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Non-Hermitian non-Abelian topological transition in the S = 1 electron spin system of a nitrogen vacancy centre in diamond

Nature Nanotechnology volume 20, pages 873–880 (2025)Cite this article

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Abstract

Non-Abelian topological transitions are well studied in Hermitian systems, exhibiting features like non-Abelian charges and edge states. Introducing non-Hermiticity gives rise to novel topological phenomena, yet non-Hermitian non-Abelian topological transitions remain experimentally unexplored. In this work we observe a non-Hermitian non-Abelian topological transition in a single electron spin system of a nitrogen vacancy centre in diamond, achieved via a dilation method with a nearby nuclear spin. While this transition cannot be detected by traditional topological numbers, we identify the transition through the measured complex eigenvalue braids. We extract the braid invariants from the relative phases between eigenvalues and thereby establish their changes as clear signatures of non-Abelian transitions. Furthermore we experimentally reveal an intriguing consequence of this transition: the creation of a third-order exceptional point through the collision of two second-order exceptional points with opposite charges. Our work unveils the dynamical interplay between exceptional points and provides guidance on the manipulation of spectral topology to achieve functionalities such as robust quantum control.

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Fig. 1: Schematics of the moving EPs in the parameter space and the braid invariants.
Fig. 2: The model of the NH Hamiltonian and experimental system of a NV centre.
Fig. 3: Observation of the non-Abelian topological transition by characterizing the change in eigenvalue braids.
Fig. 4: Measuring the charges of different EPs and the EP3 produced by collision of the EPs X and Y.

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Data availability

The data supporting the findings of this study are available within this Article and its Supplementary Information. Source data are provided with this paper. All other data such as raw data are available from the corresponding authors upon reasonable request.

Code availability

The codes that were used to generate the theoretical predictions in this study are available from the corresponding authors upon reasonable request.

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Acknowledgements

This work was supported by the Innovation Program for Quantum Science and Technology (grant no. 2021ZD0302200 (J.D.)), the National Natural Science Foundation of China (grant nos. T2388102 (X.R.), 12174373 (Y. Wu), 92265204 (Ya Wang), 12261160569 (X.R.) and 12474496 (H.H.)), National Key R&D Program of China (grant nos. 2023YFA1406704 (H.H.) and 2022YFA1405800 (H.H.)), the Chinese Academy of Sciences (grant nos. XDC07000000 (J.D.) and GJJSTD20200001 (J.D.)) and Hefei Comprehensive National Science Center (J.D.). Ya Wang and Y. Wu thank the Fundamental Research Funds for the Central Universities for their support. This work was partially carried out at the USTC Center for Micro and Nanoscale Research and Fabrication.

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Author notes

  1. These authors contributed equally: Yunhan Wang, Yang Wu.

Authors and Affiliations

  1. CAS Key Laboratory of Microscale Magnetic Resonance and School of Physical Sciences, University of Science and Technology of China, Hefei, China

    Yunhan Wang, Yang Wu, Xiangyu Ye, Chang-Kui Duan, Ya Wang, Xing Rong & Jiangfeng Du

  2. Hefei National Laboratory, University of Science and Technology of China, Hefei, China

    Yunhan Wang, Chang-Kui Duan, Ya Wang, Xing Rong & Jiangfeng Du

  3. Anhui Province Key Laboratory of Scientific Instrument Development and Application, University of Science and Technology of China, Hefei, China

    Yunhan Wang, Yang Wu, Xiangyu Ye, Chang-Kui Duan, Ya Wang, Xing Rong & Jiangfeng Du

  4. Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing, China

    Haiping Hu

  5. School of Physical Sciences, University of Chinese Academy of Sciences, Beijing, China

    Haiping Hu

  6. Institute of Quantum Sensing and School of Physics, Zhejiang University, Hangzhou, China

    Jiangfeng Du

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Contributions

J.D., X.R. and H.H. proposed the idea. J.D. and X.R. supervised the experiments. X.R., Y. Wu and Yunhan Wang designed the experiments. Yunhan Wang and Y. Wu performed the experiments. X.Y. and Ya Wang prepared the sample. Yunhan Wang, Y. Wu, H.H. and C.-K.D. carried out the calculations. All authors analysed the data, discussed the results and wrote the manuscript.

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Correspondence to Haiping Hu, Xing Rong or Jiangfeng Du.

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Nature Nanotechnology thanks Avik Dutt and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 Schematic of the experimental sequence.

The experimental realization contains three parts: the preparation, evolution under Htot and population measurements. The operations are implemented by microwave, electric field and radio-frequency pulses.

Extended Data Fig. 2 Eigenstates of the non-Hermitian Hamiltonian corresponding to the EP2 at α = 0.39, k1 = k2 = 0.

a-c, Real and imaginary parts of the measured density matrices ρ1exp(Q1) (a), ρ2exp(Q1) (b) and ρ3exp(Q1) (c) of three eigenstates (labelled by 1,2 and 3) obtained by quantum state tomography, with corresponding theoretical predictions ρ1,2,3theo(Q1). Q1 marks the EP2 at α = 0.39, k1 = k2 = 0. All errors shown are one standard deviation with 0.7 million averages.

Source data

Extended Data Fig. 3 Eigenstates of the non-Hermitian Hamiltonian corresponding to the EP2 at α = 1, k1 = 0.46, k2 = − 1.06.

a-c, Real and imaginary parts of the measured density matrices ρ1exp(Q2) (a), ρ2exp(Q2) (b) and ρ3exp(Q2) (c) of three eigenstates (labelled by 1,2 and 3) obtained by quantum state tomography, with corresponding theoretical predictions ρ1,2,3theo(Q2). Q2 marks the EP2 at α = 1, k1 = 0.46, k2 = − 1.06. All errors shown are one standard deviation with 0.7 million averages.

Source data

Extended Data Fig. 4 Eigenstates of the non-Hermitian Hamiltonian corresponding to the EP2 at α = 1, k1 = − 0.46, k2 = 1.06.

a-c, Real and imaginary parts of the measured density matrices ρ1exp(Q3) (a), ρ2exp(Q3) (b) and ρ3exp(Q3) (c) of three eigenstates (labelled by 1,2 and 3) obtained by quantum state tomography, with corresponding theoretical predictions ρ1,2,3theo(Q3). Q3 marks the EP2 at α = 1, k1 = − 0.46, k2 = 1.06. All errors shown are one standard deviation with 0.7 million averages.

Source data

Extended Data Table 1 Transformation from τ to σ for the crossings in Fig. (3d)
Full size table

Supplementary information

Supplementary Information (download PDF )

Supplementary Discussion including Sections 1–6.

Source data

Source Data Fig. 3 (download XLSX )

Source data for Fig. 3.

Source Data Fig. 4 (download XLSX )

Source data for Fig. 4.

Source Data Extended Data Fig. 2 (download XLSX )

Source data for Extended Data Fig. 2.

Source Data Extended Data Fig. 3 (download XLSX )

Source data for Extended Data Fig. 3.

Source Data Extended Data Fig. 4 (download XLSX )

Source data for Extended Data Fig. 4.

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Wang, Y., Wu, Y., Ye, X. et al. Non-Hermitian non-Abelian topological transition in the S = 1 electron spin system of a nitrogen vacancy centre in diamond. Nat. Nanotechnol. 20, 873–880 (2025). https://doi.org/10.1038/s41565-025-01913-4

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