A Topological Classification 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. 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 $N\mathbb{Z}_2$, corresponding to $2^N$ distinct phases, where $N$ is the number of sections with a non-trivial $\mathbb{Z}_2$ 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 that these zero energy states have an unusual localisation across distinct regions of the structure, determined by the manner in which the sections are connected together. We use this localisation 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 analogue of an ideal graphene ribbon, which divides into four sections and has a $4\mathbb{Z}_2$ topological classification.