Behavior of the Magnetic Vortex and Antivortex under Parity and Topological Charge Conjugation Symmetries

Basing on the novel definition of topological charge conjugation operation, it is proved that a discrete spatial symmetry, the parity transformation acting on the n=2 magnetic vortex (magnetic antivortex) with n the "spin" dimension, is equivalent to a topological charge conjugation operation which switches the magnetic vortex (magnetic antivortex) into the corresponding magnetic antivortex (magnetic vortex) marking the complete equivalence between physics and topology. It is also shown that the out-of-plane magnetization vector forming in a n=3 magnetic vortex in the "curling" configuration is decoupled from the linked Amperian current density vector under the parity transformation but the spatial symmetry is conserved. These results indicate that topological magnetic solitons and anti-solitons in the form of magnetic vortexes and antivortexes, respectively behave differently from other images of classical vortexes in nature.