Hopf solitons in modern mathematical physics

Incommensurate ferromagnet

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 incommensurate with the crystal-chemical period. 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 solitons with total Hopf index equal to zero, but the structure consists of pairs "hopfion-antihopfion". Found structures are unstable to some perturbations.

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. But it also unstable to some perturbations.

Still unsolved problems

  1. Any analytical soliton solution in 3D
  2. Existence of an absolutely stable static solitons. Necessary to perform the calculations for various types of anisotropies V(n).
  3. Accounting for the effects of magnetic dipole-dipole interaction

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