A bipartite Kronig–Penney model with Dirac-delta potential scatterers

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, topological chiral symmetry protected edge states are seen to exist in correspondence with the standard quantised Zak phase bulk invariant. This quantisation behaviour may also be observed within a 'gauge'-invariant on-diagonal matrix element of a unit eigenvalue equation. 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 cases in which chiral symmetry is absent are then seen to not host topologically protected edge states, as verified by zero bulk invariants.