Fig. 1: Sketch of the impurity model and possible behaviors.
a The Hatano–Nelson model with x = 0, 1, 2, . . ., L the number of lattice sites, e±α the non-reciprocal nearest-neighbor couplings, and impurity couplings μe±α between the sites 0 and L. Periodic and open boundary conditions (PBCs and OBCs) correspond to μ = 1 and 0, respectively, but other μ support qualitatively different phenomena. b Spatial eigenstate distributions ρ(x) = ∣ψx∣2 with different accumulating behaviors (normal and reversed scale-free accumulations, and the non-Hermitian skin effect (NHSE)), with ψx the wave-function value at x. c Complex spectra with \({\rm{Re}}[E]\) and \({\rm{Im}}[E]\) the real and imaginary parts of the energies, distinguishing the four types of eigenstates in (b). d Different regimes across the whole range μ are marked by different accumulation phenomena, with dualities relating strong and weak μ.
