Fig. 3: Edge modes in disordered systems and their robustness.

a–d Each inset shows the EPs (black crosses) and the curve of the degeneracy of the real (blue wave curves) and imaginary (red curves) parts of the edge eigenenergies in the complex wavenumber plane. Each inset corresponds to all the pairs of the edge modes with positive and negative group velocities. Because of the periodicity in the y direction, the band structure corresponds to the behavior on the \({\rm{Re}}\ {k}_{y}\) axis in the insets, whereas the EPs and the degeneracy curves in the complex wavenumber space are useful for predicting the behavior of the edge modes. a The main panel shows the obtained exceptional edge modes in the system without disorder. The parameters used are u = −1, c = 0.2, β = 0.14, \(\beta ^{\prime} =0.06\), and γ = 0.05. b The main panel shows the edge dispersion with on-site random real potentials and imaginary disorder in the coupling term. There still exist EPs and edge modes in the bulk energy gap. The noise widths are set to be W = 0.5 (W = 0.02) for the random real on-site potential (the imaginary noise in the non-Hermitian coupling). c The gap is opened and the edge modes no longer exist in the system with random Hermitian couplings. The noise width is set to be W = 0.1. d When we add imaginary on-site potentials, the edge modes are recovered even under the random Hermitian couplings. This is because the wavenumber should be real due to the periodic boundary condition in the y direction and the real axis of the wavenumber plane crosses the degeneracy curve for the real parts of the eigenenergies (cf. the green triangle in the inset). However, EPs disappear from the edge modes. The noise width is the same as in panel c and the strength of the on-site imaginary potential is g = 0.2.