Non-Hermitian Topological Zero Modes at Smooth Domain Walls: Exact Solutions

Abstract The bulk–boundary correspondence predicts the existence of boundary modes localized at the edges of topologically nontrivial systems. The wavefunctions of Hermitian boundary modes can be obtained as the eigenmodes of a modified Jackiw–Rebbi equation. The bulk–boundary correspondence has also been extended to non-Hermitian systems, which describe physical phenomena such as gain and loss in open and nonequilibrium systems. Non-Hermitian energy spectra can be complex-valued and exhibit point gaps or line gaps in the complex plane, whether the gaps can be continuously deformed into points or lines, respectively. Specifically, line-gapped non-Hermitian systems can be continuously deformed into Hermitian gapped spectra. Here, we find the analytical form of the wavefunctions of non-Hermitian boundary modes with zero energy localized at smooth domain boundaries between topologically distinct phases by solving the generalized Jackiw–Rebbi equation in the non-Hermitian regime. Moreover, we unveil a universal relation between the scalar fields and the decay rate and oscillation wavelength of the boundary modes. This relation quantifies the bulk–boundary correspondence in non-Hermitian line-gapped systems through physical quantities that are experimentally measurable. Furthermore, this relation is not affected by the specific spatial variations of the scalar fields. These results offer new insights into the localization properties of boundary modes in non-Hermitian and topologically nontrivial states of matter.