# Skyrmion

In particle theory, the skyrmion () is a hypothetical particle related originally[1] 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 number[4] 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 films[7] 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 8, 2013).