Kiral mıknatıslar ve kuantum dolaşmasında iki katmanlı skyrmiyonlar

The antiferromagnetic coupling and entanglement between skyrmion lattices in magnetic bilayer systems are studied in thesis. We first formulate the problem of large bilayer skyrmions using $\mathbb{CP}^{1}\otimes \mathbb{CP}^{1}$ theory. We have considered bilayer skyrmions under the presence of Dzyaloshinskii-Moriya (DMI) and Zeeman interactions confined in a two-dimensional chiral magnet such as Fe$_{0.5}$Co$_{0.5}$Si. We parametrize the bilayer skyrmions using $SU(4)$ representation, and represent each skyrmion and antiskyrmion using Schmidt decomposition. Moreover, we computed the reduced density matrices for skyrmion and antiskyrmion. The conditions for maximal, partial entanglement and separable bilayer skyrmions are presented. We find intimate connection between bilayer skyrmions and $SU(4)$ skyrmions in multicomponent Hall systems and Graphene from entanglement perspective. Our results can be used for generating entanglement in systems with large number of spins.