Fig. 2: Onset of the plasmon instability induced by modulated Floquet parametric driving with the modulation frequency 2ω₁. Exceptional points.

Red and blue lines show the the quasi-energy dispersion \({{{{\rm{Re}}}}}\left[\Lambda \left(q\right)\right]\) of parametrically driven plasmons (see Eq. (16) and discussion below). Far away from the critical q^*, 2D plasmons retain the characteristic square-root shape of their dispersion relation (blue lines). The quasi-energy dispersion is invariant with respect to shifts by 2ω₁—the reciprocal lattice constant in frequency space. Near q^*, where \({\omega }_{{{{{\rm{pl}}}}}}\left({q}^{*}\right)\approx {\omega }_{1}\) holds, the dispersion is strongly altered by the driving (red lines). A gap which hosts non-dispersive modes opens around q^* (yellow shading). Merging branches of the dispersion at the edges of the non-dispersive gap indicate exceptional points with diverging group velocities. We chose the modulation amplitude h to lie slightly above the instability threshold: damping is negative in a small interval around q^*, leading to an exponential growth of unstable modes (red shading). However, exceptional points and non-dispersive states appear even for subcritical driving strengths.