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Magnetic hopfions in solids

Rybakov, Filipp N., Kiselev, Nikolai S., Borisov, Aleksandr B., Döring, Lukas, Melcher, Christof, Blügel, Stefan·2022·APL Materials
DOI 10.1063/5.0099942
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Abstract

Hopfions are an intriguing class of string-like solitons, named according to a classical topological concept classifying three-dimensional direction fields. The search for hopfions in real physical systems has been ongoing for nearly half a century, starting with the seminal work of Faddeev. However, so far, realizations in bulk solids are missing. Here, we show that hopfions appear as emergent particles of the classical Heisenberg model with competing exchange interactions. This requires going beyond the model approach used in prior work and deriving a general micromagnetic energy functional directly from a spin-lattice Hamiltonian. We present a definite parameter space in which the existence of hopfions is possible. This opens a concrete vista to combine computational approaches such as density functional theory with material informatics to find magnetic crystals that can host hopfions. As proof of principle, we show how zero-field hopfions can be visualized by the means of off-axis electron holography in a transmission electron microscope.

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Figure 1 Typical structures of hopfions. a , Toroidal hopfion with Hopf index H = 1. The solid colored isolines connect points r of a vector field n ( r ) = (cos Φ sin Θ , sin Φ sin Θ , cos Θ) with fixed values of angular variables, Θ ≡ Θ( r ) and Φ ≡ Φ( r ). The color of the vectors and the corresponding isolines are defined by the angular parameter Φ and together they compose the isosurface Θ = const. b and c , Texture of toroidal hopfion at intersecting planes x =0 and z =0, respectively. d -f , Isosurfaces Θ=const for toroidal hopfion with H =2, two linked toroidallike hopfions with total H =6, and for hopfion with trefoil knot-like shape with H =7.
Figure 1 Typical structures of hopfions. a , Toroidal hopfion with Hopf index H = 1. The solid colored isolines connect points r of a vector field n ( r ) = (cos Φ sin Θ , sin Φ sin Θ , cos Θ) with fixed values of angular variables, Θ ≡ Θ( r ) and Φ ≡ Φ( r ). The color of the vectors and the corresponding isolines are defined by the angular parameter Φ and together they compose the isosurface Θ = const. b and c , Texture of toroidal hopfion at intersecting planes x =0 and z =0, respectively. d -f , Isosurfaces Θ=const for toroidal hopfion with H =2, two linked toroidallike hopfions with total H =6, and for hopfion with trefoil knot-like shape with H =7.CC-BY · From the PDF
Image 2
CC-BY · From the PDF
Figure 3 The energy of hopfions with different Hopf index. The energy of stable 3D solitons solutions as a function of the Hopf index. Hopfions have been numerically calculated in effective spin-lattice model with exchange coupling constants corresponding to three special cases: B = 0, C = 0 and C = 6 B . Inset shows the same dependencies given in logarithmic scale reproducing the power law: E ∼ | H | 3 / 4 . Parameter η ∗ = 389 . 3 · √ A· max( C , 6 B ) is defined from the fit to the numerical calculations. The morphology of solitons corresponding to each point is presented in Fig. 4.
Figure 3 The energy of hopfions with different Hopf index. The energy of stable 3D solitons solutions as a function of the Hopf index. Hopfions have been numerically calculated in effective spin-lattice model with exchange coupling constants corresponding to three special cases: B = 0, C = 0 and C = 6 B . Inset shows the same dependencies given in logarithmic scale reproducing the power law: E ∼ | H | 3 / 4 . Parameter η ∗ = 389 . 3 · √ A· max( C , 6 B ) is defined from the fit to the numerical calculations. The morphology of solitons corresponding to each point is presented in Fig. 4.CC-BY · From the PDF
Figure 4 Morphology of magnetic hopfions. Shown magnetization fields are energy minimizer of corresponding homotopy classes defined by the Hopf index H . Each row illustrates isosurfaces with Θ= π/ 2 ( n z =0). Note, the complexity of the hopfion shape and size increases with the Hopf index. For C =6 B the hopfion with H =6 is similar to the linked tori state, while for C =0 and B =0 the hopfion with H =7 are similar to trefoil knot, see Fig. 1 f.
Figure 4 Morphology of magnetic hopfions. Shown magnetization fields are energy minimizer of corresponding homotopy classes defined by the Hopf index H . Each row illustrates isosurfaces with Θ= π/ 2 ( n z =0). Note, the complexity of the hopfion shape and size increases with the Hopf index. For C =6 B the hopfion with H =6 is similar to the linked tori state, while for C =0 and B =0 the hopfion with H =7 are similar to trefoil knot, see Fig. 1 f.CC-BY · From the PDF
Figure B1. Range of existence of magnetic hopfions in minimal effective model. a , The micromagnetic parameters A , B and C are positive when the reduced parameters for simple cubic lattice ˜ J 2 / ˜ J 1 , ˜ J 3 / ˜ J 1 , and ˜ J 4 / ˜ J 1 , ( ˜ J 1 > 0) belong to the domain restricted by three planes ABC , ABD and ADC . The shaded triangles represent the sections of the domain with fixed ˜ J 2 / ˜ J 1 . b , Alternative representation of the diagram shown in a obtained by projection into the plane of ˜ J 3 / ˜ J 1 and ˜ J 4 / ˜ J 1 . As in a , the triangles corresponds to fixed values of ˜ J 2 / ˜ J 1 . The color map indicates an analytical estimate for the maximal angle δ m between nearest n vectors, see inset. The small arrow for ˜ J 3 / ˜ J 1 =0 indicates the single point found by Bogolubsky [16], for nearly isotropic case.
Figure B1. Range of existence of magnetic hopfions in minimal effective model. a , The micromagnetic parameters A , B and C are positive when the reduced parameters for simple cubic lattice ˜ J 2 / ˜ J 1 , ˜ J 3 / ˜ J 1 , and ˜ J 4 / ˜ J 1 , ( ˜ J 1 > 0) belong to the domain restricted by three planes ABC , ABD and ADC . The shaded triangles represent the sections of the domain with fixed ˜ J 2 / ˜ J 1 . b , Alternative representation of the diagram shown in a obtained by projection into the plane of ˜ J 3 / ˜ J 1 and ˜ J 4 / ˜ J 1 . As in a , the triangles corresponds to fixed values of ˜ J 2 / ˜ J 1 . The color map indicates an analytical estimate for the maximal angle δ m between nearest n vectors, see inset. The small arrow for ˜ J 3 / ˜ J 1 =0 indicates the single point found by Bogolubsky [16], for nearly isotropic case.CC-BY · From the PDF
Figure C1. Construction of infinite hopfion for spatially anisotropic Hamiltonian. a , Spin texture of ordinary 2D vortex with topological charge Q = 4 defined in zOy -plain. b , Oblique 2D vortex obtained from ordinary vortex in a by means of homogeneous rotation of each spin around its own axis parallel to the y -axis. c , Oblique 2D vortex with additional twist of each spin by π/ 2 around the own axis parallel to its asymptotic, n 0 ‖ e z . d -f , Iso-surfaces of toroidal axisymmetric hopfions, Θ = π/ 2. d , Hopfion with H = P · Q =4 constructed by rotation of oblique vortex in b around the z -axis. e , Hopfion with H =8 constructed by rotation of vortex in b around the z -axis and additional rotation of each spin by 2 π ( P = 2) around n 0 . f , Hopfion with H =32, constructed similar to that in e . Inset shows unit cell of the domain wall, with periodicity L 2 and L 3 in two directions. g and h , Schematic representation of the shape (iso-surface) of infinite hopfion. The energy contribution of curved part of hopfion can be neglected when indecies Q and P →∞ .
Figure C1. Construction of infinite hopfion for spatially anisotropic Hamiltonian. a , Spin texture of ordinary 2D vortex with topological charge Q = 4 defined in zOy -plain. b , Oblique 2D vortex obtained from ordinary vortex in a by means of homogeneous rotation of each spin around its own axis parallel to the y -axis. c , Oblique 2D vortex with additional twist of each spin by π/ 2 around the own axis parallel to its asymptotic, n 0 ‖ e z . d -f , Iso-surfaces of toroidal axisymmetric hopfions, Θ = π/ 2. d , Hopfion with H = P · Q =4 constructed by rotation of oblique vortex in b around the z -axis. e , Hopfion with H =8 constructed by rotation of vortex in b around the z -axis and additional rotation of each spin by 2 π ( P = 2) around n 0 . f , Hopfion with H =32, constructed similar to that in e . Inset shows unit cell of the domain wall, with periodicity L 2 and L 3 in two directions. g and h , Schematic representation of the shape (iso-surface) of infinite hopfion. The energy contribution of curved part of hopfion can be neglected when indecies Q and P →∞ .CC-BY · From the PDF

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Authors
Rybakov, F. N., Kiselev, N. S., Borisov, A. B., Döring, L., Melcher, C., & Blügel, S.
Venue
APL Materials
Publisher
AIP Publishing
Published
2022-11-01
Volume
10
Issue
11
ISSN
2166-532X
DOI
10.1063/5.0099942
Cited by
95
References
0
Language
en
Licence
Funders
European Research Council (856538)

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