Formation of vortex and antivortex polarization states in ferroelectric   films
  An effort based on the kinetic model allowed by Landau-Ginzburg-Devonshire theory is presented here. The formation of vortex, antivortex states in ferroelectric films are reported and theoretically explained. Phase transition in toroidal moment is observed. Vortex-antivortex-vortex (V-AV-V) triplet is generated by applying inhomogeneous transverse static localized fields in x-y plane. A specific boundary condition is used to keep the spirit of vortex state i.e. the net polarization across the boundary is always zero. %This is neither a periodic nor an open circuit electric boundary condition. 
Manas Kumar Roy and Sushanta Dattagupta
cond-mat.mtrl-sci