Bulk-edge correspondence and long-range hopping in the topological plasmonic chain

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 (BEC). Hence, it is important to gain 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 insights into the phenomena of bulk-edge breakdown. We construct BEC 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 in which BEC is preserved in the topological plasmonic chain, as well as a framework for assessing this phenomenon in other systems.