Abstract
Non-Hermitian skin effect (NHSE) is a distinctive phenomenon in non-Hermitian systems, marked by the accumulation of eigenstates at system boundaries. While well understood in one dimension, unraveling the NHSE in higher dimensions is challenging due to the diversity of lattice geometries. Here, we present a geometry-adaptive non-Bloch band theory in arbitrary dimensions, through the lens of spectral potential. Our formulation precisely determines the energy spectra and generalized Brillouin zone in the thermodynamic limit, unveiling their geometric dependency. We establish exact spectral relations that elucidate the geometric dependence of non-Bloch spectra and their connection to amoeba spectra. Moreover, we show that critical skin modes in higher dimensions exhibit scale-free localization along the boundary, thereby making the spectrum highly sensitive to the system size and boundary ratios. We demonstrate that these critical skin modes lead to spectral non-convergence and instability. Our findings pave the way toward a unified understanding of NHSE and non-Bloch bands in arbitrary dimensions.
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The data that support this study are available on Zenodo https://doi.org/10.5281/zenodo.17600810.
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The source codes that support this study are available on Zenodo https://doi.org/10.5281/zenodo.17600810.
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Acknowledgements
This work is supported by the National Key Research and Development Program of China (Grants No. 2023YFA1406704 and No. 2022YFA1405800) and National Natural Science Foundation of China (Grant No. 12547107 and No. 12474496).
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H. H. conceived the main idea, and performed the theoretical analysis with Z.-Y. X. and Y. X. Z.-Y. X. and Y. X. did the numerical calculations. All authors contribute to the writing of the paper.
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Xing, ZY., Xiong, Y. & Hu, H. Geometry-adaptive formulation of non-Bloch bands in arbitrary dimensions and spectral instability. Commun Phys (2026). https://doi.org/10.1038/s42005-026-02546-2
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DOI: https://doi.org/10.1038/s42005-026-02546-2
