# Skyrmion

In particle theory, the skyrmion () is a hypothetical particle related originally1 to baryons. It was described by Tony Skyrme and consists of a quantum superposition of baryons and resonance states.2

Skyrmions as topological objects are also important in solid state physics, especially in the emerging technology of spintronics. A two-dimensional skyrmion, as a topological object, is formed, e.g., from a 3d effective-spin "hedgehog" (in the field of micromagnetics: out of a so-called "Bloch point" singularity of homotopy degree +1) by a stereographic projection, whereby the positive northpole spin is mapped onto a far-off edge circle of a 2d-disk, while the negative southpole spin is mapped onto the center of the disk.

## Mathematical definition

In field theory, skyrmions are homotopically non-trivial classical solutions of a nonlinear sigma model with a non-trivial target manifold topology – hence, they are topological solitons. An example occurs in chiral models 3 of mesons, where the target manifold is a homogeneous space of the structure group

$\left(\frac{SU(N)_L\times SU(N)_R}{SU(N)_\text{diag}}\right)$

where SU(N)L and SU(N)R are the left and right parts of the SU(N) matrix, and SU(N)diag is the diagonal subgroup.

If spacetime has the topology S3×R, then classical configurations can be classified by an integral winding number4 because the third homotopy group

$\pi_3\left(\frac{SU(N)_L\times SU(N)_R}{SU(N)_\text{diag}}\cong SU(N)\right)$

is equivalent to the ring of integers, with the congruence sign referring to homeomorphism.

A topological term can be added to the chiral Lagrangian, whose integral depends only upon the homotopy class; this results in superselection sectors in the quantised model. A skyrmion can be approximated by a soliton of the Sine-Gordon equation; after quantisation by the Bethe ansatz or otherwise, it turns into a fermion interacting according to the massive Thirring model.

Skyrmions have been reported, but not conclusively proven, to be in Bose-Einstein condensates,5 superconductors,6 thin magnetic films7 and also chiral nematic liquid crystals.8

## Skyrmions in an emerging technology

One particular form of the skyrmions is found in magnetic materials that break the inversion symmetry and where the Dzyaloshinskii-Moriya interaction plays an important role. They form "domains" as small as a 1 nm (e.g. in Fe on Ir(111)9). The small size of magnetic skyrmions makes them a good candidate for future data storage solutions. Physicists at the University of Hamburg have managed to read and write skyrmions using scanning tunneling microscopy.10 The topological charge, representing the existence and non-existence of skyrmions, can represent the bit states "1" and "0".

## References

1. ^ At later stages the model was also related to mesons.
2. ^ Wong, Stephen (2002). "What exactly is a Skyrmion?". arXiv:hep-ph/0202250 hep/ph.
3. ^ Chiral models stress the difference between "left-handedness" and "right-handedness".
4. ^ The same classification applies to the mentioned effective-spin "hedgehog" singularity": spin upwards at the northpole, but downward at the southpole.
See also Döring, W. (1968). "Point Singularities in Micromagnetism". Journal of Applied Physics 39 (2): 1006. doi:10.1063/1.1656144.
5. ^ Al Khawaja, Usama; Stoof, Henk (2001). "Skyrmions in a ferromagnetic Bose–Einstein condensate". Nature 411 (6840): 918–20. Bibcode:2001Natur.411..918A. doi:10.1038/35082010. PMID 11418849.
6. ^ Baskaran, G. (2011). "Possibility of Skyrmion Superconductivity in Doped Antiferromagnet K$_2$Fe$_4$Se$_5$". arXiv:1108.3562 cond-mat.supr-con.
7. ^ Kiselev, N. S.; Bogdanov, A. N.; Schäfer, R.; Rößler, U. K. (2011). "Chiral skyrmions in thin magnetic films: New objects for magnetic storage technologies?". Journal of Physics D: Applied Physics 44 (39): 392001. arXiv:1102.2726. Bibcode:2011JPhD...44M2001K. doi:10.1088/0022-3727/44/39/392001.
8. ^ Fukuda, J.-I.; Žumer, S. (2011). "Quasi-two-dimensional Skyrmion lattices in a chiral nematic liquid crystal". Nature Communications 2: 246. Bibcode:2011NatCo...2E.246F. doi:10.1038/ncomms1250. PMID 21427717.
9. ^ Heinze, Stefan; Von Bergmann, Kirsten; Menzel, Matthias; Brede, Jens; Kubetzka, André; Wiesendanger, Roland; Bihlmayer, Gustav; Blügel, Stefan (2011). "Spontaneous atomic-scale magnetic skyrmion lattice in two dimensions". Nature Physics 7 (9): 713–718. doi:10.1038/NPHYS2045. Lay summary (Jul. 31, 2011).
10. ^ Romming, N.; Hanneken, C.; Menzel, M.; Bickel, J. E.; Wolter, B.; Von Bergmann, K.; Kubetzka, A.; Wiesendanger, R. (2013). "Writing and Deleting Single Magnetic Skyrmions". Science 341 (6146): 636–9. doi:10.1126/science.1240573. PMID 23929977. Lay summaryphys.org (Aug 08, 2013).

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