Nonhermitian topological zero modes at smooth domain walls: Exact solutions

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 eigenmode of a modified Jackiw-Rebbi equation. Recently, the bulk-boundary correspondence has been extended to nonhermitian systems, which describe physical phenomena such as gain and loss in open and non-equilibrium systems. Nonhermitian 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 nonhermitian systems can be continuously deformed into hermitian gapped spectra. Here, we find the analytical form of the wavefunctions of nonhermitian boundary modes with zero energy localized at smooth domain boundaries between topologically distinct phases, by solving the generalized Jackiw-Rebbi equation in the nonhermitian 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 nonhermitian line-gapped systems in terms of experimentally measurable physical quantities and is not affected by the details of the spatial dependence of the scalar fields. These findings shed some new light on the localization properties of boundary modes in nonhermitian and topologically nontrivial states of matter.