  We study the topological photonic states in one-dimensional (1-D) lattices
analogue to the Su-Schrieffer-Heeger (SSH) model beyond the dipole
approximation. The electromagnetic resonances of the lattices supported by
near-field interactions between the plasmonic nanoparticles are studied
analytically with coupled dipole--quadrupole method. The topological phase
transition in the bipartite lattices is determined by the change of Zak phase.
Our results reveal the contribution of quadrupole moments to the near-field
interactions and the band topology. It is found that the topological edge
states in non-trivial lattices have both dipolar and quadrupolar nature. The
quadrupolar edge states are not only orthogonal to the dipolar edge states, but
also spatially localized at different sublattices. Furthermore, the quadrupolar
topological edge states, which coexist at the same energy with the quadrupolar
flat band have shorter localization length and hence smaller mode volume than
the conventional dipolar edge states. The findings deepen our understanding in
topological systems that involve higher-order multipoles, or in analogy to the
wave functions in quantum systems with higher-orbital angular momentum, and may
be useful in designing topological systems for confining light robustly and
enhancing light-matter interactions.
