Phase diagram of magnetic $S^3$ Skyrmions on three-dimensional lattices and the toroidal antiSkyrmion

Abstract: Magnetic Skyrmions are planar solitons stabilized by the Dzyaloshinskii-Moriya interaction (DMI) and realized in chiral magnets. We study their natural three-dimensional generalization: a sigma model from $\mathbb{R}^3$ to $S^3$ with a four-component magnetization vector, stabilized by a one-derivative term which is a generalized DMI. We utilize two SO(3)-invariant generalized DMIs discovered recently: an "$α$-term" supporting a spherically symmetric hedgehog Skyrmion and a "$β$-term" supporting an axially symmetric Skyrmion that splits into two half-Skyrmions connected by a magnetic string of negative tension, a phenomenon we call "anti-confinement". We derive a cubic-lattice discretization that reproduces both continuum theories at long wavelengths and use Monte Carlo simulations to map the finite-temperature phase diagram. We identify spin-spiral, magnetic-string-lattice, Skyrmion-lattice, and antiSkyrmion-lattice phases, as well as a mixed-topology regime with fractional $S^3$ charges localized at string bends. We find, for the first time in the literature to the best of our knowledge, a toroidal (anti-)soliton of unit charge. Our results establish a theoretical and computational framework for three-dimensional topological magnetic textures in systems whose order-parameter manifold is $S^3$.