  The existence of topologically protected edge modes is often cited as a
highly desirable trait of topological insulators. However, these edge states
are not always present. A realistic physical treatment of long range hopping in
a one-dimensional dipolar system can break the symmetry that protects the edge
modes without affecting the bulk topological number, leading to a breakdown in
bulk-edge correspondence. It is important to find a better understanding of
where and how this occurs, as well as how to measure it. Here we examine the
behaviour of the bulk and edge modes in a dimerised chain of metallic
nanoparticles and in a simpler non-Hermitian next-nearest-neighbour model to
provide some insight into the phenomena of bulk-edge breakdown. We construct
bulk-edge correspondence phase diagrams for the simpler case and use these
ideas to devise a measure of symmetry-breaking for the plasmonic system based
on its bulk properties. This provides a parameter regime for which bulk-edge
correspondence is preserved in the topological plasmonic chain, as well as a
framework for assessing this phenomenon in other systems.
