  Using first-principles based effective Hamiltonian and finite temperature
Monte Carlo simulations we investigate cooperative responses, as well as
microscopic mechanism for vortex switching, in zero-dimensional
Pb(Zr$_{0.5}$Ti$_{0.5}$)O$_3$ nanoparticles under curled electric fields. We
find that the generally accepted domain coexistence mechanism is not valid for
toroid switching. Instead dipoles are shown to display unusual collective
behaviors by forming a new vortex with perpendicular (but not opposite) toroid
moment. The strong correlation between the new and original vortices is
revealed to be critical for reversing toroid moment. Microscopic insight for
the puzzling collective response is discussed. Based on our finding, we further
describe a technological approach that is able to drastically reduce the
magnitude of the curled electric field needed for vortex switching.
