Topology in finite chiral structures

Topological phenomena in solid state physics are conventionally studied under the assumption of an infinite or large system. This allows the study of topological properties of the bulk, which may be related to observable phenomena with the bulk-boundary correspondence. This has resulted in the very successful prediction of a large variety of unusual and novel phases in such systems. Due to their topological nature, boundary properties of such phases are resistant to disorder, however this is only exact in the infinite limit. For a finite structure there is a finite disorder where boundary phenomena are lost. We study exact topological properties in finite chiral structures, where we wish to allow arbitrary continuous hopping disorder. Our methods allow for studying topology where the bulk-boundary correspondence no longer holds. In particular, we study topological protection and topological phase transitions in such finite media. We propose a definition of topological protection when topologically robust states are not protected by a bulk. We then go on to study topological phase transitions and reveal a rich classification in finite chiral structures when allowing for arbitrary continuous hopping disorder. Allowing some hopping terms to be constrained, we then study sequences of topological phase transitions, where the number of states which close around a gap are iteratively increased. Throughout our work we experimentally verify some of our predictions using a flexible coaxial cable platform to create arbitrary real tight binding Hamiltonians. First we demonstrate this platform by experimentally observing topological protection and a phase transition in a Su-Schrieffer-Heeger chain. We also demonstrate a deep connection between the localisation properties of a topologically marginal finite chiral structure, nut graphs, and core graphs and study nut graphs experimentally, giving an experimental observation of omni-conduction.