  We study a bipartite linear chain constituted by spherical metallic
nanoparticles, where each nanoparticle supports a localized surface plasmon.
The near-field dipolar interaction between the localized surface plasmons gives
rise to collective plasmons, which are extended over the whole nanoparticle
array. We derive analytically the spectrum and the eigenstates of the
collective plasmonic excitations. At the edge of the Brillouin zone, the
spectrum is of a pseudorelativistic nature similar to that present in the
electronic band structure of polyacetylene. We find the effective Dirac
Hamiltonian for the collective plasmons and show that the corresponding spinor
eigenstates represent one-dimensional Dirac-like massive bosonic excitations.
Therefore, the plasmonic lattice exhibits similar effects to those found for
electrons in one-dimensional Dirac materials, such as the ability for
transmission with highly suppressed backscattering due to Klein tunneling. We
also show that the system is governed by a nontrivial Zak phase, which predicts
the manifestation of edge states in the chain. When two dimerized chains with
different topological phases are connected, we find the appearance of the
bosonic version of a Jackiw-Rebbi midgap state. We further investigate the
radiative and nonradiative lifetimes of the collective plasmonic excitations
and comment on the challenges for experimental realization of the topological
effects found theoretically.
