TOPOLOGICAL CLASSIFICATION OF THE MAGNETIC DOMAIN WALLS OF TUBE-AND ENVELOPE-TYPE

Abstract

The classification of the static magnetic domain wall structures of tube- and envelope-type is made in an unified way using the homotopy theory. The sets of topological classes for such two kinds of magnetic domain walls, GWn and GWn, are corresponding respectively one-by-one to the sets of homotopy classes relative to n + l base points for the S2→S2 and S3→S4 continuous maps. Either GW(n) ro GW(n), therefore, can be constructed into group isomorphic to Z, the additive group of integers. (Then we call them the tube-wall group and the envelope-wall group of type n, respectively). The ‘winding number' introduced by Slon-czewski et al. is considered anew. The sufficient and necessary conditions under which the ‘winding number' is allowed to be taken as the index of tube-wall class are obtained. Finally, the topological classification of the magnetization states with M tube-walls and N envelope-walls coexisting is discussed. It is shown that the set of the corresponding topological classes, GW(M,N), can be constructed into group isomorphic to ZM+N, the M + N dimensional lattice vector group. (It is then referred to as the mix-wall group of type [M, N] ).