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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. 2 — a, b Complex-energy spectrum for the central point gap [cf. Fig. 1f] for OBC in the z direction and for a momentum resolution of δk = 2π/800 in both k_x,y-directions (B = 0.2, δ = 1). Red tones highlight states of large surface localization, with the light gray mesh of finer resolution δk = 2π/2400 indicating the band energies along lines parallel to either the k_x or k_y axis. Concretely, we compute the inverse participation ratio (IPR), which sums the wavefunction over the lattice sites along the finite dimension z. A completely delocalized (bulk) state corresponds to vanishing IPR. A single surface band covers the point gap with a regular grid for α = π/2 (a), whereas the presence of an exceptional point at the origin for α = 0 is revealed by a 2π disclination (b) (see also Supplementary Note 3). c, d Summed density profile ∑_i∣ψ_i∣² of the eigenstates in the central point gap for open boundaries in z and y direction indicated with color scale (B = 0.2, δ = 1), showing a surface skin effect. The insets display the corresponding energy spectrum in the complex plane, with the selected states highlighted in red. The single surface band for α = π/2 shows a boundary localization in two opposite corners (c), whereas the exceptional point for α = 0 localizes all surface states in one corner (d). e Chern number (white numbers) evolution for a series of cuts in the surface BZ to illustrate the chiral charge pumping around the point gap for α = 0.9 (B = 0.5, δ = 1). The employed colormap highlights the localization of the respective eigenstates. f Combination of cuts from panel e, zoomed onto the central point gap, highlighting how the single-sheet covering in panels a, b relates to the chiral edge states in e. The inset indicates the employed color scheme for momenta in the surface BZ as well as the cuts used for diagonalization (shaded area). Only bulk states and states localized on the upper surface are displayed (α = 0.9, B = 0.5, δ = 1).