Classifying the topology of finite chiral structures using complete matchings

We present the theory and experimental demonstration of a topological classification of finite tight binding Hamiltonians with chiral symmetry. Our classification can be applied to randomly connected chiral networks, and regular lattices, giving a significant generalization of the known classification. Using the graph-theoretic notion of complete matchings, we show that many chiral tight binding structures can be divided into a number of sections, each of which has independent topological phases. Hence, the overall classification is NZ2, corresponding to 2N distinct phases, where N is the number of sections with a nontrivial Z2 classification. In our classification, distinct topological phases are separated by exact closures in the energy spectrum of the Hamiltonian, with degenerate pairs of zero-energy states. We show that these zero-energy states have an unusual localization across distinct regions of the structure, determined by the manner in which the sections are connected together. We use this localization to provide an experimental demonstration of the validity of the classification, through radio-frequency measurements on a coaxial cable network, which maps onto a tight-binding system. The structure we investigate is a cable analog of an ideal graphene ribbon, which divides into four sections and has a 4Z2 topological classification.