Exceptional deficiency of non-Hermitian systems

Nature Physics (2026)Cite this article Exceptional points are non-Hermitian singularities associated with the coalescence of individual eigenvectors accompanied by the degeneracy of their complex energies. Since their discovery, exceptional points have attracted much interest and enabled numerous advanced applications, including sensing. However, accessing exceptional points generally requires delicate parameter tuning, and any related phenomena are intrinsically restricted to a very narrow bandwidth. Here we report a generalization of the concept of an exceptional point called an exceptional deficiency, which features the complete coalescence of entire eigenspaces with identical but arbitrarily large dimensions and the coincidence of entire spectral continua. We find that an exceptional deficiency can induce the anomalous absence or presence of the non-Hermitian skin effect, which transcends the established topological bulk–edge correspondence, resulting in unexpected synergistic skin-propagative dynamics. These phenomena are experimentally observed using active mechanical lattices. We further explore how exceptional deficiencies offer a route to the reliable and flexible control of localization and propagation and how they enable a high-sensitivity broadband sensor. The experimental demonstration of exceptional deficiency provides a new perspective on non-Hermitian physics and may impact related applications, such as sensing, modal control and lasing.This is a preview of subscription content, access via your institution Access Nature and 54 other Nature Portfolio journals Get Nature+, our best-value online-access subscription $32.99 / 30 days cancel any timeSubscribe to this journal Receive 12 print issues and online access $259.00 per yearonly $21.58 per issueBuy this articleUSD 39.95Prices may be subject to local taxes which are calculated during checkoutThe data presented in this Article are available via Zenodo at https://doi.org/10.5281/zenodo.19494159 (ref. 54), and from the corresponding authors upon reasonable request. Source data are provided with this paper.The code used to generate Figs. 1–6 and Extended Data Fig. 2 is available in Source Data. 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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).Download referencesThis work was supported by the National Key R&D Program (Grant Nos. 2022YFA1404400, 2023YFA1407500), the National Natural Science Foundation of China (Grant Nos. T2525002, 12322405, 12104450), the Hong Kong Research Grants Council (Grant Nos. RFS2223-2S01, 12301822, 12300925 and JRFS2526-2S07) and the Hong Kong Baptist University (Grant Nos. RC-RSRG/23-24/SCI/01 and RC-SFCRG/23-24/R2/SCI/12). K.S. and M.S. were supported by JST CREST (Grant No. JPMJCR19T2). M.S. was supported by the JSPS (KAKENHI Grant Nos. JP24K00569, JP25H01250). K.S. was supported by JST SPRING (Grant No. JPMJSP2110) and the JSPS (KAKENHI Grant No. JP25KJ1632). Z.L. is grateful to Z. Feng for his valuable suggestions on figure organization. K.S., M.S. and G.M. thank the Simons Center for Geometry and Physics at Stony Brook University for its hospitality.Department of Physics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, ChinaZhen Li, Rundong Cai, Xulong Wang, Congwei Lu & Guancong MaCenter for Gravitational Physics and Quantum Information, Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, JapanKenji Shimomura & Masatoshi SatoDepartment of Physics, Xiamen University, Xiamen, ChinaZhesen YangAsia Pacific Center for Theoretical Physics, Pohang, Republic of KoreaZhesen YangShenzhen Institute for Research and Continuing Education, Hong Kong Baptist University, Shenzhen, ChinaGuancong MaSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarG.M. conceived and supervised the research. Z.L., K.S., C.L., Z.Y., M.S. and G.M. performed the theoretical analysis. Z.L., R.C. and X.W. performed the experiments. All authors analysed and discussed the results and contributed to the writing of the article.Correspondence to Zhesen Yang, Masatoshi Sato or Guancong Ma.The authors declare no competing interests.Nature Physics thanks the anonymous reviewers for their contribution to the peer review of this work.Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.a, The experimental lattice. The white box highlights a unit cell. b, The schematic diagram of the unit cell in system-II. The blue and orange lines represent reciprocal hopping realized by tensioned springs, and non-reciprocal hopping terms, which are facilitated by feedback-controlled DC motors, are denoted by the red arrows. c, Schematic of a single oscillator, which is composed of a brushless DC motor, a rotational arm, two springs, a position sensor, and a microcontroller board.a, schematic diagram of the design. Here, the ED lattice is the OBC system-I in the main text. The sensor detects through the variation of \(C\propto {\kappa }_{2}\), which is the interchain hopping from chain-A to B. Two leads are connected to the two ends of the lattice, and they produce voltages \({V}_{1}\) and \({V}_{2}\) that are proportional to the local moduli of the modes. When no disturbance is detected, \({\kappa }_{2}=0\) and the lattice is at ED, so all modes are extended modes (Fig. 2e) with moduli symmetric about the center of the lattice. So the voltage outputs are identical, that is, \({V}_{1}/{V}_{2}=1\). When a disturbance is detected by \(C\), it drives the lattice away from ED by introducing a non-zero \({\kappa }_{2}\). Leftward localized non-Hermitian skin modes immediately emerge in the lattice, such that \({V}_{1}/{V}_{2} > 1\). This change is picked up by the voltage divider. b, the performance of the ED sensor computed using the parameters in the main text. The device can pick up extremely small changes in \({\kappa }_{2}\) (at the order of 10−9), and maintains performance over a large dynamic range of \(\sim {10}^{10}\) (output start to plateau near \({\kappa }_{2}\cong 10\)). For example, a small downward interchain hopping at the order of 10−5 produces a change in wavefunction ratio at the order of 10. This performance exceeds traditional EP sensors, which only have high sensitivity in the immediate vicinity of the EP. Also, because all modes contribute to the detection, the sensor operates over the entire bandwidth spanned by the continuous bands of the lattice, unlike EP sensors that must function at the frequency of the EP. c, the distribution of the summed eigenmodes at selected \({\kappa }_{2}\), where \(\bar{\phi }=\frac{1}{2N-4}{\sum }_{m=1}^{2N-4}\left|{\phi }_{m}\right|\), where the summation runs over all modes in the continuous bands.Source dataSupplementary Figs. 1–11 and Discussion.This video presents the dynamical effects associated with system I.This video presents the dynamical effects associated with system II.Data and code.Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.Reprints and permissionsLi, Z., Cai, R., Wang, X. et al. Exceptional deficiency of non-Hermitian systems. Nat. Phys. (2026). https://doi.org/10.1038/s41567-026-03259-7Download citationReceived: 06 May 2025Accepted: 18 March 2026Published: 28 April 2026Version of record: 28 April 2026DOI: https://doi.org/10.1038/s41567-026-03259-7Anyone you share the following link with will be able to read this content:Sorry, a shareable link is not currently available for this article. Provided by the Springer Nature SharedIt content-sharing initiative