# Swirling of Horizontal Skyrmions into Hopfions in Bulk Cubic Helimagnets

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## Abstract

**:**

## 1. Introduction

## 2. Phenomenological Model

## 3. Internal Structure of Hopfions

## 4. Metastability of Hopfions in Bulk Helimagnets

## 5. Discussion and Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(color online) (

**a**) Magnetic structure of an isolated bimeron formed in the plane $xy$ and surrounded by the homogeneous state ${m}_{x}=1$. The field is applied along the x axis. The color indicates ${m}_{z}$-component whereas the white arrows show the magnetization projections onto the plane $xy$. The composite parts of the bimeron structure are the circular core and the crescent. (

**b**) Schematic representation of different 3D structures formed by the proliferation of bimerons: horizontal skyrmions (first panel) are bimeron tubes running perpendicular to the field direction (along z axis in the present case); “solenoids” or “springs” (second panel) are the intermediate states between the horizontal skyrmions and ordinary ISs with their axes along the field. They exist in two varieties and may have a variable radius, which subsequently squeeze into ordinary skyrmions with both polarities: hopfions (third panel) are torus-shaped 3D solitons obtained by the rotation of bimerons around the field direction. (

**c**) Interaction potential of two bimerons with the opposite topological charges (inset shows the corresponding spin structures). Due to the strong deformations of the bimeron cores facing the interior, such a bimeron pair exhibits a stronger repulsion (blue curve) as compared with its counterpart (red curve). (

**d**,

**e**) Schematics showing how to obtain the initial states for the relaxation procedures in mumax3. The characteristic points within the bimeron pair at the distance $2R$ between them are connected by circular paths with the suitable magnetization alignment. Then, two hopfion varieties with opposite Hopf indices and variable radii can easily be prepared.

**Figure 2.**Characteristic preimages for hopfions (

**a**,

**b**) and anti-hopfions (

**c**,

**d**). The toroids are formed by the ${m}_{x}=0\phantom{\rule{0.166667em}{0ex}}(\theta =\pi /2)$ isosurfaces of the spin direction (

**a**,

**c**). The color indicates ${m}_{y}$ magnetization components varying from −1 (blue) to 1 (red). The hopfion normals are co-aligned with the field (x direction). Since the hopfions and anti-hopfions have Hopf invariants ${Q}_{H}=1$ and ${Q}_{H}=-1$, respectively, each pair of preimages is linked exactly once. Preimages are also linked for $\psi =0$ and $\theta $ varying with the step $\pi /8$ from the direction along the field to the opposite direction (

**b**,

**d**). The color codes ${m}_{x}$-component in this case.

**Figure 3.**(color online) Internal structure of hopfions (upper row) and anti-hopfions (bottom row) in bulk cubic helimagnets, ${k}_{c}=0.2,\phantom{\rule{0.166667em}{0ex}}h=0.2759$. (

**a**,

**e**) Magnetization distribution in the cross-section $xy$. Color indicates ${m}_{z}$-component with white arrows being the magnetization projections onto the plane $xy$. (

**b**,

**f**) Hopfion cross-section in the plane $yz$ perpendicular to the field direction $\mathbf{h}\left|\right|x$. Hopfions, in this case, represent target-skyrmions with the magnetization rotation by the angle $2\pi $ from the center to the outskirt. (

**c**,

**g**) The contour plots of the total energy density in the plane $xy$. For hopfions, the positive energy density, which is computed with respect to the homogeneous state, is localized within the “egg”-like shell. For anti-hopfions, the positive energy density forms a belt-like pattern around the magnetization opposite to the field. (

**d**,

**h**) The contour plots of the DMI energy density for the 2D cross-section $xy$. Hopfions clearly exhibit parts with the reverse rotational sense against one chosen by the DMI, whereas anti-hopfions preserve only one rotational fashion.

**Figure 4.**(color online) (

**a**) The eigen-energies of hopfions and anti-hopfions in dependence on the hopfion radii within the isotropic model (3) (dashed black line) and including the cubic anisotropy ${k}_{c}=0.2$ (dotted blue and solid red lines). The slope of $W\left(R\right)$ curves is obviously modified by the decreasing magnetic field and the cubic anisotropy. (

**b**) The spin structures of hopfions without (first and third panels) and with (second and fourth panels) the cubic anisotropy, which leads to spatial extension of hopfions. With the pinning removed, the anti-hopfions transform into torons with two point defects. (

**c**) Toron internal structure is shown as the magnetization and energy density distributions in different 2D cross-sections. (

**d**) The simplified phase diagram including the cubic anisotropy shows the drastic decrease in the saturation field ${h}_{c2}$ of the conical phase. ${h}_{h}$ is a field of the phase transition between the helical and the FM state. In the search for the hopfion metastability, one should avoid the regions of the conical and/or helical spirals. In the former case, hopfions transform into heliknotons; in the latter one, undergo elliptical instability and elongate into helicoids. (

**e**) The slope of the energy curves along the line ${h}_{c2}$ is dependent on the cubic anisotropy value. Rather flat energy curves are reached in the vicinity of ${k}_{c}=0.2$. (

**f**) Modification of the energy curves by the additional exchange anisotropy. For some critical anisotropy values, the minimum corresponding to metastable hopfion appears.

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**MDPI and ACS Style**

Leonov, A.O.
Swirling of Horizontal Skyrmions into Hopfions in Bulk Cubic Helimagnets. *Magnetism* **2023**, *3*, 297-307.
https://doi.org/10.3390/magnetism3040023

**AMA Style**

Leonov AO.
Swirling of Horizontal Skyrmions into Hopfions in Bulk Cubic Helimagnets. *Magnetism*. 2023; 3(4):297-307.
https://doi.org/10.3390/magnetism3040023

**Chicago/Turabian Style**

Leonov, Andrey O.
2023. "Swirling of Horizontal Skyrmions into Hopfions in Bulk Cubic Helimagnets" *Magnetism* 3, no. 4: 297-307.
https://doi.org/10.3390/magnetism3040023