Nonhermitian topological zero modes at smooth domain walls: Exact solutions

TL;DR

This work extends the bulk–boundary framework to nonhermitian, line-gapped systems by deriving analytical zero-energy boundary-mode wavefunctions at smooth domain walls through a nonhermitian generalization of the modified Jackiw–Rebbi equation. It introduces nonhermitian invariants that classify left and right asymptotic regions and establishes precise conditions for the number of zero modes on infinite and semi-infinite domains, linking them to decay rates and oscillation wavelengths that are experimentally measurable. A universal relation between these local mode properties and bulk parameters is derived, surviving even when spatial variations of the domain wall are smooth, thereby broadening the scope beyond sharp interfaces. The results illuminate how zero modes exhibit featureless or hair-like behavior depending on domain-wall width and reveal a robust, measurable connection between bulk topology and localized nonhermitian states with potential implications for photonic, superconducting, and Dirac systems exhibiting gain, loss, or dissipation.

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 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 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 nonhermitian and topologically nontrivial states of matter.

Figures (5)

• Figure 1: Energy spectra of the generalized Jackiw-Rebbi equation with uniform fields $v(x)=v$, $m(x)=m$ in \ref{['eq:Hk']}. (a) Hermitian case with $v,m\in\mathbb{R}$ corresponding to a real energy spectrum with an energy gap. (b) nonhermitian case with $v,m\notin\mathbb{R}$ corresponding to a complex energy spectrum with a line gap.
• Figure 2: Zero modes with short hair ($w<\xi$) on $(-\infty,\infty)$ corresponding to a smooth domain wall given by S-shaped, symmetric or asymmetric Pöschl–Teller, or constant fields ${m}(x)$ and ${v}(x)$. Panels on different rows from top to bottom show the spatial dependence of i) the real and imaginary parts of the wavefunctions, ii) the natural logarithm of the norm of the wavefunctions, iii) the real and imaginary parts of the fields ${m}(x)$ and ${v}(x)$, iv) the quantities ${M}(x)$ and ${K}(x)$. Panels on different columns from left to right correspond to different values of ${m}_{0,1,2}$ and ${v}_{1,2}$, giving (a) single mode with exponentially damped oscillations on the right and exponential decay on the left for an S-shaped field ${m}(x)$ and a complex S-shaped field ${v}(x)$ with $\mathrm{Im}({v}_L)\neq0$ (nonhermitian), (b) single mode with exponentially damped oscillations and nonuniform complex phase on the right and exponential decay on the left for a complex S-shaped field ${m}(x)$ with a nonzero imaginary component (nonhermitian). (c) single mode with exponentially damped oscillations and nonuniform complex phase on the right and exponential decay on the left for an asymmetric Pöschl–Teller field ${m}(x)$ and a complex S-shaped field ${v}(x)$ with $\mathrm{Im}({v}_L)\neq0$ (nonhermitian), (d) single mode with exponentially damped oscillations and nonuniform complex phase on the right and exponential decay on the left for a complex asymmetric Pöschl–Teller field ${m}(x)$ with a nonzero imaginary component (nonhermitian).
• Figure 3: Zero modes with long hair ($w>\xi$) on the interval $(-\infty,\infty)$ with S-shaped, symmetric, or asymmetric Pöschl–Teller, or constant fields ${m}(x)$ and ${v}(x)$ as in \ref{['fig:infinf1']}.
• Figure 4: Zero modes with short hair ($w<\xi$) on the interval $[0,\infty)$ with a smooth domain wall given by S-shaped, symmetric or asymmetric Pöschl–Teller, or constant fields ${m}(x)$ and ${v}(x)$,. Panels on different rows are as in \ref{['fig:infinf1']}. Panels on different columns from left to right correspond to different values of ${m}_{0,1,2}$ and ${v}_{1,2}$ analogous to \ref{['fig:infinf1']}, giving (a) single mode with exponentially damped oscillations with nonuniform complex phase for an S-shaped fields ${m}(x)$ and ${v}(x)$ with $\mathrm{Im}({v}_R)\neq0$ (nonhermitian), (b) single mode with exponentially damped oscillations with nonuniform complex phase for an S-shaped field ${m}(x)$ with a nonzero imaginary component (nonhermitian). (c) single mode with exponentially damped oscillations with nonuniform complex phase for an asymmetric Pöschl–Teller field ${m}(x)$ and S-shaped field ${v}(x)$ with $\mathrm{Im}({v}_R)\neq0$ (nonhermitian), (d) single mode with exponentially damped oscillations with nonuniform complex phase for an asymmetric Pöschl–Teller field ${m}(x)$ with a nonzero imaginary component (nonhermitian).
• Figure 5: Zero modes with long hair ($w>\xi$) on the interval $[0,\infty)$ with S-shaped, symmetric or asymmetric Pöschl–Teller, or constant fields ${m}(x)$ and ${v}(x)$ as in \ref{['fig:0inf1']}.