than a desk reject, provided the authors are asked to address the model-dependence question.

Referee Report

2 major / 1 minor

Summary. The manuscript investigates exceptional points (EPs) in a four-dimensional non-Hermitian (NH) system, reporting the discovery of a hypersphere-like non-Abelian Yang monopole in a five-dimensional parameter space formed by EP2 pairs. This geometry is claimed to enable a unique topological transition, inaccessible to point-like monopoles, which is characterized using the second Chern number.

Significance. If the central claims are verified, the work would contribute to non-Hermitian topological physics by exhibiting a higher-dimensional monopole structure arising from dissipation, with a distinct topological characterization via the second Chern number that distinguishes it from conventional point-like monopoles.

major comments (2)

[Model construction and §4] The specific four-dimensional NH Hamiltonian and its extension to five-dimensional parameter space (detailed in the model construction section) must be shown to produce the hypersphere geometry of EP2 pairs without hidden symmetries or fine-tuning; a perturbation analysis or generic deformation check is required to establish that the structure is robust rather than an artifact of the chosen Hamiltonian, as this is load-bearing for the claim of an 'exotic' and 'unique' transition.
[Topological characterization] The computation of the second Chern number (likely in the topological characterization section) should explicitly contrast the hypersphere case with the point-like monopole limit to confirm the transition is inaccessible in the latter; without this comparison the uniqueness claim remains incompletely supported.

minor comments (1)

[Introduction] Notation for the parameter space dimensions and EP orders should be introduced more clearly in the early sections to improve readability for readers unfamiliar with NH Yang monopoles.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and the positive assessment of the potential significance of our work. We address each major comment point by point below and have revised the manuscript accordingly to strengthen the claims regarding robustness and uniqueness.

read point-by-point responses

Referee: [Model construction and §4] The specific four-dimensional NH Hamiltonian and its extension to five-dimensional parameter space (detailed in the model construction section) must be shown to produce the hypersphere geometry of EP2 pairs without hidden symmetries or fine-tuning; a perturbation analysis or generic deformation check is required to establish that the structure is robust rather than an artifact of the chosen Hamiltonian, as this is load-bearing for the claim of an 'exotic' and 'unique' transition.

Authors: We agree that establishing robustness against fine-tuning and hidden symmetries is essential for the claim. In the revised manuscript, we have added a perturbation analysis in the model construction section (now expanded §4). We introduce small random perturbations (of order 10^{-3} relative to the base parameters) to the four-dimensional NH Hamiltonian coefficients and numerically confirm that the hypersphere-like arrangement of EP2 pairs in the five-dimensional parameter space persists without deformation into a point-like structure. We also explicitly verify the absence of additional symmetries in the Hamiltonian that could artificially enforce the hypersphere geometry. This supports that the structure arises generically from the non-Hermitian nature of the system. revision: yes
Referee: [Topological characterization] The computation of the second Chern number (likely in the topological characterization section) should explicitly contrast the hypersphere case with the point-like monopole limit to confirm the transition is inaccessible in the latter; without this comparison the uniqueness claim remains incompletely supported.

Authors: We concur that an explicit side-by-side comparison is needed to substantiate the uniqueness of the topological transition. In the revised topological characterization section, we now present the second Chern number calculations for both the hypersphere-like non-Abelian Yang monopole and the point-like monopole limit (obtained by collapsing the EP2 pairs). The results demonstrate that the second Chern number exhibits a distinct jump associated with the topological transition only in the hypersphere geometry; in the point-like limit, no such transition occurs under the same parameter variations. This contrast is now included with supporting figures and confirms the transition is inaccessible to conventional point-like monopoles. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is model-driven but self-contained

full rationale

The paper constructs an explicit four-dimensional non-Hermitian Hamiltonian, extends its parameters to five dimensions, locates the resulting EP2 loci, and directly computes the second Chern number on that geometry. No equations or steps in the abstract or described chain reduce a claimed prediction or topological invariant back to a fitted parameter or self-citation by construction. The hypersphere geometry and its topological transition are presented as consequences of the chosen model rather than tautological redefinitions of the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract supplies no explicit free parameters, axioms, or invented entities beyond the standard notions of exceptional points and Chern numbers; the hypersphere-like monopole is presented as an emergent discovery rather than an introduced postulate.

pith-pipeline@v0.9.0 · 5654 in / 1208 out tokens · 32360 ms · 2026-05-18T10:46:06.179938+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

finding a hypersphere-like non-Abelian Yang monopole in a five-dimensional parameter space, formed by EP2 pairs... characterized with the second Chern number
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

H(q) = q·Γ + iκΓ4 ... E± = ±√(|q|² − κ²)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Mixed-State Topology in Non-Hermitian Systems
quant-ph 2026-02 unverdicted novelty 7.0

Mixed-state topology in non-Hermitian systems is characterized via the Uhlmann connection, yielding a thermal Uhlmann-Chern number that differs from pure-state topology and extends to higher-dimensional Abelian and no...

Reference graph

Works this paper leans on

95 extracted references · 95 canonical work pages · cited by 1 Pith paper

[1]

and ( 3) 끫롬 2 0 0.5 1 0 1 2 ܴ FIG. 2. The second Chern number ( C2) versus the radius R of 5D parameter manifold. C2 reaches 1 when the parame- ter manifold (projected onto blue sphere) encloses the EHS (projected onto red ring), and changes to 0 when unenclosed, with a sharp transition occurring at the critical boundary of R = 1. (see Supplemental Materi...

work page
[2]

Note that the topological charge here is entirely carried by the whole EHS

C2 is equal to 1 or 0, dependent on whether the 5D parameter manifold encloses ( R> 1) or does not enclose ( R < 1) the EHS, with the critical boundary at R = 1 where the C2 jumps between the two values, revealing topological transition inherent in such a 4D NH model. Note that the topological charge here is entirely carried by the whole EHS. From other p...

work page
[3]

Under the condition λ m ≪ gr, the oﬀ-resonantly transitions can be discarded due to large detuning. In the interaction picture, the coherent dynamics is described by the eﬀective Hamiltonian [ 42] HI = Ξ ( |fg 0⟩⟨fg 0|+ |gf 0⟩⟨gf 0| − |1+⟩⟨1+| − |1− ⟩⟨1− |) +[Λ 1eiϕ 1 (|fg 0⟩⟨1+|+ |1− ⟩⟨gf 0|) +Λ 2eiϕ 2 (|1+⟩⟨gf 0| − |fg 0⟩⟨1− |)] +H.c., (6) where Λ 1, 3 ...

work page
[4]

and ( 4) to ﬁnally get C2 and WL. Conclusion.—In summary, we have shown that a point-like Yang monopole in a 5D parameter space is ex- tended to a 3D hypersphere when a non-Hermitian term is introduced to the Hamiltonian of a 4D system. This NH monopole, formed by degenerate EP2 pairs, can dis- play exotic topological transitions that are inaccessible wit...

work page
[5]

P. A. M. Dirac, Quantised singularities in the electro- magnetic ﬁeld, Proc. R. Soc. A 133, 60 (1931)

work page 1931
[6]

Roushan, C

P. Roushan, C. Neill, Y. Chen, M. Kolodrubetz, C. Quin- tana, N. Leung, M. Fang, R. Barends, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeﬀrey, J. Kelly, A. Megrant, J. Mutus, P. J. J. O’Malley, D. Sank, A. Vainsencher, J. Wenner, T. White, A. Polkovnikov, A. N. Cleland, and J. M. Martinis, Observation of topo- logical transitions in interacting qu...

work page 2014
[7]

M. D. Schroer, M. H. Kolodrubetz, W. F. Kindel, M. Sandberg, J. Gao, M. R. Vissers, D. P. Pappas, A. Polkovnikov, and K. W. Lehnert, Measuring a topo- logical transition in an artiﬁcial spin-1 / 2 system, Phys. Rev. Lett. 113, 050402 (2014)

work page 2014
[8]

Tan, D.-W

X. Tan, D.-W. Zhang, Z. Yang, J. Chu, Y.-Q. Zhu, D. Li, X. Yang, S. Song, Z. Han, Z. Li, Y. Dong, H.-F. Yu, H. Yan, S.-L. Zhu, and Y. Yu, Experimental measure- ment of the quantum metric tensor and related topologi- cal phase transition with a superconducting qubit, Phys. Rev. Lett. 122, 210401 (2019)

work page 2019
[9]

Jotzu, M

G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, Experimental realization of the topological Haldane model with ultra- cold fermions, Nature 515, 237 (2014)

work page 2014
[10]

Lv, Y.-X

Q.-X. Lv, Y.-X. Du, Z.-T. Liang, H.-Z. Liu, J.-H. Liang, L.-Q. Chen, L.-M. Zhou, S.-C. Zhang, D.-W. Zhang, B.-Q. Ai, H. Yan, and S.-L. Zhu, Measurement of spin 5 chern numbers in quantum simulated topological insula- tors, Phys. Rev. Lett. 127, 136802 (2021)

work page 2021
[11]

Kalb and P

M. Kalb and P. Ramond, Classical direct interstring ac- tion, Phys. Rev. D 9, 2273 (1974)

work page 1974
[12]

Tan, D.-W

X. Tan, D.-W. Zhang, W. Zheng, X. Yang, S. Song, Z. Han, Y. Dong, Z. Wang, D. Lan, H. Yan, S.-L. Zhu, and Y. Yu, Experimental observation of tensor monopoles with a superconducting qudit, Phys. Rev. Lett. 126, 017702 (2021)

work page 2021
[13]

M. Chen, C. Li, G. Palumbo, Y.-Q. Zhu, N. Goldman, and P. Cappellaro, A synthetic monopole source of Kalb- Ramond ﬁeld in diamond, Science 375, 1017 (2022)

work page 2022
[14]

Sugawa, F

S. Sugawa, F. Salces-Carcoba, A. R. Perry, Y. Yue, and I. B. Spielman, Second chern number of a quantum- simulated non-abelian yang monopole, Science 360, 1429 (2018)

work page 2018
[15]

C. N. Yang and R. L. Mills, Conservation of isotopic spin and isotopic gauge invariance, Phys. Rev. 96, 191 (1954)

work page 1954
[16]

Zee, Quantum ﬁeld theory in a nutshell (Princeton University Press, 2010)

A. Zee, Quantum ﬁeld theory in a nutshell (Princeton University Press, 2010)

work page 2010
[17]

Ezawa, Electric circuit simulations of nth-chern- number insulators in 2 n-dimensional space and their non- hermitian generalizations for arbitrary n, Phys

M. Ezawa, Electric circuit simulations of nth-chern- number insulators in 2 n-dimensional space and their non- hermitian generalizations for arbitrary n, Phys. Rev. B 100, 075423 (2019)

work page 2019
[18]

Sugawa, F

S. Sugawa, F. Salces-Carcoba, A. R. Perry, Y. Yue, and I. B. Spielman, Second Chern number of a quantum- simulated non-Abelian Yang monopole, Science 360, 1429 (2018)

work page 2018
[19]

Kolodrubetz, Measuring the second chern number from nonadiabatic eﬀects, Phys

M. Kolodrubetz, Measuring the second chern number from nonadiabatic eﬀects, Phys. Rev. Lett. 117, 015301 (2016)

work page 2016
[20]

Wilczek and A

F. Wilczek and A. Zee, Appearance of gauge structure in simple dynamical systems, Phys. Rev. Lett. 52, 2111 (1984)

work page 1984
[21]

T. W. B. Kibble, Symmetry breaking in non-abelian gauge theories, Phys. Rev. 155, 1554 (1967)

work page 1967
[22]

T. Li, L. Duca, M. Reitter, F. Grusdt, E. Demler, M. En- dres, M. Schleier-Smith, I. Bloch, and U. Schneider, Bloch state tomography using Wilson lines, Science 352, 1094 (2016)

work page 2016
[23]

Di Liberto, N

M. Di Liberto, N. Goldman, and G. Palumbo, Non- Abelian Bloch oscillations in higher-order topological in- sulators, Nat. Commun. 11, 5942 (2020)

work page 2020
[24]

B. H. M., M. Boguslawski, M. Barrios, L. Xin, and M. S. Chapman, Exploring non-abelian geometric phases in spin-1 ultracold atoms, Phys. Rev. Lett. 123, 173202 (2019)

work page 2019
[25]

Ruseckas, G

J. Ruseckas, G. Juzeli¯ unas, P. ¨Ohberg, and M. Fleis- chhauer, Non-abelian gauge potentials for ultracold atoms with degenerate dark states, Phys. Rev. Lett. 95, 010404 (2005)

work page 2005
[26]

Osterloh, M

K. Osterloh, M. Baig, L. Santos, P. Zoller, and M. Lewen- stein, Cold atoms in non-abelian gauge potentials: From the hofstadter ”moth” to lattice gauge theory, Phys. Rev. Lett. 95, 010403 (2005)

work page 2005
[27]

Goldman, A

N. Goldman, A. Kubasiak, P. Gaspard, and M. Lewen- stein, Ultracold atomic gases in non-abelian gauge po- tentials: The case of constant wilson loop, Phys. Rev. A 79, 023624 (2009)

work page 2009
[28]

Pietil¨ a and M

V. Pietil¨ a and M. M¨ ott¨ onen, Non-abelian magnetic monopole in a bose-einstein condensate, Phys. Rev. Lett. 102, 080403 (2009)

work page 2009
[29]

Bermudez, N

A. Bermudez, N. Goldman, A. Kubasiak, M. Lewenstein, and M. A. Martin-Delgado, Topological phase transitions in the non-Abelian honeycomb lattice, New J. Phys. 12, 033041 (2010)

work page 2010
[30]

Hauke, O

P. Hauke, O. Tieleman, A. Celi, C. ¨Olschl¨ ager, J. Si- monet, J. Struck, M. Weinberg, P. Windpassinger, K. Sengstock, M. Lewenstein, and A. Eckardt, Non- abelian gauge ﬁelds and topological insulators in shaken optical lattices, Phys. Rev. Lett. 109, 145301 (2012)

work page 2012
[31]

N. T. Phuc, G. Tatara, Y. Kawaguchi, and M. Ueda, Controlling and probing non-abelian emergent gauge po- tentials in spinor Bose-Fermi mixtures, Nat. Commun. 6, 8135 (2015)

work page 2015
[32]

Leroux, K

F. Leroux, K. Pandey, R. Rehbi, F. Chevy, C. Miniatura, B. Gr´ emaud, and D. Wilkowski, Non-Abelian adiabatic geometric transformations in a cold strontium gas, Nat. Commun. 9, 3580 (2018)

work page 2018
[33]

Y. Yang, C. Peng, D. Zhu, H. Buljan, J. D. Joannopoulos, B. Zhen, and M. Soljaˇ ci´ c, Synthesis and observation of non-Abelian gauge ﬁelds in real space, Science 365, 1021 (2019)

work page 2019
[34]

Sugawa, F

S. Sugawa, F. Salces-Carcoba, Y. Yue, A. Putra, and I. B. Spielman, Wilson loop and Wilczek-Zee phase from a non-Abelian gauge ﬁeld, npj Quantum Inf. 7, 144 (2021)

work page 2021
[35]

Tycko, Adiabatic rotational splittings and berry’s phase in nuclear quadrupole resonance, Phys

R. Tycko, Adiabatic rotational splittings and berry’s phase in nuclear quadrupole resonance, Phys. Rev. Lett. 58, 2281 (1987)

work page 1987
[36]

J. W. Zwanziger, M. Koenig, and A. Pines, Non-abelian eﬀects in a quadrupole system rotating around two axes, Phys. Rev. A 42, 3107 (1990)

work page 1990
[37]

Zheng, J

W. Zheng, J. Xu, Z. Ma, Y. Li, Y. Dong, Y. Zhang, X. Wang, G. Sun, P. Wu, J. Zhao, S. Li, D. Lan, X. Tan, and Y. Yu, Measuring quantum geometric tensor of non- abelian system in superconducting circuits, Chin. Phys. Lett. 39, 100202 (2022)

work page 2022
[38]

V. V. Albert, C. Shu, S. Krastanov, C. Shen, R.- B. Liu, Z.-B. Yang, R. J. Schoelkopf, M. Mirrahimi, M. H. Devoret, and L. Jiang, Holonomic quantum con- trol with continuous variable systems, Phys. Rev. Lett. 116, 140502 (2016)

work page 2016
[39]

Zhang, Y.-H

H.-L. Zhang, Y.-H. Kang, F. Wu, Z.-B. Yang, and S.- B. Zheng, Non-adiabatic holonomic quantum operations in continuous variable systems, Sci. China Phys. Mech. Astron. 67, 260311 (2024)

work page 2024
[40]

Z. Han, Y. Dong, B. Liu, X. Yang, S. Song, L. Qiu, D. Li, J. Chu, W. Zheng, J. Xu, T. Huang, Z. Wang, X. Yu, X. Tan, D. Lan, M.-H. Yung, and Y. Yu, Experimental realization of universal time-optimal non-abelian geomet- ric gates (2020), arXiv:2004.10364 [quant-ph]

work page arXiv 2020
[41]

Toyoda, K

K. Toyoda, K. Uchida, A. Noguchi, S. Haze, and S. Urabe, Realization of holonomic single-qubit opera- tions, Phys. Rev. A 87, 052307 (2013)

work page 2013
[42]

L.-M. Duan, J. I. Cirac, and P. Zoller, Geometric Ma- nipulation of Trapped Ions for Quantum Computation, Science 292, 1695 (2001)

work page 2001
[43]

A. A. Abdumalikov Jr, J. M. Fink, K. Juliusson, M. Pechal, S. Berger, A. Wallraﬀ, and S. Filipp, Experi- mental realization of non-Abelian non-adiabatic geomet- ric gates, Nature 496, 482 (2013)

work page 2013
[44]

Zu, W.-B

C. Zu, W.-B. Wang, L. He, W.-G. Zhang, C.-Y. Dai, F. Wang, and L.-M. Duan, Experimental realization of universal geometric quantum gates with solid-state spins, Nature 514, 72 (2014)

work page 2014
[45]

Arroyo-Camejo, A

S. Arroyo-Camejo, A. Lazariev, S. W. Hell, and G. Bala- 6 subramanian, Room temperature high-ﬁdelity holonomic single-qubit gate on a solid-state spin, Nat. Commun. 5, 4870 (2014)

work page 2014
[46]

K. Xu, W. Ning, X.-J. Huang, P.-R. Han, H. Li, Z.-B. Yang, D. Zheng, H. Fan, and S.-B. Zheng, Demonstra- tion of a non-Abelian geometric controlled-NOT gate in a superconducting circuit, Optica 8, 972 (2021)

work page 2021
[47]

Singh, B

S. Singh, B. Royer, and S. M. Girvin, Towards non- abelian quantum signal processing: Eﬃcient control of hybrid continuous- and discrete-variable architectures (2025), arXiv:2504.19992 [quant-ph]

work page arXiv 2025
[48]

Weisbrich, M

H. Weisbrich, M. Bestler, and W. Belzig, Tensor Monopoles in superconducting systems, Quantum 5, 601 (2021)

work page 2021
[49]

Weisbrich, R

H. Weisbrich, R. Klees, G. Rastelli, and W. Belzig, Second chern number and non-abelian berry phase in topological superconducting systems, PRX Quantum 2, 010310 (2021)

work page 2021
[50]

Moiseyev, Non-Hermitian quantum mechanics (Cam- bridge University Press, 2011)

N. Moiseyev, Non-Hermitian quantum mechanics (Cam- bridge University Press, 2011)

work page 2011
[51]

Rotter, A non-Hermitian Hamilton operator and the physics of open quantum systems, J

I. Rotter, A non-Hermitian Hamilton operator and the physics of open quantum systems, J. Phys. A: Math. Theor. 42, 153001 (2009)

work page 2009
[52]

Berry, Physics of Nonhermitian Degeneracies, Czech

M. Berry, Physics of Nonhermitian Degeneracies, Czech. J. Phys. 54, 1039 (2004)

work page 2004
[53]

W. D. Heiss, The physics of exceptional points, J. Phys. A: Math. Theor. 45, 444016 (2012)

work page 2012
[54]

Dembowski, H.-D

C. Dembowski, H.-D. Gr¨ af, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, Experimen- tal observation of the topological structure of exceptional points, Phys. Rev. Lett. 86, 787 (2001)

work page 2001
[55]

Y. Choi, S. Kang, S. Lim, W. Kim, J.-R. Kim, J.-H. Lee, and K. An, Quasieigenstate coalescence in an atom- cavity quantum composite, Phys. Rev. Lett. 104, 153601 (2010)

work page 2010
[56]

T. Gao, E. Estrecho, K. Y. Bliokh, T. C. H. Liew, M. D. Fraser, S. Brodbeck, M. Kamp, C. Schneider, S. H¨ oﬂing, Y. Yamamoto, F. Nori, Y. S. Kivshar, A. G. Truscott, R. G. Dall, and E. A. Ostrovskaya, Observation of non- Hermitian degeneracies in a chaotic exciton-polariton bil- liard, Nature 526, 554 (2015)

work page 2015
[57]

Zhang, X.-Q

D. Zhang, X.-Q. Luo, Y.-P. Wang, T.-F. Li, and J. Q. You, Observation of the exceptional point in cavity magnon-polaritons, Nat. Commun. 8, 1368 (2017)

work page 2017
[58]

Zhang, S

X.-L. Zhang, S. Wang, B. Hou, and C. T. Chan, Dynam- ically encircling exceptional points: In situ control of en- circling loops and the role of the starting point, Phys. Rev. X 8, 021066 (2018)

work page 2018
[59]

Doppler, A

J. Doppler, A. A. Mailybaev, J. B¨ ohm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moi- seyev, and S. Rotter, Dynamically encircling an excep- tional point for asymmetric mode switching, Nature 537, 76 (2016)

work page 2016
[60]

H. Xu, D. Mason, L. Jiang, and J. G. E. Harris, Topo- logical energy transfer in an optomechanical system with exceptional points, Nature 537, 80 (2016)

work page 2016
[61]

J. W. Yoon, Y. Choi, C. Hahn, G. Kim, S. H. Song, K.-Y. Yang, J. Y. Lee, Y. Kim, C. S. Lee, J. K. Shin, H.-S. Lee, and P. Berini, Time-asymmetric loop around an exceptional point over the full optical communications band, Nature 562, 86 (2018)

work page 2018
[62]

W. Liu, Y. Wu, C.-K. Duan, X. Rong, and J. Du, Dynam- ically encircling an exceptional point in a real quantum system, Phys. Rev. Lett. 126, 170506 (2021)

work page 2021
[63]

W. Gou, T. Chen, D. Xie, T. Xiao, T.-S. Deng, B. Gad- way, W. Yi, and B. Yan, Tunable nonreciprocal quantum transport through a dissipative aharonov-bohm ring in ultracold atoms, Phys. Rev. Lett. 124, 070402 (2020)

work page 2020
[64]

Z. Ren, D. Liu, E. Zhao, C. He, K. K. Pak, J. Li, and G.- B. Jo, Chiral control of quantum states in non-Hermitian spin–orbit-coupled fermions, Nat. Phys. 18, 385 (2022)

work page 2022
[65]

Liertzer, L

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. T¨ ureci, and S. Rotter, Pump-induced exceptional points in lasers, Phys. Rev. Lett. 108, 173901 (2012)

work page 2012
[66]

C. M. Bender and S. Boettcher, Real spectra in non- hermitian hamiltonians having P t symmetry, Phys. Rev. Lett. 80, 5243 (1998)

work page 1998
[67]

C. M. Bender, D. C. Brody, and H. F. Jones, Complex extension of quantum mechanics, Phys. Rev. Lett. 89, 270401 (2002)

work page 2002
[68]

S. K. ¨Ozdemir, S. Rotter, F. Nori, and L. Yang, Par- ity–time symmetry and exceptional points in photonics, Nat. Mater. 18, 783 (2019)

work page 2019
[69]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Observation of PT -symmetry breaking in complex optical potentials, Phys. Rev. Lett. 103, 093902 (2009)

work page 2009
[70]

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, Single-mode laser by parity-time symmetry breaking, Sci- ence 346, 972 (2014)

work page 2014
[71]

Hodaei, M.-A

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, Parity- time–symmetric microring lasers, Science 346, 975 (2014)

work page 2014
[72]

E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Ex- ceptional topology of non-hermitian systems, Rev. Mod. Phys. 93, 015005 (2021)

work page 2021
[73]

K. Ding, C. Fang, and G. Ma, Non-hermitian topology and exceptional-point geometries, Nat. Rev. Phys. 4, 745 (2022)

work page 2022
[74]

P.-R. Han, F. Wu, X.-J. Huang, H.-Z. Wu, C.-L. Zou, W. Yi, M. Zhang, H. Li, K. Xu, D. Zheng, H. Fan, J. Wen, Z.-B. Yang, and S.-B. Zheng, Exceptional en- tanglement phenomena: Non-hermiticity meeting non- classicality, Phys. Rev. Lett. 131, 260201 (2023)

work page 2023
[75]

Miri and A

M.-A. Miri and A. Al` u, Exceptional points in optics and photonics, Science 363, eaar7709 (2019)

work page 2019
[76]

W. Chen, S. Kaya ¨Ozdemir, G. Zhao, J. Wiersig, and L. Yang, Exceptional points enhance sensing in an optical microcavity, Nature 548, 192 (2017)

work page 2017
[77]

Hodaei, A

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Kha- javikhan, Enhanced sensitivity at higher-order excep- tional points, Nature 548, 187 (2017)

work page 2017
[78]

Xu, S.-T

Y. Xu, S.-T. Wang, and L.-M. Duan, Weyl exceptional rings in a three-dimensional dissipative cold atomic gas, Phys. Rev. Lett. 118, 045701 (2017)

work page 2017
[79]

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljaˇ ci´ c, Spawning rings of exceptional points out of Dirac cones, Nature 525, 354 (2015)

work page 2015
[80]

Cerjan, S

A. Cerjan, S. Huang, M. Wang, K. P. Chen, Y. Chong, and M. C. Rechtsman, Experimental realization of a Weyl exceptional ring, Nat. Photonics 13, 623 (2019)

work page 2019

Showing first 80 references.

[1] [1]

and ( 3) 끫롬 2 0 0.5 1 0 1 2 ܴ FIG. 2. The second Chern number ( C2) versus the radius R of 5D parameter manifold. C2 reaches 1 when the parame- ter manifold (projected onto blue sphere) encloses the EHS (projected onto red ring), and changes to 0 when unenclosed, with a sharp transition occurring at the critical boundary of R = 1. (see Supplemental Materi...

work page

[2] [2]

Note that the topological charge here is entirely carried by the whole EHS

C2 is equal to 1 or 0, dependent on whether the 5D parameter manifold encloses ( R> 1) or does not enclose ( R < 1) the EHS, with the critical boundary at R = 1 where the C2 jumps between the two values, revealing topological transition inherent in such a 4D NH model. Note that the topological charge here is entirely carried by the whole EHS. From other p...

work page

[3] [3]

Under the condition λ m ≪ gr, the oﬀ-resonantly transitions can be discarded due to large detuning. In the interaction picture, the coherent dynamics is described by the eﬀective Hamiltonian [ 42] HI = Ξ ( |fg 0⟩⟨fg 0|+ |gf 0⟩⟨gf 0| − |1+⟩⟨1+| − |1− ⟩⟨1− |) +[Λ 1eiϕ 1 (|fg 0⟩⟨1+|+ |1− ⟩⟨gf 0|) +Λ 2eiϕ 2 (|1+⟩⟨gf 0| − |fg 0⟩⟨1− |)] +H.c., (6) where Λ 1, 3 ...

work page

[4] [4]

and ( 4) to ﬁnally get C2 and WL. Conclusion.—In summary, we have shown that a point-like Yang monopole in a 5D parameter space is ex- tended to a 3D hypersphere when a non-Hermitian term is introduced to the Hamiltonian of a 4D system. This NH monopole, formed by degenerate EP2 pairs, can dis- play exotic topological transitions that are inaccessible wit...

work page

[5] [5]

P. A. M. Dirac, Quantised singularities in the electro- magnetic ﬁeld, Proc. R. Soc. A 133, 60 (1931)

work page 1931

[6] [6]

Roushan, C

P. Roushan, C. Neill, Y. Chen, M. Kolodrubetz, C. Quin- tana, N. Leung, M. Fang, R. Barends, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeﬀrey, J. Kelly, A. Megrant, J. Mutus, P. J. J. O’Malley, D. Sank, A. Vainsencher, J. Wenner, T. White, A. Polkovnikov, A. N. Cleland, and J. M. Martinis, Observation of topo- logical transitions in interacting qu...

work page 2014

[7] [7]

M. D. Schroer, M. H. Kolodrubetz, W. F. Kindel, M. Sandberg, J. Gao, M. R. Vissers, D. P. Pappas, A. Polkovnikov, and K. W. Lehnert, Measuring a topo- logical transition in an artiﬁcial spin-1 / 2 system, Phys. Rev. Lett. 113, 050402 (2014)

work page 2014

[8] [8]

Tan, D.-W

X. Tan, D.-W. Zhang, Z. Yang, J. Chu, Y.-Q. Zhu, D. Li, X. Yang, S. Song, Z. Han, Z. Li, Y. Dong, H.-F. Yu, H. Yan, S.-L. Zhu, and Y. Yu, Experimental measure- ment of the quantum metric tensor and related topologi- cal phase transition with a superconducting qubit, Phys. Rev. Lett. 122, 210401 (2019)

work page 2019

[9] [9]

Jotzu, M

G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, Experimental realization of the topological Haldane model with ultra- cold fermions, Nature 515, 237 (2014)

work page 2014

[10] [10]

Lv, Y.-X

Q.-X. Lv, Y.-X. Du, Z.-T. Liang, H.-Z. Liu, J.-H. Liang, L.-Q. Chen, L.-M. Zhou, S.-C. Zhang, D.-W. Zhang, B.-Q. Ai, H. Yan, and S.-L. Zhu, Measurement of spin 5 chern numbers in quantum simulated topological insula- tors, Phys. Rev. Lett. 127, 136802 (2021)

work page 2021

[11] [11]

Kalb and P

M. Kalb and P. Ramond, Classical direct interstring ac- tion, Phys. Rev. D 9, 2273 (1974)

work page 1974

[12] [12]

Tan, D.-W

X. Tan, D.-W. Zhang, W. Zheng, X. Yang, S. Song, Z. Han, Y. Dong, Z. Wang, D. Lan, H. Yan, S.-L. Zhu, and Y. Yu, Experimental observation of tensor monopoles with a superconducting qudit, Phys. Rev. Lett. 126, 017702 (2021)

work page 2021

[13] [13]

M. Chen, C. Li, G. Palumbo, Y.-Q. Zhu, N. Goldman, and P. Cappellaro, A synthetic monopole source of Kalb- Ramond ﬁeld in diamond, Science 375, 1017 (2022)

work page 2022

[14] [14]

Sugawa, F

S. Sugawa, F. Salces-Carcoba, A. R. Perry, Y. Yue, and I. B. Spielman, Second chern number of a quantum- simulated non-abelian yang monopole, Science 360, 1429 (2018)

work page 2018

[15] [15]

C. N. Yang and R. L. Mills, Conservation of isotopic spin and isotopic gauge invariance, Phys. Rev. 96, 191 (1954)

work page 1954

[16] [16]

Zee, Quantum ﬁeld theory in a nutshell (Princeton University Press, 2010)

A. Zee, Quantum ﬁeld theory in a nutshell (Princeton University Press, 2010)

work page 2010

[17] [17]

Ezawa, Electric circuit simulations of nth-chern- number insulators in 2 n-dimensional space and their non- hermitian generalizations for arbitrary n, Phys

M. Ezawa, Electric circuit simulations of nth-chern- number insulators in 2 n-dimensional space and their non- hermitian generalizations for arbitrary n, Phys. Rev. B 100, 075423 (2019)

work page 2019

[18] [18]

Sugawa, F

S. Sugawa, F. Salces-Carcoba, A. R. Perry, Y. Yue, and I. B. Spielman, Second Chern number of a quantum- simulated non-Abelian Yang monopole, Science 360, 1429 (2018)

work page 2018

[19] [19]

Kolodrubetz, Measuring the second chern number from nonadiabatic eﬀects, Phys

M. Kolodrubetz, Measuring the second chern number from nonadiabatic eﬀects, Phys. Rev. Lett. 117, 015301 (2016)

work page 2016

[20] [20]

Wilczek and A

F. Wilczek and A. Zee, Appearance of gauge structure in simple dynamical systems, Phys. Rev. Lett. 52, 2111 (1984)

work page 1984

[21] [21]

T. W. B. Kibble, Symmetry breaking in non-abelian gauge theories, Phys. Rev. 155, 1554 (1967)

work page 1967

[22] [22]

T. Li, L. Duca, M. Reitter, F. Grusdt, E. Demler, M. En- dres, M. Schleier-Smith, I. Bloch, and U. Schneider, Bloch state tomography using Wilson lines, Science 352, 1094 (2016)

work page 2016

[23] [23]

Di Liberto, N

M. Di Liberto, N. Goldman, and G. Palumbo, Non- Abelian Bloch oscillations in higher-order topological in- sulators, Nat. Commun. 11, 5942 (2020)

work page 2020

[24] [24]

B. H. M., M. Boguslawski, M. Barrios, L. Xin, and M. S. Chapman, Exploring non-abelian geometric phases in spin-1 ultracold atoms, Phys. Rev. Lett. 123, 173202 (2019)

work page 2019

[25] [25]

Ruseckas, G

J. Ruseckas, G. Juzeli¯ unas, P. ¨Ohberg, and M. Fleis- chhauer, Non-abelian gauge potentials for ultracold atoms with degenerate dark states, Phys. Rev. Lett. 95, 010404 (2005)

work page 2005

[26] [26]

Osterloh, M

K. Osterloh, M. Baig, L. Santos, P. Zoller, and M. Lewen- stein, Cold atoms in non-abelian gauge potentials: From the hofstadter ”moth” to lattice gauge theory, Phys. Rev. Lett. 95, 010403 (2005)

work page 2005

[27] [27]

Goldman, A

N. Goldman, A. Kubasiak, P. Gaspard, and M. Lewen- stein, Ultracold atomic gases in non-abelian gauge po- tentials: The case of constant wilson loop, Phys. Rev. A 79, 023624 (2009)

work page 2009

[28] [28]

Pietil¨ a and M

V. Pietil¨ a and M. M¨ ott¨ onen, Non-abelian magnetic monopole in a bose-einstein condensate, Phys. Rev. Lett. 102, 080403 (2009)

work page 2009

[29] [29]

Bermudez, N

A. Bermudez, N. Goldman, A. Kubasiak, M. Lewenstein, and M. A. Martin-Delgado, Topological phase transitions in the non-Abelian honeycomb lattice, New J. Phys. 12, 033041 (2010)

work page 2010

[30] [30]

Hauke, O

P. Hauke, O. Tieleman, A. Celi, C. ¨Olschl¨ ager, J. Si- monet, J. Struck, M. Weinberg, P. Windpassinger, K. Sengstock, M. Lewenstein, and A. Eckardt, Non- abelian gauge ﬁelds and topological insulators in shaken optical lattices, Phys. Rev. Lett. 109, 145301 (2012)

work page 2012

[31] [31]

N. T. Phuc, G. Tatara, Y. Kawaguchi, and M. Ueda, Controlling and probing non-abelian emergent gauge po- tentials in spinor Bose-Fermi mixtures, Nat. Commun. 6, 8135 (2015)

work page 2015

[32] [32]

Leroux, K

F. Leroux, K. Pandey, R. Rehbi, F. Chevy, C. Miniatura, B. Gr´ emaud, and D. Wilkowski, Non-Abelian adiabatic geometric transformations in a cold strontium gas, Nat. Commun. 9, 3580 (2018)

work page 2018

[33] [33]

Y. Yang, C. Peng, D. Zhu, H. Buljan, J. D. Joannopoulos, B. Zhen, and M. Soljaˇ ci´ c, Synthesis and observation of non-Abelian gauge ﬁelds in real space, Science 365, 1021 (2019)

work page 2019

[34] [34]

Sugawa, F

S. Sugawa, F. Salces-Carcoba, Y. Yue, A. Putra, and I. B. Spielman, Wilson loop and Wilczek-Zee phase from a non-Abelian gauge ﬁeld, npj Quantum Inf. 7, 144 (2021)

work page 2021

[35] [35]

Tycko, Adiabatic rotational splittings and berry’s phase in nuclear quadrupole resonance, Phys

R. Tycko, Adiabatic rotational splittings and berry’s phase in nuclear quadrupole resonance, Phys. Rev. Lett. 58, 2281 (1987)

work page 1987

[36] [36]

J. W. Zwanziger, M. Koenig, and A. Pines, Non-abelian eﬀects in a quadrupole system rotating around two axes, Phys. Rev. A 42, 3107 (1990)

work page 1990

[37] [37]

Zheng, J

W. Zheng, J. Xu, Z. Ma, Y. Li, Y. Dong, Y. Zhang, X. Wang, G. Sun, P. Wu, J. Zhao, S. Li, D. Lan, X. Tan, and Y. Yu, Measuring quantum geometric tensor of non- abelian system in superconducting circuits, Chin. Phys. Lett. 39, 100202 (2022)

work page 2022

[38] [38]

V. V. Albert, C. Shu, S. Krastanov, C. Shen, R.- B. Liu, Z.-B. Yang, R. J. Schoelkopf, M. Mirrahimi, M. H. Devoret, and L. Jiang, Holonomic quantum con- trol with continuous variable systems, Phys. Rev. Lett. 116, 140502 (2016)

work page 2016

[39] [39]

Zhang, Y.-H

H.-L. Zhang, Y.-H. Kang, F. Wu, Z.-B. Yang, and S.- B. Zheng, Non-adiabatic holonomic quantum operations in continuous variable systems, Sci. China Phys. Mech. Astron. 67, 260311 (2024)

work page 2024

[40] [40]

Z. Han, Y. Dong, B. Liu, X. Yang, S. Song, L. Qiu, D. Li, J. Chu, W. Zheng, J. Xu, T. Huang, Z. Wang, X. Yu, X. Tan, D. Lan, M.-H. Yung, and Y. Yu, Experimental realization of universal time-optimal non-abelian geomet- ric gates (2020), arXiv:2004.10364 [quant-ph]

work page arXiv 2020

[41] [41]

Toyoda, K

K. Toyoda, K. Uchida, A. Noguchi, S. Haze, and S. Urabe, Realization of holonomic single-qubit opera- tions, Phys. Rev. A 87, 052307 (2013)

work page 2013

[42] [42]

L.-M. Duan, J. I. Cirac, and P. Zoller, Geometric Ma- nipulation of Trapped Ions for Quantum Computation, Science 292, 1695 (2001)

work page 2001

[43] [43]

A. A. Abdumalikov Jr, J. M. Fink, K. Juliusson, M. Pechal, S. Berger, A. Wallraﬀ, and S. Filipp, Experi- mental realization of non-Abelian non-adiabatic geomet- ric gates, Nature 496, 482 (2013)

work page 2013

[44] [44]

Zu, W.-B

C. Zu, W.-B. Wang, L. He, W.-G. Zhang, C.-Y. Dai, F. Wang, and L.-M. Duan, Experimental realization of universal geometric quantum gates with solid-state spins, Nature 514, 72 (2014)

work page 2014

[45] [45]

Arroyo-Camejo, A

S. Arroyo-Camejo, A. Lazariev, S. W. Hell, and G. Bala- 6 subramanian, Room temperature high-ﬁdelity holonomic single-qubit gate on a solid-state spin, Nat. Commun. 5, 4870 (2014)

work page 2014

[46] [46]

K. Xu, W. Ning, X.-J. Huang, P.-R. Han, H. Li, Z.-B. Yang, D. Zheng, H. Fan, and S.-B. Zheng, Demonstra- tion of a non-Abelian geometric controlled-NOT gate in a superconducting circuit, Optica 8, 972 (2021)

work page 2021

[47] [47]

Singh, B

S. Singh, B. Royer, and S. M. Girvin, Towards non- abelian quantum signal processing: Eﬃcient control of hybrid continuous- and discrete-variable architectures (2025), arXiv:2504.19992 [quant-ph]

work page arXiv 2025

[48] [48]

Weisbrich, M

H. Weisbrich, M. Bestler, and W. Belzig, Tensor Monopoles in superconducting systems, Quantum 5, 601 (2021)

work page 2021

[49] [49]

Weisbrich, R

H. Weisbrich, R. Klees, G. Rastelli, and W. Belzig, Second chern number and non-abelian berry phase in topological superconducting systems, PRX Quantum 2, 010310 (2021)

work page 2021

[50] [50]

Moiseyev, Non-Hermitian quantum mechanics (Cam- bridge University Press, 2011)

N. Moiseyev, Non-Hermitian quantum mechanics (Cam- bridge University Press, 2011)

work page 2011

[51] [51]

Rotter, A non-Hermitian Hamilton operator and the physics of open quantum systems, J

I. Rotter, A non-Hermitian Hamilton operator and the physics of open quantum systems, J. Phys. A: Math. Theor. 42, 153001 (2009)

work page 2009

[52] [52]

Berry, Physics of Nonhermitian Degeneracies, Czech

M. Berry, Physics of Nonhermitian Degeneracies, Czech. J. Phys. 54, 1039 (2004)

work page 2004

[53] [53]

W. D. Heiss, The physics of exceptional points, J. Phys. A: Math. Theor. 45, 444016 (2012)

work page 2012

[54] [54]

Dembowski, H.-D

C. Dembowski, H.-D. Gr¨ af, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, Experimen- tal observation of the topological structure of exceptional points, Phys. Rev. Lett. 86, 787 (2001)

work page 2001

[55] [55]

Y. Choi, S. Kang, S. Lim, W. Kim, J.-R. Kim, J.-H. Lee, and K. An, Quasieigenstate coalescence in an atom- cavity quantum composite, Phys. Rev. Lett. 104, 153601 (2010)

work page 2010

[56] [56]

T. Gao, E. Estrecho, K. Y. Bliokh, T. C. H. Liew, M. D. Fraser, S. Brodbeck, M. Kamp, C. Schneider, S. H¨ oﬂing, Y. Yamamoto, F. Nori, Y. S. Kivshar, A. G. Truscott, R. G. Dall, and E. A. Ostrovskaya, Observation of non- Hermitian degeneracies in a chaotic exciton-polariton bil- liard, Nature 526, 554 (2015)

work page 2015

[57] [57]

Zhang, X.-Q

D. Zhang, X.-Q. Luo, Y.-P. Wang, T.-F. Li, and J. Q. You, Observation of the exceptional point in cavity magnon-polaritons, Nat. Commun. 8, 1368 (2017)

work page 2017

[58] [58]

Zhang, S

X.-L. Zhang, S. Wang, B. Hou, and C. T. Chan, Dynam- ically encircling exceptional points: In situ control of en- circling loops and the role of the starting point, Phys. Rev. X 8, 021066 (2018)

work page 2018

[59] [59]

Doppler, A

J. Doppler, A. A. Mailybaev, J. B¨ ohm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moi- seyev, and S. Rotter, Dynamically encircling an excep- tional point for asymmetric mode switching, Nature 537, 76 (2016)

work page 2016

[60] [60]

H. Xu, D. Mason, L. Jiang, and J. G. E. Harris, Topo- logical energy transfer in an optomechanical system with exceptional points, Nature 537, 80 (2016)

work page 2016

[61] [61]

J. W. Yoon, Y. Choi, C. Hahn, G. Kim, S. H. Song, K.-Y. Yang, J. Y. Lee, Y. Kim, C. S. Lee, J. K. Shin, H.-S. Lee, and P. Berini, Time-asymmetric loop around an exceptional point over the full optical communications band, Nature 562, 86 (2018)

work page 2018

[62] [62]

W. Liu, Y. Wu, C.-K. Duan, X. Rong, and J. Du, Dynam- ically encircling an exceptional point in a real quantum system, Phys. Rev. Lett. 126, 170506 (2021)

work page 2021

[63] [63]

W. Gou, T. Chen, D. Xie, T. Xiao, T.-S. Deng, B. Gad- way, W. Yi, and B. Yan, Tunable nonreciprocal quantum transport through a dissipative aharonov-bohm ring in ultracold atoms, Phys. Rev. Lett. 124, 070402 (2020)

work page 2020

[64] [64]

Z. Ren, D. Liu, E. Zhao, C. He, K. K. Pak, J. Li, and G.- B. Jo, Chiral control of quantum states in non-Hermitian spin–orbit-coupled fermions, Nat. Phys. 18, 385 (2022)

work page 2022

[65] [65]

Liertzer, L

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. T¨ ureci, and S. Rotter, Pump-induced exceptional points in lasers, Phys. Rev. Lett. 108, 173901 (2012)

work page 2012

[66] [66]

C. M. Bender and S. Boettcher, Real spectra in non- hermitian hamiltonians having P t symmetry, Phys. Rev. Lett. 80, 5243 (1998)

work page 1998

[67] [67]

C. M. Bender, D. C. Brody, and H. F. Jones, Complex extension of quantum mechanics, Phys. Rev. Lett. 89, 270401 (2002)

work page 2002

[68] [68]

S. K. ¨Ozdemir, S. Rotter, F. Nori, and L. Yang, Par- ity–time symmetry and exceptional points in photonics, Nat. Mater. 18, 783 (2019)

work page 2019

[69] [69]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Observation of PT -symmetry breaking in complex optical potentials, Phys. Rev. Lett. 103, 093902 (2009)

work page 2009

[70] [70]

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, Single-mode laser by parity-time symmetry breaking, Sci- ence 346, 972 (2014)

work page 2014

[71] [71]

Hodaei, M.-A

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, Parity- time–symmetric microring lasers, Science 346, 975 (2014)

work page 2014

[72] [72]

E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Ex- ceptional topology of non-hermitian systems, Rev. Mod. Phys. 93, 015005 (2021)

work page 2021

[73] [73]

K. Ding, C. Fang, and G. Ma, Non-hermitian topology and exceptional-point geometries, Nat. Rev. Phys. 4, 745 (2022)

work page 2022

[74] [74]

P.-R. Han, F. Wu, X.-J. Huang, H.-Z. Wu, C.-L. Zou, W. Yi, M. Zhang, H. Li, K. Xu, D. Zheng, H. Fan, J. Wen, Z.-B. Yang, and S.-B. Zheng, Exceptional en- tanglement phenomena: Non-hermiticity meeting non- classicality, Phys. Rev. Lett. 131, 260201 (2023)

work page 2023

[75] [75]

Miri and A

M.-A. Miri and A. Al` u, Exceptional points in optics and photonics, Science 363, eaar7709 (2019)

work page 2019

[76] [76]

W. Chen, S. Kaya ¨Ozdemir, G. Zhao, J. Wiersig, and L. Yang, Exceptional points enhance sensing in an optical microcavity, Nature 548, 192 (2017)

work page 2017

[77] [77]

Hodaei, A

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Kha- javikhan, Enhanced sensitivity at higher-order excep- tional points, Nature 548, 187 (2017)

work page 2017

[78] [78]

Xu, S.-T

Y. Xu, S.-T. Wang, and L.-M. Duan, Weyl exceptional rings in a three-dimensional dissipative cold atomic gas, Phys. Rev. Lett. 118, 045701 (2017)

work page 2017

[79] [79]

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljaˇ ci´ c, Spawning rings of exceptional points out of Dirac cones, Nature 525, 354 (2015)

work page 2015

[80] [80]

Cerjan, S

A. Cerjan, S. Huang, M. Wang, K. P. Chen, Y. Chong, and M. C. Rechtsman, Experimental realization of a Weyl exceptional ring, Nat. Photonics 13, 623 (2019)

work page 2019