Topology in Chiral Transmission Line Networks

The aim of the project is to investigate theoretically, and potentially experimentally, the use of networks of transmission lines to study topological physics. We have established that a such a network, most simply made from sections of coaxial cable, maps onto a tight binding Hamiltonian with chiral symmetry, provided that the transmission time through all the sections is the same (and any loops contain an even number of sections). The strengths of the couplings in the Hamiltonian depend on the impedance of the cables used, so we can make structures where the couplings vary, either systematically or randomly, without breaking the chiral symmetry. The fact that the cables are joined by simple connectors makes it possible to assemble and measure a large number of structures in a short time. This makes it an ideal system for exploring topological protection: for example, we have previously demonstrated experimentally the topological protection of an interface state in a one-dimensional SSH model.

The main thrust of the project is to look at other systems which will be accessible to these sorts of measurements. In one dimension, we will look at random sequences of cables of different impedances, where we should be able to see a topological phase transition, with concomitant localisation-delocalisation transition, by measuring the transmission through an ensemble of cables. We will also look at two-dimensional lattices such as graphene, where a similar transition is predicted when vacancies are randomly created in the structure. We can take this further by adding circuitry at the interfaces, for example making, for example, an analogue of a spin-Hall system. One of the features of the cable system is that we are not limited by Euclidean geometry: we can also make physically `impossible' lattices, such as periodic octagonal tilings of the hyperbolic plane, and structures with more than three spatial dimensions.

Though the project is mainly theoretical, we will look to do any promising experiments which arise from the work. These are relatively straightforward, and the measurements can be carried out with a cheap vector network analyser, so there are possible impacts in creating undergraduate experiments demonstrating aspects of bandstructures and topological physics. We are also looking at applications for novel GHz components in telecommunications systems.