Band structure engineering and reconstruction in electric circuit networks

We develop an approach to design, engineer, and measure band structures in a synthetic crystal composed of electric circuit elements. Starting from a nodal analysis of a circuit lattice in terms of currents and voltages, our Laplacian formalism for synthetic matter allows us to investigate arbitrary tight-binding models in terms of wave-number-resolved Laplacian eigenmodes, yielding an admittance band structure of the circuit. For illustration, we model and measure a honeycomb circuit featuring a Dirac cone admittance bulk dispersion as well as flat band admittance edge modes at its bearded and zigzag terminations. We further employ our circuit band analysis to measure a topological phase transition in the topolectrical Su-Schrieffer-Heeger circuit.