Topological spectral features and tight-binding Hamiltonian in the thermal noise of transmission-line networks

We investigate topological features of Johnson–Nyquist noise in networks of coaxial transmission lines. These networks exhibit well-known topological effects, such as bandgaps and protected edge states. In contrast to other topolectrical circuit systems, the transmission-line networks analyzed herein are intrinsically excited, and since constituent elements are comparable in size to the resonant wavelength, the Hamiltonian formalism is motivated by these distinctions. The noise spectrum can be predicted by a Hamiltonian analogous to that of electrons in a tight-binding model, and a newly introduced variable characterizes the symmetry of each mode and predicts its frequency and width. Certain networks sustain a dark mode that cannot be externally detected or excited, but whose presence is nonetheless guaranteed by the fluctuation–dissipation theorem. When the geometrical uniformity of the network is slightly broken, this previously invisible mode appears at the frequency predicted by the Hamiltonian. Transmission-line networks represent a promising platform for studies of topological physics since the resonant frequency, coupling strength, and connectivity can be easily modified. Owing to chiral symmetry, these eigenmodes are topologically protected in the face of small perturbations arising from component imperfections and intentional symmetry breaking.