Abstract

D.M.Whittaker and M.M. McCarthy We show that disordered networks of coaxial cables can exhibit very high electromagnetic transmission (theoretically 100%) at certainfrequencies. This high transmission is a signature that a network is at a boundary between two topological phases. We make our networks by connecting together sections of coaxial cable, all with the same electrical length. They can be disordered either byrandomness in these connections, or by randomly including cables with different electrical impedances. In this talk, I focus on linearstructures, made by connecting, end-to-end, sections of cable with 50 and 93 ohm impedance. Using a vector network analyser, we can measuretransmission through the structure and the local density of states at any point. For certain sequences of impedance, we find that that thetransmission at a particular frequency, which we call the chiral frequency, is close to 100%. Our key theoretical result is to show that a network can be represented by a tight-binding matrix Hamiltonian describing thevoltages at the cable junctions, with the hopping matrix elements determined by the impedances of the corresponding cables. As theseHamiltonians exhibit chiral (sublattice) symmetry, one of the central symmetries of random matrix theory, the networks are an excellentsystem in which to investigate topological physics. Our random linear structures map onto the SSH (Su-Schrieffer-Heegler) model with disordered hopping. The chiral frequency corresponds to thezero of energy in the model. Disordered SSH chains can exist in two topological phases, depending on the sequence of impedances. By                                     swapping cables around, we can quickly investigate the topological phase space, using transmssion and density of states measurements as aprobe. To distinguish the phases, we introduce a topological invariant related to the eigenvalues of the transfer matrix describingtransmission through a structure. When there is a unit eigenvalue at the chiral frequency, the structure is intermediate between the twophases. However, this is also the condition for perfect transmission, which establishes the connection between a structure beingtopologically marginal and exhibiting high transmssion.