Nullclines entanglement induced topological transitions in driven liquid crystal cells

Physical systems can exhibit transitions between states with and without topological properties. These transitions have been observed in conductors to superconductors, fluids to superfluids, chiral magnets, and liquid crystal phases. Likewise, topological transitions have been observed by driving homogeneous liquid crystal cells. However, the mechanism of the formation of topological lattices has not been established. Here, we reveal an out-of-equilibrium topological transition mechanism. A vortex triplet is trapped in a nematic liquid crystal layer by the combined action of a magnetic ring and an oscillating electric field. By lowering the frequency, we observed that the nullcline curves of the vortex touch each other, creating new vortices. This nullcline entanglement process generates the emergence of a vortex lattice. Theoretically, based on a phenomenological amplitude equation valid close to the reorientation instability with the phenomenological inclusion of inertial effects, we observe a trapped vortex, oscillations, and entanglement of its nullcline curves, which generate new vortices and lattices in agreement with experimental findings.