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We propose a method to parametrically excite low frequency collective modesin an interacting many body system using a Floquet driving at opticalfrequencies with a modulated amplitude. We demonstrate that it can be used todesign plasmonic time-varying media with singular dispersions. Plasmons nearresonance with half the modulation frequency exhibit two lines of exceptionalpoints connected by dispersionless states. Above a critical driving strength,resonant plasmon modes become unstable and arrange themselves in a crystal-likestructure stabilized by interactions and nonlinearities. This new state breaksthe discrete time translational symmetry of the drive as well as thetranslational and rotational spatial symmetries of the system and exhibitssoft, Goldstone-like phononic excitations.

Inducing plasmonic exceptional points and pattern formation with modulated Floquet parametric driving