Swirling of Horizontal Skyrmions into Hopfions in Bulk Cubic Helimagnets
Magnetic hopfions are three-dimensional topological solitons embedded into a homogeneously magnetized background. The internal structure of hopfions is distinguished by the linked preimages—closed loops with a single orientation of the magnetization on the target space <inline-formula><math...
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| Format: | Article |
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MDPI AG
2023-10-01
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| Series: | Magnetism |
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| Online Access: | https://www.mdpi.com/2673-8724/3/4/23 |
| _version_ | 1797380264656109568 |
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| author | Andrey O. Leonov |
| author_facet | Andrey O. Leonov |
| author_sort | Andrey O. Leonov |
| collection | DOAJ |
| description | Magnetic hopfions are three-dimensional topological solitons embedded into a homogeneously magnetized background. The internal structure of hopfions is distinguished by the linked preimages—closed loops with a single orientation of the magnetization on the target space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>S</mi><mn>2</mn></msup></semantics></math></inline-formula>—and is thus characterized by the integer Hopf index <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>Q</mi><mi>H</mi></msub></semantics></math></inline-formula>. Alternatively, hopfions can be visualized as a result of the swirling of two-dimensional bimerons around the direction of an applied magnetic field. Since the bimeron consists of a circular core and an anti-skyrmion crescent, two hopfion varieties can be achieved with either bimeron constituent facing the hopfion interior. In bulk cubic helimagnets, however, the applied magnetic field leads to a spontaneous collapse of hopfions, i.e., the eigen-energy of hopfions has the minimum for zero hopfion radius <i>R</i>. Anti-hopfions with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Q</mi><mi>H</mi></msub><mo>=</mo><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula>, in this case, pass through the intermediate toron state with two-point defects. Here, we demonstrate that the competing cubic and exchange anisotropies inherent in cubic non-centrosymmetric magnets (e.g., in the Mott insulator Cu<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>2</mn></msub></semantics></math></inline-formula>OSeO<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>3</mn></msub></semantics></math></inline-formula>) as a third level of the hierarchy of energy scales following the exchange and Dzyaloshinskii–Moriya interactions, may shift the energy minimum into the region of finite hopfion radii. |
| first_indexed | 2024-03-08T20:34:55Z |
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| language | English |
| last_indexed | 2024-03-08T20:34:55Z |
| publishDate | 2023-10-01 |
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| spelling | doaj.art-801800c2a5e14b56af8d2f38dd49b7f62023-12-22T14:22:05ZengMDPI AGMagnetism2673-87242023-10-013429730710.3390/magnetism3040023Swirling of Horizontal Skyrmions into Hopfions in Bulk Cubic HelimagnetsAndrey O. Leonov0International Institute for Sustainability with Knotted Chiral Meta Matter, Kagamiyama, Higashihiroshima 739-8511, Hiroshima, JapanMagnetic hopfions are three-dimensional topological solitons embedded into a homogeneously magnetized background. The internal structure of hopfions is distinguished by the linked preimages—closed loops with a single orientation of the magnetization on the target space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>S</mi><mn>2</mn></msup></semantics></math></inline-formula>—and is thus characterized by the integer Hopf index <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>Q</mi><mi>H</mi></msub></semantics></math></inline-formula>. Alternatively, hopfions can be visualized as a result of the swirling of two-dimensional bimerons around the direction of an applied magnetic field. Since the bimeron consists of a circular core and an anti-skyrmion crescent, two hopfion varieties can be achieved with either bimeron constituent facing the hopfion interior. In bulk cubic helimagnets, however, the applied magnetic field leads to a spontaneous collapse of hopfions, i.e., the eigen-energy of hopfions has the minimum for zero hopfion radius <i>R</i>. Anti-hopfions with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Q</mi><mi>H</mi></msub><mo>=</mo><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula>, in this case, pass through the intermediate toron state with two-point defects. Here, we demonstrate that the competing cubic and exchange anisotropies inherent in cubic non-centrosymmetric magnets (e.g., in the Mott insulator Cu<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>2</mn></msub></semantics></math></inline-formula>OSeO<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>3</mn></msub></semantics></math></inline-formula>) as a third level of the hierarchy of energy scales following the exchange and Dzyaloshinskii–Moriya interactions, may shift the energy minimum into the region of finite hopfion radii.https://www.mdpi.com/2673-8724/3/4/23skyrmionhopfionchiral magnetsDzyaloshinskii–Moriya interactionCu<sub>2</sub>OSeO<sub>3</sub> |
| spellingShingle | Andrey O. Leonov Swirling of Horizontal Skyrmions into Hopfions in Bulk Cubic Helimagnets Magnetism skyrmion hopfion chiral magnets Dzyaloshinskii–Moriya interaction Cu<sub>2</sub>OSeO<sub>3</sub> |
| title | Swirling of Horizontal Skyrmions into Hopfions in Bulk Cubic Helimagnets |
| title_full | Swirling of Horizontal Skyrmions into Hopfions in Bulk Cubic Helimagnets |
| title_fullStr | Swirling of Horizontal Skyrmions into Hopfions in Bulk Cubic Helimagnets |
| title_full_unstemmed | Swirling of Horizontal Skyrmions into Hopfions in Bulk Cubic Helimagnets |
| title_short | Swirling of Horizontal Skyrmions into Hopfions in Bulk Cubic Helimagnets |
| title_sort | swirling of horizontal skyrmions into hopfions in bulk cubic helimagnets |
| topic | skyrmion hopfion chiral magnets Dzyaloshinskii–Moriya interaction Cu<sub>2</sub>OSeO<sub>3</sub> |
| url | https://www.mdpi.com/2673-8724/3/4/23 |
| work_keys_str_mv | AT andreyoleonov swirlingofhorizontalskyrmionsintohopfionsinbulkcubichelimagnets |