Flat-Band Localization in Electrical Circuits from One to Three Dimensions.

Flat bands exhibit vanishing group velocity and marked sensitivity to lattice geometry, making them a useful setting for studying localization driven by destructive interference. In this work, electrical-circuit simulations are employed to investigate flat-band systems in one, two, and three dimensions. A one-dimensional two-band circuit is first considered, and its flat-band response is characterized through node-to-ground impedance spectra and steady-state voltage distributions. The analysis is then extended to two- and three-dimensional Lieb lattice circuits characterized by sublattice imbalance. In the two-dimensional Lieb circuit, the flat band touches the dispersive bands at a Dirac point, so hybridization with dispersive modes affects the observed localization. Under periodic boundary conditions, wave vector quantization also produces responses that depend on whether the number of unit cells is even or odd. By contrast, in the three-dimensional Lieb circuit, the flat band is spectrally isolated from the dispersive bands, allowing stronger spatial confinement and clearer sublattice selectivity. The one-dimensional, two-dimensional, and three-dimensional models therefore represent three different situations: a singular flat band, a flat band that touches dispersive bands, and a spectrally isolated flat band. Comparing these cases shows how different degeneracy conditions shape impedance responses and localization patterns in electrical circuit systems. At the flat band frequency, the localized voltage response can also be used to generate spatial patterns in both two-dimensional and three-dimensional circuits, pointing to a possible route for spatial mode control of compact localized states in electrical systems.