Emergent non-Hermitian edge polarisation in an Hermitian tight-binding model

We study a bipartite Kronig–Penney model with negative Dirac-delta potentials that may be used, amongst other models, to interpret plasmon propagation in nanoparticle arrays. Such a system can be mapped into a Su–Schrieffer–Heeger-like model however, in general, the overlap between 'atomic' wavefunctions of neighbouring sites is not negligible. In such a case, the edge states of the finite system, which retain their topological protection, appear to be either attenuated or amplified. This phenomenon, called "edge polarisation", is usually associated with an underlying non-Hermitian topology. By investigating the bulk system, we show that the resulting tight-binding eigenvalue problem may be made to appear non-Hermitian in this physical 'atomic' (lattice-site) basis. The resulting effective bulk Hamiltonian possesses $PT$-symmetry and its topological invariant, interpreted in terms of a non-Hermitian classification, is found to be given by a bulk winding number of $Z$-type. The observation of edge polarisation is then interpreted as an emerging non-Hermitian skin-effect of the effective bulk Hamiltonian, through the established bulk-boundary correspondence. Therefore, this novel phenomenon of Hermitian edge polarisation occurs as a result of the overlap matrix in a tight-binding model.