Hopf solitons in modern mathematical physics

Magnets with next-neighbor interaction (and beyond), including frustrated magnets

In [I.L. Bogolubsky "Three-dimensional topological solitons in the lattice model of a magnet with competing interactions", Phys. Lett. A 126, 511 (1988) ref] the hopfion with H=1 was found numerically for the magnet with competing interactions of Heisenberg type on simple cubic lattice, by considering nearest-neighbor interaction, next-nearest-neighbor interaction and fourth-neighbor interaction.

In [P. Sutcliffe "Skyrmion knots in frustrated magnets", arXiv:1705.10966v1; Phys. Rev. Lett. 118, 247203 (2017) ref] the hopfions were found numerically for the phenomenological model of isotropic frustrated magnet where external magnetic field should be applied first to suppress the helical texture of the ground state.

In [F.N. Rybakov, N.S. Kiselev, A.B. Borisov, L. Doring, C. Melcher, S. Blugel "Magnetic hopfions in solids", arXiv:1904.00250v1] the hopfions were found numerically for the general case of competing interactions in cubic Heisenberg magnets with ferromagnetic ground state. A straightforward criterion was suggested in the paper, which makes it possible to immediately conclude on whether stable magnetic hopfions are possible in a given crystal or not. To check the criterion, it is necessary to make typical calculations of energies for static spin spirals, using density functional theory (DFT) methods. Is seems that this criterion should be important for the computational search for realistic magnetic materials hosting true hopfions. The work [Y. Liu, W. Hou, X. Han, J. Zang "Three-Dimensional Dynamics of a Magnetic Hopfion Driven by Spin Transfer Torque", arXiv:2001.00417v1; Phys. Rev. Lett. 124, 127204 (2020) ref] predicts rich dynamics of such hopfions driven by spin transfer torque.

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