  We investigate the magnetization dynamics of a pair of ferromagnetic
insulators (FMIs) deposited on the surface of a topological insulator (TI). Due
to the nonlinear nature of the underlying physics and intrinsic dynamics, the
FMIs can exhibit oscillatory behaviors even under a constant applied voltage.
The motion of the surface electrons of the TI, which obeys relativistic quantum
mechanics, provides a mechanism of direct coupling between the FMIs. In
particular, the spin polarized current of the TI surface electrons can affect
the magnetization of the two FMIs, which in turn modulates the electron
transport, giving rise to a hybrid relativistic quantum/classical nonlinear
dynamical system. We find robust phase and anti-phase locking between the
magnetization dynamics. As driving the surface electrons of a TI only requires
extremely low power, our finding suggests that nano FMIs coupled by a spin
polarized current on the surface of TI have the potential to serve as the
fundamental building blocks of unconventional, low-power computing paradigms.
