Liquid crystal skyrmions as elastic multipoles

Solitons in liquid crystals are spatially localised stable configuration of the liquid crystal orientational order parameter that exhibit emergent particle-like properties such as mutual interaction, translational motion and reconfigurable self-asembling behaviour in oscillating electric fields. Understanding solitons' out-of-equilibrium interactions remains a significant challenge. In this study, we present a minimalist model capable to address this challenge. We derive analytically elastic multipole interaction potentials between two solitons assuming two dimensional LC configurations. We determine the soliton's elastic dipole and quadrupole moments based on the equilibrium director textures obtained as minimizers of the Frank-Oseen free energy functional. Incorporating the time dependence of the soliton's multipole moments in response to electric field oscillations, we carry out particle-based simulations that demonstrate the formation of soliton chains and two dimensional clusters by varying the amplitude and/or the frequency of the electric field, in agreement with experiments. The presented formalism may serve as a benchmark model for future theoretical developments incorporating, e.g. higher order elastic multipoles or/and many body interactions.