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

Cooperative response of Pb(ZrTi)O$_3$ nanoparticles to curled electric fields