nLab Hopfion

Idea

By Hopfions one refers to those solitons on 3-dimensional Euclidean space $\mathbb{R}^3$ whose topological configurations are classified by homotopy classes of maps to the (Riemann) 2-sphere $S^2 \simeq \mathbb{C}P^1$, hence (by the defining vanishing at infinity of solitons) by homotopy classes of pointed maps from the one-point compactification $\mathbb{R}^3_{\cup\{\infty\}} \simeq S^3$ to $S^2$, hence by the 3rd homotopy group of $S^2$, which is the integers

$$\pi_3(S^2) \,\simeq\, \mathbb{Z} \,.$$

Since the unit generator of this group is represented by the class of the Hopf fibration $h \colon S^3 \longrightarrow S^2$ this means that Hopfions have classifying maps given by (multiples of) the Hopf fibration, whence the name.

To note the difference to but similarity with Skyrmions in 3d, which are instead classified by maps $\mathbb{R}^3_{\cup \{\infty\}} \longrightarrow S^3$ to the 3-sphere, and to magnetic Skyrmions in 2d, which are instead classified by maps $\mathbb{R}^2_{\cup \{\infty\}} \longrightarrow S^2$ — albeit both also being classified by integers, now via the Hopf degree theorem: $\pi_n(S^n) \simeq \mathbb{Z}$.

If one identified the “core” locus of a Hopfion with the preimage under its classifying map of the antipode of the chosen base point in $S^2$, then this is, by the Pontrjagin theorem, the (cobordism class) of a normally framed submanifold of codimension=2 hence of dimension=1, hence is (the cobordism class) of a (framed) link.

The traditional Langrangian field realization of Hopfions are unit-norm vector fields in the Skyrme-Fadeev model (Fadeev 1975, Fadeev & Niemi 1997, review in Fadeev 2002, Manton & Sutcliffe 2004 §9.11).

Another field theory where such maps appear is the 3-dimensional $\mathbb{C}P^1$ sigma-model (cf. Radu, Tchrakian & Yang 2013).

Related concepts

• Hopf-Wess-Zumino term

• Skyrmion

References

The original articles on the (Skyrme-)Fadeev model:

• Ludwig D. Faddeev: Quantization of Solitons, talk at ICHEP 76 (1975) [inSpire:2631]

• Ludwig D. Faddeev, Antti J. Niemi: Knots and Particles, Nature 387 (1997) 58–61 [arXiv:hep-th/9610193, doi:10.1038/387058a0]

Related early discussion identifying Hopfions in 1+2 dimensions with anyon worldlines:

• Frank Wilczek, Anthony Zee: Linking Numbers, Spin, and Statistics of Solitons, Phys. Rev. Lett. 51 (1983) 2250 [doi:10.1103/PhysRevLett.51.2250]

Review:

• Ludwig D. Faddeev: Knotted solitons, Proceedings of the ICM Beijing 2002, vol. 1, Higher Education Press (2002) 235-244 [arXiv:math-ph/0212079, pdf]

• Paul Jennings: The Skyrme-Faddeev model, talk notes (2014) [pdf, pdf]

Textbook account:

• Nicholas Manton, Paul Sutcliffe: Topological Solitons, Cambridge University Press (2004) [doi:10.1017/CBO9780511617034, inSpire:660150, pdf]

See also:

• Wikipedia: Hopfion

Further discussion:

• Paul Sutcliffe: Knots in the Skyrme-Faddeev model, Proc. R. Soc. A 463 (2007) 3001–3020 [arXiv:0705.1468, doi:10.1098/rspa.2007.0038]

Relation to homotopy of rational maps:

• Wang-Chang Su: Faddeev-Skyrme Model and Rational Maps, Chin. J. Phys. 40 (2002) 516-525 [arXiv:hep-th/0107187, inSpire:560464]

Relation to the $\mathbb{C}P^1$ sigma-model:

• Eugen Radu, D. H. Tchrakian, Yisong Yang: Abelian Hopfions of the $\mathbb{CP}^n$ model on $\mathbb{R}^{2n+1}$ and a fractionally powered topological lower bound, Nuclear Physics B 875 2 (2013) 388-407 [arXiv:1305.4784, doi:10.1016/j.nuclphysb.2013.07.006]

Variants:

• Yuki Amari, Nobuyuki Sawado: $SU(3)$ Knot Solitons: Hopfions in the $F_2$ Skyrme-Faddeev-Niemi model, Physics Letters B 784 (2018) 294-300 [arXiv:1805.10008, doi:10.1016/j.physletb.2018.08.020]