• Bender, C. M. & Boettcher, S. Real spectra in non-Hermitian Hamiltonians having PT symmetry. Phys. Rev. Lett. 80, 5243–5246 (1998).

    Article 
    ADS 

    Google Scholar
     

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

    Article 
    ADS 

    Google Scholar
     

  • Chen, H.-Z. et al. Revealing the missing dimension at an exceptional point. Nat. Phys. 16, 571–578 (2020).

    Article 

    Google Scholar
     

  • Liu, T. et al. Chirality-switchable acoustic vortex emission via non-Hermitian selective excitation at an exceptional point. Sci. Bull. 67, 1131–1136 (2022).

    Article 

    Google Scholar
     

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

    Article 
    ADS 

    Google Scholar
     

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

    Article 
    ADS 

    Google Scholar
     

  • Ding, K., Fang, C. & Ma, G. Non-Hermitian topology and exceptional-point geometries. Nat. Rev. Phys. 4, 745–760 (2022).

    Article 

    Google Scholar
     

  • Bender, C. M. & Hook, D. W. PT-symmetric quantum mechanics. Rev. Mod. Phys. 96, 045002 (2024).

    Article 
    ADS 

    Google Scholar
     

  • Feng, L., El-Ganainy, R. & Ge, L. Non-Hermitian photonics based on parity–time symmetry. Nat. Photonics 11, 752–762 (2017).

    Article 
    ADS 

    Google Scholar
     

  • Özdemir, Ş. K., Rotter, S., Nori, F. & Yang, L. Parity–time symmetry and exceptional points in photonics. Nat. Mater. 18, 783–798 (2019).

    Article 
    ADS 

    Google Scholar
     

  • Miri, M.-A. & Alù, A. Exceptional points in optics and photonics. Science 363, eaar7709 (2019).

    Article 

    Google Scholar
     

  • Huang, L. et al. Acoustic resonances in non-Hermitian open systems. Nat. Rev. Phys. 6, 11–27 (2024).

    Article 
    ADS 

    Google Scholar
     

  • Zhong, Q., Kou, J., Özdemir, Ş. K. & El-Ganainy, R. Hierarchical construction of higher-order exceptional points. Phys. Rev. Lett. 125, 203602 (2020).

    Article 
    ADS 

    Google Scholar
     

  • Wiersig, J. Revisiting the hierarchical construction of higher-order exceptional points. Phys. Rev. A 106, 063526 (2022).

    Article 
    ADS 

    Google Scholar
     

  • Wiersig, J. & Chen, W. Higher-order exceptional points in composite non-Hermitian systems. Phys. Rev. Res. 7, 033034 (2025).

    Article 

    Google Scholar
     

  • Okuma, N. & Sato, M. Non-Hermitian topological phenomena: a review. Annu. Rev. Condens. Matter Phys. 14, 83–107 (2023).

    Article 
    ADS 

    Google Scholar
     

  • Lin, R., Tai, T., Li, L. & Lee, C. H. Topological Non-Hermitian skin effect. Front. Phys. 18, 53605 (2023).

    Article 
    ADS 

    Google Scholar
     

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

    Article 
    ADS 

    Google Scholar
     

  • 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 

    Google Scholar
     

  • 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 

    Google Scholar
     

  • Seyranian, A. P. & Mailybaev, A. A. Multiparameter Stability Theory with Mechanical Applications (World Scientific, 2003).

  • Radjavi, H. & Rosenthal, P. Invariant Subspaces (Springer-Verlag, 1973).

  • Zhang, L. et al. Acoustic non-Hermitian skin effect from twisted winding topology. Nat. Commun. 12, 6297 (2021).

    Article 
    ADS 

    Google Scholar
     

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

    Article 
    ADS 

    Google Scholar
     

  • Wang, W., Wang, X. & Ma, G. Non-Hermitian morphing of topological modes. Nature 608, 50–55 (2022).

    Article 
    ADS 

    Google Scholar
     

  • Wang, W., Wang, X. & Ma, G. Extended state in a localized continuum. Phys. Rev. Lett. 129, 264301 (2022).

    Article 
    ADS 

    Google Scholar
     

  • 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 

    Google Scholar
     

  • Cui, X. et al. Experimental realization of stable exceptional chains protected by non-Hermitian latent symmetries unique to mechanical systems. Phys. Rev. Lett. 131, 237201 (2023).

    Article 
    ADS 

    Google Scholar
     

  • Li, Z. et al. Observation of dynamic non-Hermitian skin effects. Nat. Commun. 15, 6544 (2024).

    Article 
    ADS 

    Google Scholar
     

  • Wang, W., Wang, X. & Ma, G. Anderson transition at complex energies in one-dimensional parity-time-symmetric disordered systems. Phys. Rev. Lett. 134, 066301 (2025).

    Article 
    ADS 

    Google Scholar
     

  • Wiersig, J. Response strengths of open systems at exceptional points. Phys. Rev. Res. 4, 023121 (2022).

    Article 

    Google Scholar
     

  • Wiersig, J. Moving along an exceptional surface towards a higher-order exceptional point. Phys. Rev. A 108, 033501 (2023).

    Article 
    ADS 

    Google Scholar
     

  • Okuma, N., Kawabata, K., Shiozaki, K. & Sato, M. Topological origin of non-Hermitian skin effects. Phys. Rev. Lett. 124, 086801 (2020).

    Article 
    ADS 

    Google Scholar
     

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


    Google Scholar
     

  • Gao, H., Zhu, W., Xue, H., Ma, G. & Su, Z. Controlling acoustic non-Hermitian skin effect via synthetic magnetic fields. Appl. Phys. Rev. 11, 031410 (2024).

    Article 
    ADS 

    Google Scholar
     

  • Teo, H. T., Mandal, S., Long, Y., Xue, H. & Zhang, B. Pseudomagnetic suppression of non-Hermitian skin effect. Sci. Bull. 69, 1667–1673 (2024).

    Article 

    Google Scholar
     

  • Hatano, N. & Nelson, D. R. Localization transitions in non-Hermitian quantum mechanics. Phys. Rev. Lett. 77, 570–573 (1996).

    Article 
    ADS 

    Google Scholar
     

  • Shi, Y. B., Zhang, K. L. & Song, Z. Exceptional spectrum and dynamic magnetization. J. Phys.: Condens. Matter 34, 485401 (2022).

    ADS 

    Google Scholar
     

  • Wang, S. et al. Arbitrary order exceptional point induced by photonic spin–orbit interaction in coupled resonators. Nat. Commun. 10, 832 (2019).

    Article 

    Google Scholar
     

  • Chen, Z. et al. Sound non-reciprocity based on synthetic magnetism. Sci. Bull. 68, 2164–2169 (2023).

    Article 

    Google Scholar
     

  • Su, L. et al. Observation of size-dependent boundary effects in non-Hermitian electric circuits. Chin. Phys. B 32, 038401 (2023).

    Article 
    ADS 

    Google Scholar
     

  • Guo, C.-X. et al. Scale-tailored localization and its observation in non-Hermitian electrical circuits. Nat. Commun. 15, 9120 (2024).

    Article 
    ADS 

    Google Scholar
     

  • Chen, W., Kaya Özdemir, Ş., Zhao, G., Wiersig, J. & Yang, L. Exceptional points enhance sensing in an optical microcavity. Nature 548, 192–196 (2017).

    Article 
    ADS 

    Google Scholar
     

  • Doppler, J. et al. Dynamically encircling an exceptional point for asymmetric mode switching. Nature 537, 76–79 (2016).

    Article 
    ADS 

    Google Scholar
     

  • Feng, L., Wong, Z. J., Ma, R.-M., Wang, Y. & Zhang, X. Single-mode laser by parity-time symmetry breaking. Science 346, 972–975 (2014).

    Article 
    ADS 

    Google Scholar
     

  • Kawabata, K., Bessho, T. & Sato, M. Classification of exceptional points and non-Hermitian topological semimetals. Phys. Rev. Lett. 123, 066405 (2019).

    Article 
    ADS 

    Google Scholar
     

  • Wojcik, C. C., Sun, X.-Q., Bzdušek, T. & Fan, S. Homotopy characterization of non-Hermitian Hamiltonians. Phys. Rev. B 101, 205417 (2020).

    Article 
    ADS 

    Google Scholar
     

  • Li, Z. & Mong, R. S. K. Homotopical characterization of non-Hermitian band structures. Phys. Rev. B 103, 155129 (2021).

    Article 
    ADS 

    Google Scholar
     

  • Tang, W. et al. Exceptional nexus with a hybrid topological invariant. Science 370, 1077–1080 (2020).

    Article 
    ADS 

    Google Scholar
     

  • Tang, W., Ding, K. & Ma, G. Experimental realization of non-abelian permutations in a three-state non-Hermitian system. Natl Sci. Rev. 9, nwac010 (2022).

    Article 
    ADS 

    Google Scholar
     

  • Tang, W., Ding, K. & Ma, G. Realization and topological properties of third-order exceptional lines embedded in exceptional surfaces. Nat. Commun. 14, 6660 (2023).

    Article 
    ADS 

    Google Scholar
     

  • Yang, K. et al. Homotopy, symmetry, and non-Hermitian band topology. Rep. Prog. Phys. 87, 078002 (2024).

    Article 
    ADS 

    Google Scholar
     

  • Lee, C. H. Exceptional bound states and negative entanglement entropy. Phys. Rev. Lett. 128, 010402 (2022).

    Article 
    ADS 

    Google Scholar
     

  • Li, Z. et al. Exceptional deficiency of non-Hermitian systems. Zenodo https://doi.org/10.5281/zenodo.19494159 (2026).