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Two-dimensional non-Hermitian skin effect in an ultracold Fermi gas

Nature volume 637, pages 565–573 (2025)Cite this article

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Abstract

The concept of non-Hermiticity has expanded the understanding of band topology, leading to the emergence of counter-intuitive phenomena. An example is the non-Hermitian skin effect (NHSE)1,2,3,4,5,6,7, which involves the concentration of eigenstates at the boundary. However, despite the potential insights that can be gained from high-dimensional non-Hermitian quantum systems in areas such as curved space8,9,10, high-order topological phases11,12 and black holes13,14, the realization of this effect in high dimensions remains unexplored. Here we create a two-dimensional (2D) non-Hermitian topological band for ultracold fermions in spin–orbit-coupled optical lattices with tunable dissipation, which exhibits the NHSE. We first experimentally demonstrate pronounced nonzero spectral winding numbers in the complex energy plane with nonzero dissipation, which establishes the existence of 2D skin effect. Furthermore, we observe the real-space dynamical signature of NHSE in real space by monitoring the centre of mass motion of atoms. Finally, we also demonstrate that a pair of exceptional points are created in the momentum space, connected by an open-ended bulk Fermi arc, in contrast to closed loops found in Hermitian systems. The associated exceptional points emerge and shift with increasing dissipation, leading to the formation of the Fermi arc. Our work sets the stage for further investigation into simulating non-Hermitian physics in high dimensions and paves the way for understanding the interplay of quantum statistics with NHSE.

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Fig. 1: Hermitian and non-Hermitian systems in an optical lattice with spin–orbit coupling (SOC).
Fig. 2: Momentum-dependent Rabi oscillation to resolve the band gap closing.
Fig. 3: Observation of non-Hermitian exceptional points and Fermi arc.
Fig. 4: Signature of NHSE in momentum space.
Fig. 5: Signature of NHSE in real space.

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Data availability

The data that support the findings of this work are available at https://doi.org/10.5281/zenodo.13951808.

References

  1. Yao, S. & Wang, Z. Edge states and topological invariants of non-Hermitian systems. Phys. Rev. Lett. 121, 086803 (2018).

    Article  ADS  CAS  PubMed  MATH  Google Scholar 

  2. Yao, S., Song, F. & Wang, Z. Non-hermitian Chern bands. Phys. Rev. Lett. 121, 136802 (2018).

    Article  ADS  CAS  PubMed  MATH  Google Scholar 

  3. Kunst, F. K., Edvardsson, E., Budich, J. C. & Bergholtz, E. J. Biorthogonal bulk-boundary correspondence in non-Hermitian systems. Phys. Rev. Lett. 121, 026808 (2018).

    Article  ADS  CAS  PubMed  MATH  Google Scholar 

  4. Yokomizo, K. & Murakami, S. Non-Bloch band theory of non-Hermitian systems. Phys. Rev. Lett. 123, 066404 (2019).

    Article  ADS  MathSciNet  CAS  PubMed  MATH  Google Scholar 

  5. Guo, S., Dong, C., Zhang, F., Hu, J. & Yang, Z. Theoretical prediction of a non-Hermitian skin effect in ultracold-atom systems. Phys. Rev. A 106, L061302 (2022).

    Article  ADS  CAS  MATH  Google Scholar 

  6. Zhou, L., Li, H., Yi, W. & Cui, X. Engineering non-Hermitian skin effect with band topology in ultracold gases. Commun. Phys. 5, 252 (2022).

    Article  MATH  Google Scholar 

  7. Li, H. & Yi, W. Dissipative two-dimensional Raman lattice. Phys. Rev. A 107, 013306 (2023).

    Article  ADS  CAS  MATH  Google Scholar 

  8. Wang, S.-X. & Wan, S. Duality between the generalized non-Hermitian Hatano-Nelson model in flat space and a Hermitian system in curved space. Phys. Rev. B 106, 075112 (2022).

    Article  ADS  CAS  MATH  Google Scholar 

  9. Ju, C.-Y. et al. Einstein’s quantum elevator: Hermitization of non-Hermitian Hamiltonians via a generalized vielbein formalism. Phys. Rev. Res. 4, 023070 (2022).

    Article  CAS  Google Scholar 

  10. Chenwei, L., Ren, Z., Zhengzheng, Z. & Qi, Z. Curving the space by non-Hermiticity. Nat. Commun. 13, 2184 (2022).

    Article  MATH  Google Scholar 

  11. Ryo, O., Ryo, T. & Kazuki, Y. Second-order topological non-Hermitian skin effects. Phys. Rev. B 102, 241202 (2020).

    Article  MATH  Google Scholar 

  12. Kohei, K., Masatoshi, S. & Ken, S. Higher-order non-Hermitian skin effect. Phys. Rev. B 102, 205118 (2020).

    Article  ADS  Google Scholar 

  13. Ikhlef, Y., Jacobsen, J. L. & Saleur, H. Integrable spin chain for the SL(2, R)/U(1) black hole sigma model. Phys. Rev. Lett. 108, 081601 (2012).

    Article  ADS  PubMed  MATH  Google Scholar 

  14. Bijan, B. & Sauvik, S. Artificial Hawking radiation, weak pseudo-Hermiticity, and Weyl semimetal blackhole analogy. J. Math. Phys. 63, 122102 (2022).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Ashida, Y., Gong, Z. & Ueda, M. Non-hermitian physics. Adv. Phys. 69, 249–435 (2020).

    Article  ADS  MATH  Google Scholar 

  16. El-Ganainy, R. et al. Non-Hermitian physics and PT symmetry. Nat. Phys. 14, 11–19 (2018).

    Article  CAS  MATH  Google Scholar 

  17. Helbig, T. et al. Generalized bulk-boundary correspondence in non-Hermitian topolectrical circuits. Nat. Phys. 16, 747–750 (2020).

    Article  CAS  MATH  Google Scholar 

  18. Ezawa, M. Non-Hermitian higher-order topological states in nonreciprocal and reciprocal systems with their electric-circuit realization. Phys. Rev. B 99, 201411 (2019).

    Article  ADS  CAS  MATH  Google Scholar 

  19. Zhou, H. et al. Observation of bulk Fermi arc and polarization half charge from paired exceptional points. Science 359, 1009–1012 (2018).

    Article  ADS  MathSciNet  CAS  PubMed  MATH  Google Scholar 

  20. Ghatak, A., Brandenbourger, M., Van Wezel, J. & Coulais, C. Observation of non-Hermitian topology and its bulk-edge correspondence in an active mechanical metamaterial. Proc. Natl Acad. Sci. USA 117, 29561 (2020).

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  21. Liu, S. et al. Non-Hermitian skin effect in a non-Hermitian electrical circuit. Research 2021, 5608038 (2021).

    Article  ADS  PubMed  PubMed Central  MATH  Google Scholar 

  22. Kozii, V. & Fu, L. Non-Hermitian topological theory of finite-lifetime quasiparticles: Prediction of bulk Fermi arc due to exceptional point. Phys. Rev. B 109, 235139 (2024).

    Article  CAS  Google Scholar 

  23. Shen, H., Zhen, B. & Fu, L. Topological band theory for non-Hermitian Hamiltonians. Phys. Rev. Lett. 120, 146402 (2018).

    Article  ADS  MathSciNet  CAS  PubMed  MATH  Google Scholar 

  24. Öztürk, F. E. et al. Observation of a non-Hermitian phase transition in an optical quantum gas. Science 372, 88–91 (2021).

    Article  ADS  PubMed  MATH  Google Scholar 

  25. Bergholtz, E. J., Budich, J. C. & Kunst, F. K. Exceptional topology of non-Hermitian systems. Rev. Mod. Phys. 93, 015005 (2021).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Kawabata, K., Shiozaki, K., Ueda, M. & Sato, M. Symmetry and topology in non-Hermitian physics. Phys. Rev. X 9, 041015 (2019).

    CAS  MATH  Google Scholar 

  27. Zhang, K., Yang, Z. & Fang, C. Universal non-Hermitian skin effect in two and higher dimensions. Nat. Commun. 13, 2496 (2022).

    Article  ADS  CAS  PubMed  PubMed Central  MATH  Google Scholar 

  28. Liang, Q. et al. Dynamic signatures of non-Hermitian skin effect and topology in ultracold atoms. Phys. Rev. Lett. 129, 070401 (2022).

    Article  ADS  CAS  PubMed  Google Scholar 

  29. Weidemann, S. et al. Topological funneling of light. Science 368, 311–314 (2020).

    Article  ADS  MathSciNet  CAS  PubMed  MATH  Google Scholar 

  30. Xiao, L. et al. Non-Hermitian bulk-boundary correspondence in quantum dynamics. Nat. Phys. 16, 761–766 (2020).

    Article  CAS  MATH  Google Scholar 

  31. Nobuyuki, O., Kohei, K., Ken, S. & Masatoshi, S. Topological origin of non-Hermitian skin effects. Phys. Rev. Lett. 124, 086801 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  32. Yang, Z., Zhang, K., Fang, C. & Hu, J. Non-Hermitian bulk-boundary correspondence and auxiliary generalized Brillouin zone theory. Phys. Rev. Lett. 125, 226402 (2020).

    Article  ADS  MathSciNet  CAS  PubMed  MATH  Google Scholar 

  33. Hofmann, T. et al. Reciprocal skin effect and its realization in a topolectrical circuit. Phys. Rev. Res. 2, 023265 (2020).

    Article  CAS  MATH  Google Scholar 

  34. Zhang, W. et al. Observation of non-Hermitian aggregation effects induced by strong interactions. Phys. Rev. B 105, 195131 (2022).

    Article  ADS  CAS  Google Scholar 

  35. Zou, D. et al. Observation of hybrid higher-order skin-topological effect in non-Hermitian topolectrical circuits. Nat. Commun. 12, 7201 (2021).

    Article  ADS  CAS  PubMed  PubMed Central  MATH  Google Scholar 

  36. Shang, C. et al. Experimental identification of the second-order non-Hermitian skin effect with physics-graph-informed machine learning. Adv. Sci. 9, 2202922 (2022).

    Article  MATH  Google Scholar 

  37. Zhang, X., Tian, Y., Jiang, J.-H., Lu, M.-H. & Chen, Y.-F. Observation of higher-order non-Hermitian skin effect. Nat. Commun. 12, 5377 (2021).

    Article  ADS  CAS  PubMed  PubMed Central  MATH  Google Scholar 

  38. Zhou, Q. et al. Observation of geometry-dependent skin effect in non-Hermitian phononic crystals with exceptional points. Nat. Commun. 14, 4569 (2023).

    Article  ADS  CAS  PubMed  PubMed Central  MATH  Google Scholar 

  39. Wang, W., Hu, M., Wang, X., Ma, G. & Ding, K. Experimental realization of geometry-dependent skin effect in a reciprocal two-dimensional lattice. Phys. Rev. Lett. 131, 207201 (2023).

    Article  ADS  CAS  PubMed  MATH  Google Scholar 

  40. Wan, T., Zhang, K., Li, J., Yang, Z. & Yang, Z. Observation of the geometry-dependent skin effect and dynamical degeneracy splitting. Sci. Bull. 68, 2330–2335 (2023).

    Article  MATH  Google Scholar 

  41. Gopalakrishnan, S. & Gullans, M. J. Entanglement and purification transitions in non-Hermitian quantum mechanics. Phys. Rev. Lett. 126, 170503 (2021).

    Article  ADS  MathSciNet  CAS  PubMed  PubMed Central  MATH  Google Scholar 

  42. Okuma, N. & Sato, M. Quantum anomaly, non-Hermitian skin effects, and entanglement entropy in open systems. Phys. Rev. B 103, 085428 (2021).

    Article  ADS  CAS  MATH  Google Scholar 

  43. Kawabata, K., Numasawa, T. & Ryu, S. Entanglement phase transition induced by the non-Hermitian skin effect. Phys. Rev. X 13, 021007 (2023).

    CAS  MATH  Google Scholar 

  44. Song, B. et al. Observation of symmetry-protected topological band with ultracold fermions. Sci. Adv. 4, eaao4748 (2018).

    Article  ADS  PubMed  PubMed Central  Google Scholar 

  45. Liu, X.-J., Liu, Z.-X. & Cheng, M. Manipulating topological edge spins in a one-dimensional optical lattice. Phys. Rev. Lett. 110, 076401 (2013).

    Article  ADS  PubMed  MATH  Google Scholar 

  46. Ren, Z. et al. Chiral control of quantum states in non-Hermitian spin-orbit-coupled fermions. Nat. Phys. 18, 385–389 (2022).

    Article  CAS  MATH  Google Scholar 

  47. Zhang, K., Yang, Z. & Fang, C. Correspondence between winding numbers and skin modes in non-Hermitian systems. Phys. Rev. Lett. 125, 126402 (2020).

    Article  ADS  MathSciNet  CAS  PubMed  MATH  Google Scholar 

  48. Liu, X.-J., Law, K. T. & Ng, T. K. Realization of 2D spin-orbit interaction and exotic topological orders in cold atoms. Phys. Rev. Lett. 112, 086401 (2014).

    Article  ADS  MATH  Google Scholar 

  49. Song, B. et al. Observation of nodal-line semimetal with ultracold fermions in an optical lattice. Nat. Phys. 15, 911–916 (2019).

    Article  CAS  MATH  Google Scholar 

  50. Liang, M.-C. et al. Realization of Qi-Wu-Zhang model in spin-orbit-coupled ultracold fermions. Phys. Rev. Res. 5, L012006 (2023).

    Article  CAS  Google Scholar 

  51. Zhen, B. et al. Spawning rings of exceptional points out of Dirac cones. Nature 525, 354–358 (2015).

    Article  ADS  CAS  PubMed  MATH  Google Scholar 

  52. Yi, Y. & Yang, Z. Non-Hermitian skin modes induced by on-site dissipations and chiral tunneling effect. Phys. Rev. Lett. 125, 186802 (2020).

    Article  ADS  MathSciNet  CAS  PubMed  MATH  Google Scholar 

  53. Wang, H.-Y., Song, F. & Wang, Z. Amoeba formulation of non-Bloch band theory in arbitrary dimensions. Phys. Rev. X 14, 021011 (2024).

    CAS  MATH  Google Scholar 

  54. Zhang, K., Yang, Z. & Sun, K. Edge theory of non-Hermitian skin modes in higher dimensions. Phys. Rev. B 109, 165127 (2024).

    Article  ADS  CAS  Google Scholar 

  55. Asteria, L., Zahn, H. P., Kosch, M. N., Sengstock, K. & Weitenberg, C. Quantum gas magnifier for sub-lattice-resolved imaging of 3D quantum systems. Nature 599, 571–575 (2021).

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  56. Wang, Z. Y., Hong, J. S. & Liu, X. J. Symmetric non-Hermitian skin effect with emergent nonlocal correspondence. Phys. Rev. B 108, L060204 (2023).

    Article  ADS  CAS  MATH  Google Scholar 

  57. Kaluza, T. On the unification problem in physics. Int. J. Mod. Phys. D 27, 1870001 (2018).

    Article  ADS  MATH  Google Scholar 

  58. Klein, O. Quantentheorie und fünfdimensionale Relativitätstheorie. Z. Phys. 37, 895–906 (1926).

    Article  ADS  MATH  Google Scholar 

  59. Klein, O. The atomicity of electricity as a quantum theory law. Nature 118, 516 (1926).

    Article  ADS  CAS  MATH  Google Scholar 

  60. Song, B. et al. Spin-orbit-coupled two-electron Fermi gases of ytterbium atoms. Phys. Rev. A 94, 061604 (2016).

    Article  ADS  Google Scholar 

  61. Murthy, P. et al. Matter-wave Fourier optics with a strongly interacting two-dimensional Fermi gas. Phys. Rev. A 90, 043611 (2014).

    Article  ADS  Google Scholar 

  62. Holten, M. et al. Observation of Cooper pairs in a mesoscopic two-dimensional Fermi gas. Nature 606, 287–291 (2022).

    Article  ADS  CAS  PubMed  MATH  Google Scholar 

  63. Mao, L., Deng, T. & Zhang, P. Boundary condition independence of non-Hermitian Hamiltonian dynamics. Phys. Rev. B 104, 125435 (2021).

    Article  ADS  CAS  MATH  Google Scholar 

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Acknowledgements

This work was mainly supported by the Research Grants Council (RGC) of Hong Kong (RFS2122-6S04) and the National Key Research and Development Program of China (2021YFA1400900), respectively. G.-B.J. also acknowledges support from the RGC through 16306119, 16302420, 16302821, 16306321, 16306922, C6009-20G and N-HKUST636-22. X.-J.L. was further supported by the National Natural Science Foundation of China (grant nos. 12425401, 12261160368 and 11921005), and the Innovation Program for Quantum Science and Technology (grant no. 2021ZD0302000).

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Author notes

  1. These authors contributed equally: Entong Zhao, Zhiyuan Wang

Authors and Affiliations

  1. Department of Physics, The Hong Kong University of Science and Technology, Kowloon, Hong Kong, China

    Entong Zhao, Chengdong He, Ka Kwan Pak, Yu-Jun Liu, Peng Ren & Gyu-Boong Jo

  2. International Center for Quantum Materials, School of Physics, Peking University, Beijing, China

    Zhiyuan Wang, Ting Fung Jeffrey Poon & Xiong-Jun Liu

  3. Hefei National Laboratory, Hefei, China

    Xiong-Jun Liu

  4. International Quantum Academy, Shenzhen, China

    Xiong-Jun Liu

  5. Center for Complex Quantum Systems, The Hong Kong University of Science and Technology, Hong Kong, China

    Gyu-Boong Jo

  6. Department of Physics and Astronomy, Rice University, Houston, TX, USA

    Gyu-Boong Jo

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  1. Entong Zhao
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Contributions

E.Z., C.H., K.K.P., Y.-J.L. and P.R. carried out the experiment. E.Z. analysed the data and helped in numerical calculations. Z.W. and T.F.P. performed the theoretical modelling and numerical calculations. G.-B.J. and X.-J.L. conceived the project and supervised the research. E.Z., Z.W., X.-J.L. and G.-B.J. wrote the paper.

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Correspondence to Xiong-Jun Liu or Gyu-Boong Jo.

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Zhao, E., Wang, Z., He, C. et al. Two-dimensional non-Hermitian skin effect in an ultracold Fermi gas. Nature 637, 565–573 (2025). https://doi.org/10.1038/s41586-024-08347-3

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