Comprehensive Description of Spacetime Crystal Bandgaps

The dispersion diagram is a graphical representation of wave scattering that summarizes the most fundamental properties of crystals: it includes all the information on the frequency dependence of the phase and group velocities and on the attenuation of the medium. Here, we provide exact formulas for the positions of spacetime-crystal bandgaps taking into account the asymmetry of the uncoupled harmonics, and unveil the complex nature of the frequencies in these bandgaps, in addition to the usual complex wavenumbers.