  Topological phases such as polar skyrmions have been a fertile playground for
ferroelectric oxide superlattices, with exotic physical phenomena such as
negative capacitance. Herein, using phase-field simulations, we demonstrate the
local control of the skyrmion phase with electric potential applied through a
top electrode. Under a relatively small electric potential, the skyrmions
underneath the electrode can be erased and recovered reversibly. A
topologically protected transition from the symmetric to asymmetric skyrmion
bubbles is observed at the edge of the electrode. While a topological
transition to a labyrinthine domain requires a high applied potential, it can
switch back to the skyrmion state with a relatively small electric potential.
The topological transition from +1 to 0 occurs before the full destruction of
the bubble state. It is shown that the shrinking and bursting of the skyrmions
leads to a large reduction in the dielectric permittivity, the magnitude of
which depends on the size of the electrode.
