In particle theory, the skyrmion () is a topologically stable field configuration of a certain class of non-linear sigma models. The term was first used in 1979 to name a model of the nucleon by proposed in 1961 by Tony Skyrme. As a topological soliton in the pion field, it has the remarkable property of being able to model, with reasonable accuracy, multiple low-energy properties of the nucleon, simply by fixing the nucleon radius. It has since found application in solid-state physics, as well as having ties to certain areas of string theory. Skyrmions as topological objects are 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. In a spinor field such as for example photonic or polariton fluids the skyrmion topology corresponds to a full Poincaré beam (a spin vortex comprising all the states of polarization mapped by a stereographic projection of the Poincaré sphere to the real plane). A dynamical pseudospin skyrmion results from the stereographic projection of a rotating polariton Bloch sphere in the case of dynamical full Bloch beams. Skyrmions have been reported, but not conclusively proven, to appear in Bose–Einstein condensates, thin magnetic films, and chiral nematic liquid crystals, as well as in free-space optics. As a model of the nucleon, the topological stability of the skyrmion can be interpreted as a statement that the baryon number is conserved; i.e. that the proton does not decay. The Skyrme Lagrangian is essentially a one-parameter model of the nucleon. Fixing the parameter fixes the proton radius, and also fixes all other low-energy properties, which appear to be correct to about 30%, a significant level of predictive power. Hollowed-out skyrmions form the basis for the chiral bag model (Cheshire Cat model) of the nucleon. The exact results for the duality between the fermion spectrum and the topological winding number of the non-linear sigma model have been obtained by Dan Freed. This can be interpreted as a foundation for the duality between a quantum chromodynamics (QCD) description of the nucleon (but consisting only of quarks, and without gluons) and the Skyrme model for the nucleon. The skyrmion can be quantized to form a quantum superposition of baryons and resonance states. It could be predicted from some nuclear matter properties.
Topological soliton 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 of mesons, where the target manifold is a homogeneous space of the structure group
( SU ( N ) L × SU ( N ) R SU ( N ) diag ) , {\displaystyle \left({\frac {\operatorname {SU} (N)_{L}\times \operatorname {SU} (N)_{R}}{\operatorname {SU} (N)_{\text{diag}}}}\right),}
where SU(N)L and SU(N)R are the left and right chiral symmetries, and SU(N)diag is the diagonal subgroup. In nuclear physics, for N = 2, the chiral symmetries are understood to be the isospin symmetry of the nucleon. For N = 3, the isoflavor symmetry between the up, down and strange quarks is more broken, and the skyrmion models are less successful or accurate. If spacetime has the topology S3×R, then classical configurations can be classified by an integral winding number because the third homotopy group
π 3 ( SU ( N ) L × SU ( N ) R SU ( N ) diag ≅ SU ( N ) ) {\displaystyle \pi _{3}\left({\frac {\operatorname {SU} (N)_{L}\times \operatorname {SU} (N)_{R}}{\operatorname {SU} (N)_{\text{diag}}}}\cong \operatorname {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 quantized model. In (1 + 1)-dimensional spacetime, a skyrmion can be approximated by a soliton of the Sine–Gordon equation; after quantization by the Bethe ansatz or otherwise, it turns into a fermion interacting according to the massive Thirring model.
Lagrangian The Lagrangian for the skyrmion, as written for the original chiral SU(2) effective Lagrangian of the nucleon-nucleon interaction (in (3 + 1)-dimensional spacetime), can be written as
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