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. 2021 Sep 28;12:5681. doi: 10.1038/s41467-021-25947-z

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© The Author(s) 2021

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

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Fig. 1 — a Schematic band structure near an exceptional point (green). b Real-space cubic lattice model of the 3D TI with s (p) orbitals depicted in blue (red). c Bulk spectrum of the Hermitian 3D TI in the nontrivial phase (1 < M < 3) along one momentum direction k with a superimposed Dirac surface state (blue). d Bulk spectrum of the Hermitian 3D TI at the transition point (M = 3) along one momentum direction k. e Bulk spectrum in the complex plane [Re(E), Im(E)] of the non-Hermitian model in Eq. (1) under PBC for all momenta k_x, k_y, k_z at zero magnetic field (B = 0, M = 2.3, λ = 1, and δ = 0.5). As ∣M − 3∣ > δ, a line gap opens along the imaginary axis, with six hole regions with corresponding point-gap invariants w_3D. f The bulk spectrum for M = 3 and δ = 1 for the non-Hermitian model shows five hole regions, with corresponding point-gap invariants w_3D.