# A Bipartite Kronig-Penney Model with Dirac Potential Scatterers

## TL;DR

This paper extends the classic Kronig-Penney model to include bipartite features, enabling the study of topologically protected edge states through a scattering formalism without tight-binding approximations.

## Contribution

It introduces a bipartite Kronig-Penney model with chiral symmetry, providing a simple 1D system to analyze topological properties via reflection coefficient winding.

## Key findings

- Presence of topologically protected edge states when chiral symmetry is present
- Absence of edge states when chiral symmetry is broken
- Topological invariant linked to the winding of the reflection coefficient

## Abstract

Here we present a simple extension to the age-old Kronig-Penney model, which is made to be bipartite by varying either the scatterer separations or the potential heights. In doing so, chiral (sublattice) symmetry can be introduced. When such a symmetry is present, topologically protected edge states are seen to exist. The solution proceeds through the conventional scattering formalism used to study the Kronig-Penney model, which does not require further tight-binding approximations or mapping into a Su-Schrieffer-Heeger model. The topological invariant for this specific system is found to be the winding of the reflection coefficient, ultimately linked to the system wavefunction. The solution of such a simple and illustrative 1D problem, whose topological content is extracted without requiring further tight-binding approximations, represents the novel aspect of our paper. The cases in which chiral symmetry is absent are then seen to not host topologically protected edge states, as verified by the behaviour of the reflection coefficient and the absence of winding.