Fig. 1: Point gaps from Weyl points and the ensuing Fermi arcs.

a Line and point gaps in one-dimensional (1D) systems. b Point gaps (PGs) formed by bulk and surface states in 2D or 3D systems. \({L}_{r}\) and \({L}_{i}\) respectively denote the \({\mbox{Re}}\left(E\right)\) and \({\mbox{Im}}\left(E\right)\) line gaps, while \({{{\rm{PG}}}}_{\pm }\) stands for the PG with its subscript labeling the eigenvalue winding number herein. c Schematics of the unit cell and Brillouin zone of a representative tight-binding model supporting Weyl points and Fermi arcs (FAs). The lattice constants in all \({xyz}\) directions are set to 1 for simplicity. Two pairs of Weyl points at the \({{\rm{K}}}\) (\({{{\rm{K}}}}^{{\prime} }\)) and \({{\rm{H}}}\) (\({{{\rm{H}}}}^{{\prime} }\)) points with oppositely topological charges when γ =0  labeled in the Brillouin zone by blue and red spheres. d A rectangular block setup with the blue and yellow (gray) surfaces being the zigzag (armchair) boundaries, wherein the blue and yellow solid lines sketch the typical surface states on the FAs. e Schematics of the bulk and surface bands when \(\gamma \ne 0\). The yellow and blue lines represent the FAs, which connect the exceptional point (EP) ring with distinct \({\mbox{Re}}\left(E\right)\) and have the distinct \({\mbox{Im}}\left(E\right)\) as differentiated by color. f PG formed by bulk states and FA surface states at \({k}_{z}=\pm 0.5\pi\). g The equal frequency contour in the \({k}_{x}\)-\({k}_{z}\) plane at \({\mbox{Re}}\left(E\right)=0\) with its color showing Im (E). h The spectral function \(A\left(\omega =0,{{\boldsymbol{k}}}\right)\) in the \({k}_{x}\)-\({k}_{z}\) plane. i Diagrammatical representation of the hinge skin effect under different lifetime layouts at a fixed \({k}_{z}=-0.5\pi\). The orientation and color of the arrows denote the propagation direction and lifetime of the corresponding surface waves. j The corresponding calculated eigenstate. The parameters used are t₀ = −1, \({t}_{z}=-0.5\), and \(\gamma =\)0.15.