SQUID metamaterials on a Lieb lattice: From flat-band to nonlinear localization

The dynamic equations for the fluxes through the superconducting quantum interference devices (SQUIDs) that form a two-dimensional metamaterial on a Lieb lattice are derived and then linearized around zero flux to obtain the linear frequency spectrum according to the standard procedure. That spectrum due to the Lieb lattice geometry possesses a frequency band structure exhibiting two characteristic features: two dispersive bands, which form a Dirac cone at the corners of the first Brillouin zone and a flat band crossing the Dirac points. It is demonstrated numerically that localized states can be excited in the system when it is initialized with single-site excitations; depending on the amplitude of those initial states, the localization is either due to the flat-band or to nonlinear effects. Flat-band localized states are formed in the nearly linear regime, whereas localized excitations of the discrete breather type are formed in the nonlinear regime. These two regimes are separated by an intermediate turbulent regime for which no localization is observed. Notably, initial single-site excitations of only edge SQUIDs of a unit cell may end up in flat-band localized states; no such states are formed for initial single-site excitations of a corner SQUID of a unit cell. The degree of localization of the resulting states is in any case quantified using well-established measures, such as the energetic participation ratio and the second moment.