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Topology in Magnetism - André Thiaville

Laboratoire de Physique des Solides, Université Paris-Sud, Orsay

After recalling what topology is, the seminal paper by Ernst Feldtkeller [1] that introduced topology in magnetism will be described. His arguments were generalized and systematized, first by Maurice Kléman and coworkers [2], into what is now known as the topological theory of defects for all types of order parameter and all samples’ dimensionality and topology [3]. The tools of this theory can also be used to define the topologically stable continuous magnetic structures, prominent examples being the vortex and the skyrmion [4]. The associated topological number (a.k.a. Chern number, Pontryagin index, skyrmion number, etc.) will be recalled and its signification discussed through examples.
We shall then describe the links of topology with magnetization dynamics, following the work of Alfred Thiele [5]. Finally, introducing chirality, it will be shown that topology is not enough to describe all the relevant characteristics of magnetic textures.

[1] E. Feldtkeller, Mikromagnetisch stetige ung unstetige Magnetisierungskonfigurationen (in german), Z. angew. Phys. 19, 530-536 (1965).
[2] G. Toulouse, M. Kléman, Principles of a classification of defects in ordered media, J. Physique Lett. 37, L149-151 (1976).
[3] N.D. Mermin, The topological theory of defects in ordered media, Rev. Mod. Phys. 51, 591-648 (1979) ; M. Kléman, Points, lines and walls (Wiley, New York, 1983).
[4] T.H.R. Skyrme, A non-linear theory of strong interactions, Proc. Roy. Soc. A 247, 260-278 (1958) ; A unified field theory of mesons and baryons, Nucl. Phys. 31, 556-569 (1962).
[5] A.A. Thiele, Steady-state motion of magnetic domains, Phys. Rev. Lett. 30, 230-233 (1973) ; Application of the gyrocoupling vector and dissipation dyadic in the dynamics of magnetic domains, J. Appl. Phys. 45, 377-393 (1974).