  We theoretically investigated two kinds of density oscillations: the Friedel
oscillation and collective excitation in the silicene and germanene within
random phase approximation, and found that the tunable spin-valley coupled band
structure could lead to some exotic properties in such two phenomena. Based on
the exact analytical and numerical analysis, we demonstrated that the beating
of screened potential as well as the undamped plasmon mode can be taken as
fingerprints of topological phase transition in doped silicene and doped
germanene. Thus our proposal here establishes the connection between the
topological phase transition and the density oscillations that can be accessed
by a variety of experimental techniques.
