  We study topologically-protected four-wave mixing (FWM) interactions in a
plasmonic metasurface consisting of a periodic array of nanoholes in a graphene
sheet, which exhibits a wide topological bandgap at terahertz frequencies upon
the breaking of time-reversal symmetry by a static magnetic field. We
demonstrate that due to the significant nonlinearity enhancement and large
lifetime of graphene plasmons in specific configurations, a net gain of FWM
interaction of plasmonic edge states within the topological bandgap can be
achieved with pump power of less than 10 nW. In particular, we find that the
effective waveguide nonlinearity coefficient is about 1.1x10^13 1/(Wm), i.e.,
more than ten orders of magnitude larger than that of commonly used, highly
nonlinear silicon photonic nanowires. These findings could pave a new way for
developing ultra-low-power-consumption, highly-integrated and robust active
photonic systems at deep-subwavelength scale for applications in quantum
communications and information processing.
