Skyrmion

In particle theory, the skyrmion (/ˈskɜrmi.ɒn/) 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 magnetic 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 north-pole spin is mapped onto a far-off edge circle of a 2D-disk, while the negative south-pole 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

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

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

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