Fig. 1. Typical spin topology defects in magnetic materials: (a) Domain wall^[42]; (b) flux-closure pattern^[42]; (c) vortex^[43]; (d) anti-vortex^[43]; (e) center-divergent pattern^[43]; (f) center-convergent pattern^[43]; (g) meron^[43,70]; (h) skyrmion^[43,70].

Fig. 2. Typical polar topologies in ferroelectric materials: (a) Polar vortex in nanodisks^[22,71]; (b) polar vortex in nanorods^[22,71]; (c) polar vortex in nanodots^[74]; (d) vortex in BTO nanoislands^[75]; (e) vortex domain in PZT nanodots^[28]; (f) anti-vortex domain in BFO films^[76,77]; (g) center-divergent domain in BFO films^[76-78]; (h) flux-closure pattern in BTO crystal^[64,79]; (i), (j) center-divergent (convergent) domain in BFO nanoislands^[29-31].

Fig. 3. Mobility of flux-closed topological domains in ferroelectric materials: (a) Bundles-like domain structures at the edges of the PZNPT single crystal lamella^[26]; (b) approach, coalesce and separate of the vertices after delivery of a prepoling field pulse^[27].

Fig. 4. Conductivity of polar topological domains in ferroelectric thin films. Creation (a) and conductivity (b) of the flux-closure domain in BFO films^[65,66]; (c) flux- closure domain and center-divergent (convergent) domain in BiFeO₃ films and (d) their conductivity^[24,68].

Fig. 5. Observation of the polar topological domains in ferroelectric thin films: (a) Flux-closure domains in ferroelectric PZT^[34]; (b) vortex domains in ferroelectric BFO ultrathin films^[82]; (c) flux-closure domains in ferroelectric BFO ultrathin films^[37].

Fig. 6. Observation of the polar bubble-like domains in ferroelectric thin films: (a) Polar bubble domains in PZT thin films; (b) structure of the bubble domains; (c) merging and coarsening of the polar bubble domains^[83]; (d) erasuring and recreation of the polar bubble domains^[84].

Fig. 7. Polar topological domains in PTO/STO superlattices: (a) Flux-closure domain arrays in a PTO/STO superlattices on GdScO₃ substrate^[35]; (b) polar vortex domain arrays in PTO/STO superlattices on DSO substrate^[39,90]; (c) a calculated phase diagram for PTO_m/STO_n illustrating the length scales within which different topological states can be stabilized^[40]; (d) polar skyrmion bubbles in a PTO/STO superlattices on STO substrate^[41].

Fig. 8. Topological mixed phase structure and field control in ferroelectric superlattice: (a) Lateral piezoresponse force studies revealing the distribution of a₁/a₂ and vortex phases^[95]; (b) dark field TEM image showing ferroelectric vortices and a₁/a₂-domain coexistence^[96]; (c) phase field model of the a₁/a₂-domain/vortex boundary^[96]; (d) reversible electric-field control of ferroelectric and vortex phases^[95,97]; (e) temperature-dependent synchrotron X-ray diffraction on reversible switching of ferroelectric and vortex phases^[95,97]; (f) reversible sub-picosecond optical pulses control of ferroelectric mixture and supercrystal structure^[95,97].

Fig. 9. Topological mixed phase structure and field control in ferroelectric superlattice: (a) Theoretical guidelines to create polar skyrmions^[98]; (b) topoligical transition between polar vortex and skyrmion in ferroelectric nanocomposites^[88]; (c) phase field model of the topoligical transition between polar vortex and skyrmion in ferroelectric PTO/STO superlattices^[58]; (d) manipulating topological transformations of polar vortices in ferroelectric superlattices^[99].