Fig. 4: Topological state and topological charge vs electric potential phase diagram of skyrmion bag.

From phase-field simulations: a–f Planar views of the polarization distribution after electrical potential with −1 V, −2 V, −3 V, −3.5 V, −5.5 V, and −8 V, respectively. The boundary appears when potential is below −3 V, and inner skyrmions begin to melt into strips when voltage is below −5 V. g Absolute value of topological charge as a function of the absolute value of voltage (electrical potential), demonstrating the variation in topological charge under different voltages. From PFM measurements: h PFM phase images reveal the experimental observation of skyrmion bags containing varying numbers of skyrmions and stripe domains within the bag, depending on the magnitude of the applied bias voltage for same duration of 30 s. i Schematics of the energy landscape for the formation of high-order topologies. j Evolution of the total energy after the application and removal of the external field (−4.5 V) from phase-field simulations, showing that the skyrmion bag and skyrmion have similar total energy, and the external electric field could help overcome the energy barrier between the polar skyrmion and polar skyrmion bag.