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

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.

Emergent Non-Hermitian Edge Polarisation in an Hermitian Tight-binding Model

“…This therefore mimics the system as initially studied in Ref. 37. The second is to consider the opposite case in which the distances are held constant whilst varying the baseline potentials between the delta potentials.…”

“…Then, in order to mimic Ref. 37, we take natural units of = m = 1, a Dirac-delta strength of V = −10 and vary the distances between the Dirac-deltas as v = a and w = d − a such that the unit-cell length, which will be taken to be d = 1, remains constant. Then, using Eq.…”

“…Yet, as will be seen, this non-Hermiticity is highly non-trivial with respect to the behaviours of the edge states. Indeed, it must be stressed that this to-be-seen behaviour is a physical effect of the system and not an artefact of the tight-binding approximation; the exact solution to the problem exhibits this same phenomenon 37 .…”

“…The system we consider herein is a bipartite Kronig-Penney 34,35 model with negative-strength Dirac-delta potentials, as shown in figure 1, that is constructed in order to effectively mimic the SSH model 36,37 . The bipartition may be achieved by either varying the distances or the baseline potentials between the two Dirac-deltas within and without the unit-cell.…”

Topological Surface-plasmon-polaritons on Corrugated Metal-dielectric Surfaces

“…As such, they may only exist at a dielectric-metal interface whereat the sign of the dielectric function changes; a fact which has been recently shown to be of topological origin 37 . Work has been conducted on the appearance of SPPs at the surface of 3D topological insulators [38][39][40][41] whilst study into their own potential topological characteristics has begun in ernest [42][43][44] . Herein, we consider an extended corrugated surface that is made to be bipartite in nature through a periodic variation of the radii of curvature between neighbouring peaks and troughs.…”