Hopf solitons in modern mathematical physics

Chiral magnet

The model description

There are many magnetic crystals without an inversion center, where exchange-relativistic interactions lead to the formation of long-period magnetic structures whose periods are significantly longer than interatomic distances. These structures are spirals and helicoids of a certain chirality. Such materials are commonly called chiral magnets or incommensurate magnets. In [V.G. Bar'yakhtar and E.P. Stefanovsky Fiz. Tverd. Tela 11, 1946 (1969) copy; Sov. Phys. Solid State 11, 1566 (1970)], [P. Bak and M.H. Jensen "Theory of helical magnetic structures and phase transitions in MnSi and FeGe", J. Phys. C 13 L881 (1980) ref] was proposed the energy functional (1.1) (1.2) In [A. Bogdanov "New localized solutions of the nonlinear field equations", JETP Lett. 62, 247 (1995) ref] was proved that Hobart-Derrick theorem does not prohibit the existence of three-dimensional localized solutions for the energy functional {1.1}. The main difference between this model from "Heisenberg model" and "Uniaxial model" that it admits the possibility of a completely static three-dimensional solitons.

The results of recent years

In [A.B. Borisov, F.N. Rybakov "Three-dimensional static vortex solitons in incommensurate magnetic crystals", Low Temp. Phys. 36, 766 (2010) ref; arXiv:1108.4330v1], the authors found numerically static solutions with total Hopf index equal to zero, but the structure consists of pairs "hopfion-antihopfion". Found structures are unstable to some perturbations, because they are solutions of a saddle type, and not local minima. Later authors have reported [A B. Borisov, F.N. Rybakov "Three-dimensional solitons in incommensurate ferromagnets", Int. Workshop "Spin Chirality and Dzyaloshinskii-Moriya Interaction" DMI 2011, St.-Petersburg (2011), Progr. and Abstr., p.40 , ref] on the results of calculations of toroidal hopfion-type texture. But it is also unstable to some perturbations and, therefore, is not a true hopfion.

Numerical simulations shows that hopfion-type textures can be stabilized in confined geometries - films and discs of chiral magnet with additional artificial layers on top/bottom surfaces responsible for pinning of spins [J.-S.B. Tai, I.I. Smalyukh "Hopf solitons and knotted emergent fields in solid-state non-centrosymmetric magnets" arXiv:1806.00453v1 (2018)], [Y. Liu, R. Lake, J. Zang "Binding a Hopfion in Chiral Magnet Nanodisk" arXiv:1806.01682v1 (2018)], [P. Sutcliffe "Hopfions in chiral magnets", arXiv:1806.06458v1 (2018)]. Such solutions are not true hopfions, because stability is due to the geometrical confinement, while solitonic features (i.e. ability of motion, collisions and also other features inherent for particle-like states) are impossible in principle. It is more correct to name the studied states: the imprinting of Hopf fibration.

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