Zero energy modes with Gaussian, exponential, or polynomial decay: Exact solutions in hermitian and nonhermitian regimes

TL;DR

This paper derives exact analytical solutions for zero energy modes with various decay profiles in both hermitian and nonhermitian systems, enhancing the understanding of topological boundary modes in insulators and superconductors.

Contribution

It introduces inverse methods to explicitly solve for zero modes with Gaussian, exponential, and polynomial decay, extending the analysis to nonhermitian regimes.

Findings

Explicit forms of zero modes with different decay behaviors

Asymptotic mass term determines mode decay properties

Extension of solutions to nonhermitian systems

Abstract

Topological zero modes in topological insulators or superconductors are exponentially localized at the phase transition between a topologically trivial and nontrivial phase. These modes are solutions of a Jackiw-Rebbi equation modified with an additional term which is quadratic in the momentum. Moreover, localized fermionic modes can also be induced by harmonic potentials in superfluids and superconductors or in atomic nuclei. Here, by using inverse methods, we consider in the same framework exponentially-localized zero modes, as well as Gaussian modes induced by harmonic potentials (with superexponential decay) and polynomially decaying modes (with subexponential decay), and derive the explicit and analytical form of the modified Jackiw-Rebbi equation (and of the Schr\"odinger equation) which admits these modes as solutions. We find that the asymptotic behavior of the mass term is crucial in determining the decay properties of the modes. Furthermore, these considerations naturally extend to the nonhermitian regime. These findings allow us to classify and understand topological and nontopological boundary modes in topological insulators and superconductors.