Extended Data Fig. 1: Numerical calculation of the field-induced dynamics of toroidal polar topology.

a, The first row displays the time evolution of the polarization field and the spatial distribution of the elevation (out-of-plane) angle δ, which aligns well with experimental results. The second row shows the evolution of charge distributions (σ_b) on DWs and at the droplet’s centre, with corresponding total electric field (E = E₀ + E_depol) donated by orange arrows. The direction of the applied d.c. electric field (E₀) is shown by a red arrow. b, The evolutions of spatial distributions of the depolarization free energy density f_depol, the total electrostatic energy density f_elect, the flexoelectric free energy density f_flexo, and the total free energy density f_total, respectively. c, Time evolutions of the total free energy total (black solid line) and the average out-of-plane angle \(\bar\delta\) of the polarization (dark-yellow solid line) in an area with a clear out-of-plane tilting (circled by a green box in the right-bottom corner figure of (b)). The bottom of (c) shows time evolutions of various integrated free energy terms in the circled region, including the splay elastic energy (blue dotted line), twist elastic energy (red dotted line), bend elastic energy (light-yellow dotted line), polarization gradient energy (purple long-dash line), flexoelectric energy (pink dot-dash line and electric) and electrostatic energy (green scattered triangles connected by a long-dash line). The free energies are defined in Eqs. (S53)−(S55). Details about simulation parameters are provided in Methods and Supplementary Discussion 4.