Corresponding Author 2
<EMAIL_ADDRESS>
# Heliknoton in a film of cubic chiral magnet

Vladyslav M. Kuchkin 1,2,3,∗, Nikolai S. Kiselev 2,∗, Filipp N. Rybakov 4 Igor
S. Lobanov 5, Stefan Blügel 2, and Valery M. Uzdin 6

###### Abstract

## 1

Cubic chiral magnets exhibit a remarkable diversity of two-dimensional
topological magnetic textures, including skyrmions. However, the experimental
confirmation of topological states localized in all three spatial dimensions
remains challenging. In this paper, we investigate a three-dimensional
topological state called a heliknoton, which is a hopfion embedded into a
helix or conic background. We explore the range of parameters at which the
heliknoton can be stabilized under realistic conditions using micromagnetic
modeling, harmonic transition state theory, and stochastic spin dynamics
simulations. We present theoretical Lorentz TEM images of the heliknoton,
which can be used for experimental comparison. Additionally, we discuss the
stability of the heliknoton at finite temperatures and the mechanism of its
collapse. Our study offers a pathway for future experimental investigations of
three-dimensional topological solitons in magnetic crystals.

##  2 Keywords:

heliknoton, hopfion, topological magnetic soliton, chiral magnets, stochastic
LLG dynamics, micromagnetic simulations, minimum energy path calculations,
lifetime calculations

##  3 Introduction

Cubic chiral magnets have attracted significant theoretical and experimental
attention due to the vast diversity of experimentally observed topological
magnetic solitons. These materials include various Si- and Ge-based alloys
with B20-type crystal structures, such as Fe1-xCoxSi (52, 31), FeGe (51, 22,
11, 50), MnSi (49), and others (39). The competition between the Heisenberg
exchange interaction and the chiral Dzyaloshinskii-Moriya interaction (13, 29)
(DMI) stabilizes topological solitons in these materials. The most extensively
studied types of magnetic solitons in these systems are magnetic skyrmions (5,
45). Skyrmions are vortex-like strings or tubes characterized by a topological
index $Q=-1$. Experimental observations have revealed that clusters of
skyrmion tubes can form complex three-dimensional superstructures, such as
skyrmion braids (56). Additionally, the skyrmion antiparticle, antiskyrmions,
with topological charge $Q=+1$, have been observed in thin films of FeGe (54).
Skyrmion bags with arbitrary topological charge and skyrmions with chiral
kinks are other solitons that have been reported (34, 16, 23). Recently, the
experimental observation of skyrmion bags with positive topological charge and
their current-induced motion has been reported by Tang et al. (43).

Skyrmions, antiskyrmions, and skyrmion bags are two-dimensional (2D)
topological solitons that are localized in the plane of the sample and
confined by the free surfaces of the sample in the third dimension. The
homotopy classification of these solitons is based on the continuous mapping
between two spheres, $\mathbb{S}^{2}\\!\rightarrow\\!\mathbb{S}^{2}$. Where
the pre-image is a sphere that is homeomorphic to the two-dimensional region
where the skyrmion is localized, while the image is a sphere that corresponds
to a parameter space of the magnetization unit field, given by
$\mathbf{n}(\mathbf{r})=\mathbf{M}(\mathbf{r})/M_{\mathrm{s}}$. For the
magnetic texture in $xy$-plane the corresponding topological charge (12):

$Q=\dfrac{1}{4\pi}\int(\mathbf{F}\cdot\hat{\mathbf{e}}_{z})\,\mathrm{d}x\mathrm{d}y,$
(1)

where the vector field

$\mathbf{F}=\left(\begin{array}[]{c}\mathbf{n}\cdot[\partial_{y}\mathbf{n}\times\partial_{z}\mathbf{n}]\\\
\mathbf{n}\cdot[\partial_{z}\mathbf{n}\times\partial_{x}\mathbf{n}]\\\
\mathbf{n}\cdot[\partial_{x}\mathbf{n}\times\partial_{y}\mathbf{n}]\end{array}\right)$
(2)

is the curvature vector of the vector field $\mathbf{n}$ and is frame-
invariant, meaning it does not depend on the choice of coordinate frame (1,
2). For an unambiguous definition of $Q$, the direction of the $z$-axis should
be chosen consistently with the magnetization $\mathbf{n}_{0}$ around the
region of texture localization such that
$\hat{\mathbf{e}}_{z}\cdot\mathbf{n}_{0}>0$, see details in Refs. (36, 53).

Three-dimensional (3D) topological magnetic solitons belong to a distinct
class of solutions that are localized in all three spatial dimensions. The
classification of 3D magnetic solitons is based on the mapping
$\mathbb{S}^{3}\\!\rightarrow\\!\mathbb{S}^{2}$, which was originally proposed
by Hopf (19). The Hopf index $H$, which is the 3D analog of the topological
invariant $Q$, was derived by Whitehead (48) and is given by:

$H=-\frac{1}{16\pi^{2}}\int(\mathbf{F}\cdot\mathbf{V})\mathrm{d}x\mathrm{d}y\mathrm{d}z,$
(3)

where $\mathbf{V}$ is a vector potential satisfying
$\nabla\times\mathbf{V}=\mathbf{F}$. For localized 3D magnetic textures
without singularities, the Hopf index $H$ is an integer that has the following
meaning: two morphologically distinct magnetic textures with identical $H$ can
be continuously transformed into each other without the appearance of
singularities, whereas continuous transformations between magnetic
configurations with different $H$ are impossible. Statically stable structures
with non-zero Hopf index are commonly referred to as Hopf solitons or
hopfions. It should be noted that the topological charge $Q$ is zero for any
hopfion cross-section. An example of a hopfion texture with $H=1$ is depicted
in the first column of Fig. 1.

Theoretical studies of hopfions in various models of magnetic crystals began
in the 1970s and continue to the present day (14, 6, 7, 41, 24, 47, 35, 37,
27). There is also some limited experimental evidence for direct observation
of such magnetic textures (20, 53). An intriguing type of hopfion has been
predicted in cubic chiral magnets (47), but has not been observed so far. This
hopfion is embedded not in a collinear ferromagnetic state but in a helical
spin spiral state. We refer to it as a heliknoton following the terminology
used in Refs.(42, 47).

The primary goal of this study is to investigate the stability of the
heliknoton under realistic conditions, taking into account the demagnetizing
field effects, and estimate its stability range in films of different
thicknesses. Additionally, we explore how the contrast in theoretical Lorentz
transmission electron microscopy (TEM) images changes with film thickness and
suggest optimal conditions for experimental observations of the heliknoton.
Furthermore, we estimate the stability of the heliknoton at finite
temperatures using two complementary methods: harmonic transition state theory
(HTST)(3, 4) and stochastic spin dynamics (44).

##  4 Model

To estimate the stability of heliknoton in the films of finite thickness, we
use a micromagnetic model, which contains the Heisenberg exchange term, the
DMI, the Zeeman interaction, and the self-energy of the demagnetizing field
(56, 10):

$\mathcal{E}\\!=\\!\\!\int\limits_{\mathbb{R}^{3}}\\!\\!\left[\\!\mathcal{A}|\nabla\mathbf{n}|^{2}\\!+\\!\mathcal{D}\,\mathbf{n}\\!\cdot\\!(\nabla\\!\\!\times\\!\mathbf{n})-\\!M_{\mathrm{s}}\,\mathbf{n}\\!\cdot\\!\mathbf{B}\\!+\\!\dfrac{|\nabla\\!\\!\times\\!\mathbf{A}|^{2}}{2\mu_{0}}\\!\right]\\!d\mathbf{r}\,,$
(4)

where ${\mathbf{n}(\mathbf{r})}$ is the magnetization unit vector field,
$M_{\mathrm{s}}$ is the saturation magnetization, $\mathcal{A}$ is the
exchange stiffness constant, $\mathcal{D}$ is the constant of isotropic bulk
DMI, and $\mu_{0}$ is the vacuum permeability. The magnetic field
$\mathbf{B}(\mathbf{r})$ in (4) represents the sum of a homogeneous external
magnetic field $\mathbf{B}_{\mathrm{ext}}$ and the demagnetizing field
produced by the sample itself,
$\mathbf{B}=\mathbf{B}_{\mathrm{ext}}+\nabla\\!\times\\!\mathbf{A},$ where
${\mathbf{A}(\mathbf{r})}$ is the magnetic vector potential due to the
presence of the magnetization field
$\mathbf{M}(\mathbf{r})=M_{\mathrm{s}}\mathbf{n}$. It should be noted that
while the unit vector $\mathbf{n}$ is defined only within the volume of the
sample, the vector potential $\mathbf{A}(\mathbf{r})$ is defined in the entire
space of $\mathbb{R}^{3}$. As a result, the integration for the first three
terms in Eq. (4) must be carried out only within the sample volume. For the
present analysis, we will consider an infinite plate lying in the $xy$-plane
and assume that the external magnetic field is parallel to the $z$-axis.

The static equilibrium heliknoton was obtained by numerically minimizing (4)
with respect to the pair of fields $\mathbf{n}$ and $\mathbf{A}$ using a
nonlinear conjugate gradient method (for details, see Ref. (56)). The
simulated domain was discretized on a regular mesh with $256\times 256\times
181$ nodes along the $x$, $y$, and $z$ axes, respectively. To simulate an
extended film, periodic boundary conditions were applied in the $xy$-plane.
The results presented below for the micromagnetic calculations were obtained
using GPU-accelerated software Excalibur (32). Additionally, the results were
verified with the publicly available software MuMax3 (46). For definiteness,
we used the material parameters for FeGe (55): $\mathcal{A}=4.75$ pJm-1,
$\mathcal{D}=0.853$ mJm-2, and $M_{\mathrm{s}}=384$ kAm-1. To ensure that our
results are generally applicable, we utilize reduced units for the external
magnetic fields and distances throughout the paper. Specifically, we employ
units concerning the cone saturation field
$B_{\mathrm{D}}=\mathcal{D}^{2}/2M_{\mathrm{s}}\mathcal{A}$ and the
equilibrium period of the spin spiral at the ground state
$L_{\mathrm{D}}=4\pi\mathcal{A}/\mathcal{D}$ (70nm for FeGe). As has been
shown earlier (55), when accounting for demagnetizing fields, the critical
field for cone saturation changes to
$B_{\mathrm{c}}=B_{\mathrm{D}}+\mu_{0}M_{\mathrm{s}}$ (682 mT for FeGe), and
$B_{\mathrm{c}}=B_{\mathrm{D}}$ (199 mT for FeGe) only if the demagnetizing
field is neglected.

##  5 Results

###  5.1 Initial guess

The initial configuration for heliknoton can be obtained in two steps. First,
we insert in the simulated domain the classical hedgehog-based ansatz (19, 40)
for a hopfion in the ferromagnetic background:

$\mathbf{n}=\dfrac{2\sin^{2}G}{r^{2}}\left(\begin{array}[]{c}xz\\\ -yz\\\
z^{2}\\\ \end{array}\right)+\dfrac{1}{r}\left(\begin{array}[]{c}y\sin 2G\\\
x\sin 2G\\\ r\cos 2G\\\ \end{array}\right),$ (5)

where function $G=G(r)$ with $r=\sqrt{x^{2}+y^{2}+z^{2}}$ describes the
skyrmion-like profile: $G(r=0)=\pi$ and $G(r\rightarrow\infty)=0$. In our
simulations we use $G(r)=2\arctan(\exp(-2r)/r)$. The first column in Fig. 1
illustrates the spin texture obtained with the ansatz (5). Note the opposite
chirality of the spin texture in $x$ and $y$ directions.

In the second step, we implemented the spiralization procedure, which involves
rotating all spins by an angle dependent on their coordinate along the
wavevector $\mathbf{k}$ of the spiral. This transforms the ferromagnetic
background into a helical one. Note that we applied this procedure to the
entire volume, including the area containing the hopfion. For the case
$\mathbf{k}||\mathbf{e}_{\mathrm{y}}$ the spiralization procedure can be
written as

$\mathbf{n}^{\prime}=\left(\begin{array}[]{ccc}\cos\varphi&0&\sin\varphi\\\
0&1&0\\\
-\sin\varphi&0&\cos\varphi\end{array}\right)\mathbf{n},\,\,\,\,\,\varphi=\dfrac{2\pi}{L_{\mathrm{D}}}y.$
(6)

where $\mathbf{n}$ corresponds to the ansatz (5) and $\mathbf{n}^{\prime}$ is
the ansatz for the hopfion in the helical background. The second column in
Fig. 1 illustrates the spin texture obtained after applying the spiralization
(6) to the initial hopfion ansatz (5). The spin texture after energy
minimization is shown in the third column of Fig. 1. The quality of our ansatz
is seen from the comparison of the spin textures before and after energy
minimization. To better visualize the spin texture of the heliknoton, we
utilize the despiralization procedure, which is illustrated in the last column
of Fig. 1. This procedure serves as the inverse of the spiralization operation
(6). After the energy minimization or capturing a snapshot of the magnetic
texture, we apply the transformation (6) with $\varphi\mapsto-\varphi$ to
obtain the despiralized spin texture. By using this procedure, one can gain a
better understanding of the heliknoton’s spin texture.

###  5.2 Demagnetizing field effect

To estimate the demagnetizing field effect on the stability of heliknoton in a
film of finite thickness, we performed systematic energy minimization at
different external magnetic fields and film thickness starting with the ansatz
(5)-(6). The results of these calculations performed with and without
demagnetizing fields are presented in Fig. 2. In these calculations, we keep
the size of the simulated domain in $xy$-plane fixed, $4L_{\mathrm{D}}\times
4L_{\mathrm{D}}$, and vary only the film thickness between $0.5L_{\mathrm{D}}$
and $3L_{\mathrm{D}}$. Since the absolute values of the critical fields for
the cases with and without demagnetizing fields are very different, we provide
the critical fields in reduced units with respect to the saturation field of
conical phase, $B_{\mathrm{c}}$. This approach objectively estimates the range
of heliknoton stability in both cases.

As follows from the diagram in Fig. 2, taking into account the demagnetizing
fields reduces the heliknoton stability range down to $\sim
0.3B_{\mathrm{c}}$, which for the material parameters of FeGe corresponds to
$\sim 200$ mT. This is a reasonably high range of fields accessible in most of
the experimental setups.

At the film thickness below $L_{\mathrm{D}}$, the heliknoton does not fit the
size of the film and becomes unstable in both cases, with and without
demagnetizing fields. With increasing thickness, the critical heliknoton
fields tend to saturate in both cases. Near the thicknesses commensurate to
the period of helical modulations, e.g., $L/L_{\mathrm{D}}=2$, the critical
fields show minor picks. The most distinguishing is the behavior of the
heliknoton in the thickness range between $1$ and $1.5L_{\mathrm{D}}$ without
demagnetizing fields, where the solution remains stable only in the presence
of the external field. At realistic conditions, with demagnetizing fields,
however, it is not the case, and the heliknoton is stable in the whole range
of $L>1.1L_{\mathrm{D}}$ even at zero magnetic fields. The thickness of $\sim
1.1L_{\mathrm{D}}$ can be thought of as the lower bound for experimental
observations of heliknotons.

###  5.3 Theoretical analysis of Lorentz TEM contrast

According to the diagram shown in Figure 2, there is no upper bound limit for
the sample thickness, and even in a bulk crystal, a heliknoton can exist.
However, the thickness of the sample plays a crucial role in experimental
observation. In particular, the sample’s thickness is a significant factor for
the applicability of Lorentz TEM. In our previous studies (56), we estimated
the upper limit for the thickness of the FeGe sample to be approximately 300
nm. Above this thickness, the sample becomes no longer transparent for
electrons. In the case of thicker samples beyond 300 nm, one must use a TEM
setup with an acceleration voltage above 300 kV, the standard voltage for the
most modern TEM instruments. Besides that, it is well known that for some
magnetic textures, the Lorentz deflection forces can be completely canceled
and thus give no contrast in non-tilted samples (8, 9).

Since the heliknoton is the texture localized in all three dimensions, it is
natural to expect that the contrast it produces in Lorentz TEM is thickness
dependent. Figure 3 shows the Lorentz TEM contrast provided by the heliknoton
in films of different thicknesses, which we calculated assuming material
parameters for FeGe. For these calculations, we used a well-established method
based on the phase object approximation (17). For details of implementation,
see Ref. (56).

As seen from Fig. 3(A), above the thickness of $\geq 2.5L_{\mathrm{D}}$ ($\sim
170$ nm for FeGe), the characteristic features in both over-focus and under-
focus images of heliknoton are hardly seen. The quality of the contrast can be
improved by varying the defocus distance, as shown in Fig. 3(B). However, in
real experiments this is not always possible, mainly due to diffraction from
the crystal structure, which usually also contributes to the Lorentz TEM
contrast. Based on a comparison of the theoretical Lorentz TEM images and the
stability diagram, the optimal thickness range for observing heliknotons in a
TEM experiment is between $1.5L_{\mathrm{D}}$ and $2L_{\mathrm{D}}$ (around
100 nm and 140 nm, respectively, for FeGe).

###  5.4 Minimum energy path calculations and HTST analysis

To estimate the stability of heliknoton at finite temperature within HTST, we
use an effective atomistic spin Hamiltonian defined on a simple cubic lattice:

$\displaystyle E=\\!-J\\!\sum_{\left\langle
i,j\right\rangle}\\!\mathbf{n}_{i}\cdot\mathbf{n}_{j}-\sum_{\left\langle
i,j\right\rangle}\\!\mathbf{D}_{ij}\cdot[\mathbf{n}_{i}\\!\times\\!\mathbf{n}_{j}]-\mu\sum_{i}\mathbf{B}_{\mathrm{ext}}\cdot\mathbf{n}_{i},$
(7)

were $\mathbf{n}_{i}$ is the unit vector along the magnetic moment at the
lattice site $i$, $J$ and $\mathbf{D}=D\hat{\mathbf{r}}_{ij}$ are the
Heisenberg exchange constant and Dzyaloshinskii-Moriya (DM) vector,
respectively, $\hat{\mathbf{r}}_{ij}$ is the unit vector along the segment
connecting sites $i$ and $j$, $\mu$ is the magnitude of the magnetic moment at
each site. The symbol $\left\langle i,j\right\rangle$ denotes the summation
over the nearest neighbor pairs over all lattice sites.

Here we ignore the demagnetizing field effect and introduce characteristic
parameters $L_{\mathrm{D}}=2\pi Ja/D$ and $B_{\mathrm{D}}=D^{2}/(J\mu)$ for
the atomistic model (7), where $a$ is the lattice constant.

First, we apply HTST to estimate the lifetime of a particular magnetic state
and the possible mechanisms of its collapse. Within the HTST, the rate of
transition between states $X$ and $Y$ at temperature $T$ is described by the
Arrhenius law,

$\displaystyle k^{X\rightarrow Y}=\nu^{X\rightarrow
Y}\exp{\left(-\dfrac{\Delta E^{X\rightarrow Y}}{k_{\mathrm{B}}T}\right)},$ (8)

where the energy barrier $\Delta E^{X\rightarrow Y}$ can be computed knowing
the minimal energy path (MEP) connecting $X$ and $Y$ states on the energy
surface as the energy difference between the highest point along the MEP – the
first-order saddle point ($SP$) on the energy surface of the system – and the
minimum at $X$. The pre-exponential factor $\nu^{X\rightarrow Y}$ incorporates
dynamical $\nu_{dyn}$ and entropic $\nu_{ent}$ contributions to the transition
rate

$\nu^{X\rightarrow Y}={\frac{1}{2\pi}}\nu_{dyn}\nu_{ent}.$

Within the harmonic approximation, the energy of the system in the vicinity of
a stationary state is approximated by a quadratic form that allows the
prefactors to be determined explicitly (26). The quadratic forms for the
minimum $\mathbf{n}^{X}$ and the saddle point $\mathbf{n}^{SP}$ are determined
by the corresponding Hessian matrices $\mathcal{H}^{X}$ and
$\mathcal{H}^{SP}$, respectively. The entropy of the states is expressed in
terms of determinants of the matrices. If the energy is an invariant of some
transformation, e.g., the energy of the hopfion is preserved during
translations, then some eigenvalues of the Hessians are zero, and the
corresponding modes are called zero modes. In the harmonic approximation, the
contribution of zero-modes to the state $\mathbf{n}^{X}$ and saddle point
$\mathbf{n}^{SP}$ which have $Z^{X}$ and $Z^{SP}$, respectively, the zero-
modes are estimated by their volumes $V^{X}$ and $V^{SP}$. The entropy
prefactor is given by

$\nu_{ent}=(2\pi
k_{B}T)^{\frac{Z^{X}-Z^{SP}}{2}}\frac{V^{SP}}{V^{X}}\sqrt{\frac{\det\mathcal{H}^{X}}{|\det\mathcal{H}^{SP}|}},$

where the Hessians are restricted to subspaces consisting of non-zero modes.

The dynamical prefactor is expressed in terms of the negative eigenvalue
$\zeta$ of the Hessian $\mathcal{H}^{\mathrm{SP}}$ with the corresponding
eigenvector $e$,

$\nu_{dyn}=\sqrt{\frac{\mathbf{b}\cdot\mathcal{H}^{\mathrm{SP}}\mathbf{b}}{|\zeta|}},\quad\it
b_{i}=\frac{\gamma\zeta}{\mu}\mathbf{n}_{i}^{\mathrm{SP}}\times e_{i}.$

The index $i$ numbers the lattice sites.

The heliknoton in bulk has at least two zero modes corresponding to
translations in directions orthogonal to the helical axis. We eliminate one of
the zero modes by pinning the spins at the boundaries $x=\mathrm{const}$ and
$y=\mathrm{const}$ in such a way that the direction of the magnetic moments
coincides with one of the helix. The pinned spins were not affected by the
optimization procedures and were not included as degrees of freedom to the
Hessians. On the surfaces $z=\mathrm{const}$ the boundary conditions are free,
which allows us to simulate the escape through the boundary. Translation along
the $z$-axis becomes a quasi-zero mode because of the effect of the boundary.

The computation of MEP was performed in Cartesian coordinates with constraints
on the magnetic moments length taking into account by introduction of Lagrange
multipliers, which allows us to avoid singularities and the necessity to
transform coordinates while changing the map in the atlas (25). The string
method (15) with stable tangent estimate (18) was used for MEP calculation.

The most challenging part of the computation of the transition rate is the
calculation of the determinants. When a standard LU or QR decomposition is
used, the complexity of the problem is $\mathcal{O}(N^{3})$, where $N$ is the
number of spins. Taking into account the block band structure of the Hessian
matrix for close-range interactions in the absence of the demagnetization
field, the complexity can be reduced to $\mathcal{O}(N^{7/3})$ (26).
Nevertheless, the problem remains time-consuming, especially for the domain of
size $5L_{\mathrm{D}}\times 5L_{\mathrm{D}}\times 2L_{\mathrm{D}}$ used in our
calculations.

To find the optimal mesh density, We computed MEPs calculated at different
mesh densities of $L_{\mathrm{D}}$ equal to $12a$, $24a$, $36a$, and $48a$.
While the coarsest grid of $12a$ gives significantly different values of both
the Heliknoton energy and the activation barrier, the results for the three
densest grids (with $24a$, $36a$, and $48a$) are only slightly different (see
right inset in Fig. 4A). Therefore, we used the discretization density
$L_{\mathrm{D}}=24a$ for the transition rate computation.

The results of the HTST analysis are summarized in Figure 4. The MEPs
presented in Figure 4A correspond to two distinct mechanisms for heliknoton
collapse, which we refer to as decay and escape. The decay mechanism is
similar to the collapse of hopfions in a model of a frustrated magnet (37,
27). In this case, the heliknoton collapses via the emergence of a pair of
magnetic singularities, namely Bloch points of opposite signs. Representative
images along the MEP corresponding to heliknoton decay are shown in Figure 4C.

As seen from the images in Figure 4D, when the heliknoton escapes through the
boundary, magnetic singularities do not appear at any point along the MEP.
Surprisingly, the escape mechanism for the heliknoton collapse is
characterized by a higher energy barrier compared to heliknoton decay for
sufficiently dense lattices. Thereby, unlike the case of hopfions in the model
of frustrated magnets (35), the escape through the boundary for the heliknoton
requires overcoming some activation barrier, which we attribute to the
emergence of the chiral surface twist effect (33). The latter suggests that
the heliknoton in a thin plate of a chiral magnet may be more promising for
experimental observations due to its finite energy barrier for escape.

The total transition rate for a metastable state is the sum of transition
rates for all mechanisms of the collapse. The main contributions to the mean
lifetime (inverse to the transition rate) of the heliknoton are the decay and
escape mechanisms, with decay dominating:

$\tau=\frac{1}{k^{decay}+k^{escape}}\approx\frac{1}{k^{decay}}.$

The dependence of the heliknoton lifetime on temperature is shown in Figure
4B. The inverse to the transition rate for the decay mechanism gives an
estimate of the lifetime of the heliknoton in thick films because the escape
transition rate is inversely proportional to the film thickness.

Overall, our HTST analysis sheds light on the decay mechanisms of the
heliknoton and highlights potential avenues for its experimental observation,
especially at low temperatures. For instance, at $T=0.455J/k_{\mathrm{B}}$ (or
$0.35T_{\mathrm{c}}$), the lifetime for heliknoton is $\tau\sim 10^{6}$ in
dimensionless time units scaled by $J\gamma\mu^{-1}$, see Fig. 4B. For $J=112$
meV and $\mu=1030\mu_{\mathrm{B}}$, we obtain the heliknoton lifetime of $\sim
1000$ seconds, which is quite reasonable for experimental observation.
Assuming every model spin $\mathbf{n}_{i}$ corresponds to a cluster of $7^{3}$
atoms, where each atom has a magnetic moment of $3\mu_{B}$ and a lattice
constant of $5$Å, the diameter of the observable heliknotons should be in the
order of $100$ nm. Note that the temperature of $0.35T_{\mathrm{c}}$ for FeGe
corresponds to $\sim 95$ K, which is accessible for the experiment performed
at liquid nitrogen temperature.

###  5.5 Stochastic LLG dynamics

The stochastic LLG equation can be written as

$\displaystyle\dfrac{\partial\mathbf{n}_{i}}{\partial
t}=\mathbf{n}_{i}\times\left(\dfrac{1}{J}\frac{\partial
E}{\partial\mathbf{n}_{i}}-\mathbf{B}_{\mathrm{fluc}}^{i}\right)+\alpha\mathbf{n}_{i}\times\frac{\partial\mathbf{n}_{i}}{\partial
t},$ (9)

where $t$ is a dimensionless time scaled by $J\gamma\mu^{-1}$, with $\gamma$
being the gyromagnetic ratio, $\alpha$ is the Gilbert damping parameter,
$\mathbf{B}_{\mathrm{fluc}}^{i}$ is the fluctuating field representing
uncorrelated Gaussian white noise with a prefactor proportional to the
temperature, $T$. For the numerical integration of Eq. (9), we use the semi-
implicit method provided in Ref. (28) implemented in the publicly available
software Magnoom (21, 38). For the chosen coupling parameters, we estimate the
critical temperature, $T_{\mathrm{c}}\simeq 1.345J/{k_{\mathrm{B}}}$ (see Ref.
(30)). The simulations were performed with $\alpha=0.3$ and a fixed time step
of $\Delta t=0.01$. We perform ten independent simulations on a domain with
$L_{\mathrm{x}}=L_{\mathrm{y}}=4L_{\mathrm{D}}$,
$L_{\mathrm{z}}=2L_{\mathrm{D}}$ and discretization density
$L_{\mathrm{D}}/a=32$. We use periodic boundary conditions in the $xy$-plane.
To increase the probability of heliknoton collapse in a reasonable time, the
simulations were performed at an elevated temperature of
$T=1.0J/{k_{\mathrm{B}}}$ ($\sim 0.74T_{\mathrm{c}}$). The total simulation
time of each run is $t=10^{5}$ ($10^{7}$ iterations).

To visualize the heliknoton isosurface $n_{\mathrm{z}}=0$ in the presence of
strong thermal fluctuations, we first apply Fourier filtering as described in
Ref. (30). After that, we apply the despiralization procedure as introduced
above, Fig. 5. Among the few independent runs of the LLG simulations, we chose
the two most representative examples illustrated in Fig. 5A, B and
Supplementary Movie 1, 2. In agreement with the results of HTST presented in
the previous section, we observe two mechanisms of heliknoton collapse. In
particular, Fig. 5A shows the snapshots of the systems during the heliknoton
collapse via the decay, accompanied by the Bloch points nucleation. Fig. 5B
illustrates the case when heliknoton collapses via escape through the top free
surface of the plate. These are the only two mechanisms observed in stochastic
LLG simulations. In the first case, the heliknoton shrinks until the tor hole
completely disappears. At this moment, a pair of Bloch points of the opposite
charge appears, and we see the transition from the heliknoton to the dipole
string or toron (30). The dipole string represents a non-stable state at these
parameters and collapses through shrinking. In the second case, we observe the
shifting of the heliknoton to the open surface as a result of its Brownian
motion. The escaping happens if the distance between the heliknoton and the
surface is smaller than the critical one. It is worth noting that despite the
temporal fluctuations, the escape through the free surface represents a smooth
transition. Remarkably, the heliknoton remains stable near the surface of the
plate for a noticeably long time. That might indicate the existence of a
metastable state resembling surface modulations. Details of the investigation
of such localized states will be provided elsewhere.

Among ten independent simulations, we observed the collapse of the heliknoton
in the times range between $2\times 10^{4}$ to $8\times 10^{4}$ in reduced
time units. Using these results, we estimated the lifetime of the heliknoton
at $T=1.0J/k_{\mathrm{B}}$ to be approximately $\tau\sim 10^{4}$. This result
agrees with the findings of the HTST calculations presented in Fig. 4B, which
suggest a lifetime of $\tau\sim 10^{3}$ for this temperature. The limited
sampling (only ten independent runs) may account for the observed discrepancy.
However, given the variation of the heliknoton’s lifetime by ten orders of
magnitude for different temperatures, we consider the agreement between the
simulations and HTST calculations to be satisfactory.

##  6 Conclusions

In this study, we investigated the stability and decay mechanisms of the
heliknoton in realistic conditions using direct energy minimization,
stochastic LLG simulations, and HTST calculations. Our results showed that the
heliknoton is stable in a wide range of magnetic fields and plate thicknesses,
as demonstrated by the stability diagram calculated at zero temperature.
However, our calculations indicated that for plate thicknesses above
$2.5L_{\mathrm{D}}$, the magnetic contrast in Lorentz TEM becomes too weak for
reliable observations.

Based on our findings, we estimate the optimal plate thickness for heliknoton
observations to be $2L_{\mathrm{D}}\pm 0.5L_{\mathrm{D}}$ (140 $\pm$ 35 nm for
FeGe parameters). Our analysis using the GNEB method revealed two main
mechanisms of heliknoton decay: collapse via the formation of Bloch points and
escape through the plate surface. The LLG simulations at finite temperatures
supported these observations and showed reasonable agreement with the
estimation of heliknoton lifetime, at least at elevated temperatures.
Moreover, the HTST calculations allowed us to estimate heliknoton lifetimes at
different temperatures. Overall, our results suggest that the heliknoton
should be stable with a reasonably long lifetime at the standard measurement
temperature of liquid nitrogen ($T=95$ K) used in TEM experiments. We
anticipate that these findings will stimulate further experimental
observations of heliknotons in magnetic systems.

## Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any
commercial or financial relationships that could be construed as a potential
conflict of interest.

## Author Contributions

V.M.K., N.S.K. and F.N.R. performed micromagnetic and LLG simulations, I.S.L.
performed minimum energy path and lifetime calculations. N.S.K. wrote the
first draft of the manuscript. S.B. and V.M.U. participated in the
interpretation of the results, review and editing, supervision. All authors
contributed to the manuscript writing.

## Funding

V.M.K. acknowledges financial support from the Icelandic Research Fund (Grant
No. 217750). This project has received funding from the European Research
Council under the European Union’s Horizon 2020 Research and Innovation
Programme (Grant No. 856538 - project “3D MAGiC”). N.S.K. and S.B. acknowledge
financial support from the Deutsche Forschungsgemeinschaft through SPP 2137
“Skyrmionics”, Grant No. KI 2078/1-1 and Grant No. BL 444/16, respectively.
F.N.R. acknowledges support from the Swedish Research Council.

## Acknowledgments

## Supplemental Data

Movies 1 and 2 illustrate two mechanisms of heliknoton collapse by decay and
escape through the surface, respectively, observed in stochastic LLG
simulations.

## References

  * Aminov (1969) Aminov, J. A. (1969).  Sources of curvature of a vector field.  _Mathematics of the USSR-Sbornik_ 9, 199–211.  10.1070/sm1969v009n02abeh001128
  * Aminov (2000) Aminov, Y. (2000).  _The Geometry of Vector Fields_ (Gordon and Breach, Amsterdam)
  * Bessarab et al. (2012) Bessarab, P. F., Uzdin, V. M., and Jónsson, H. (2012).  Harmonic transition-state theory of thermal spin transitions.  _Phys. Rev. B_ 85, 184409.  10.1103/PhysRevB.85.184409
  * Bessarab et al. (2013) Bessarab, P. F., Uzdin, V. M., and Jónsson, H. (2013).  Potential energy surfaces and rates of spin transitions.  _Zeitschrift für Physikalische Chemie_ , 13070800031000810.1524/zpch.2013.0403
  * Bogdanov and Yablonskii (1989) Bogdanov, A. N. and Yablonskii, D. A. (1989).  Thermodynamically stable “vortices” in magnetically ordered crystals. the mixed state of magnets.  _Sov. Phys. JETP_ 68
  * Bogolubsky (1988) Bogolubsky, I. (1988).  Three-dimensional topological solitons in the lattice model of a magnet with competing interactions.  _Physics Letters A_ 126, 511–514.  10.1016/0375-9601(88)90049-7
  * Cooper (1999) Cooper, N. R. (1999).  Propagating magnetic vortex rings in ferromagnets.  _Phys. Rev. Lett._ 82, 1554–1557.  10.1103/PhysRevLett.82.1554
  * Denneulin et al. (2021a) Denneulin, T., Caron, J., Hoffmann, M., Lin, M., Tan, H., Kovács, A., et al. (2021a).  Off-axis electron holography of néel-type skyrmions in multilayers of heavy metals and ferromagnets.  _Ultramicroscopy_ 220, 113155.  10.1016/j.ultramic.2020.113155
  * Denneulin et al. (2021b) Denneulin, T., Caron, J., Müller-Caspary, K., Boulle, O., Kovács, A., and Dunin-Borkowski, R. E. (2021b).  Visibility and apparent size of néel-type magnetic skyrmions in fresnel defocus images of multilayer films.  _Microscopy and Microanalysis_ 27, 1356–1365.  10.1017/s1431927621012927
  * Di Fratta et al. (2020) Di Fratta, G., Muratov, C. B., Rybakov, F. N., and Slastikov, V. V. (2020).  Variational principles of micromagnetics revisited.  _SIAM Journal on Mathematical Analysis_ 52, 3580–3599.  10.1137/19M1261365
  * Du et al. (2018) Du, H., Zhao, X., Rybakov, F. N., Borisov, A. B., Wang, S., Tang, J., et al. (2018).  Interaction of individual skyrmions in a nanostructured cubic chiral magnet.  _Phys. Rev. Lett._ 120, 197203.  10.1103/PhysRevLett.120.197203
  * Dubrovin et al. (1985) Dubrovin, B. A., Fomenko, A. T., and Novikov, S. P. (1985).  _Modern Geometry – Methods and Applications. Part II. The Geometry and Topology of Manifolds_ (Springer-Verlag, New York).  10.1007/978-1-4612-1100-6
  * Dzyaloshinsky (1958) Dzyaloshinsky, I. (1958).  A thermodynamic theory of “weak” ferromagnetism of antiferromagnetics.  _Journal of Physics and Chemistry of Solids_ 4, 241–255.  https://doi.org/10.1016/0022-3697(58)90076-3
  * Dzyloshinskii and Ivanov (1979) Dzyloshinskii, I. E. and Ivanov, B. A. (1979).  Localized topological solitons in a ferromagnet.  _JETP Lett._ 29, 540
  * E et al. (2007) E, W., Ren, W., and Vanden-Eijnden, E. (2007).  Simplified and improved string method for computing the minimum energy paths in barrier crossing events.  _J. Chem. Phys._ 126\.  10.1063/1.2720838
  * Foster et al. (2019) Foster, D., Kind, C., Ackerman, P. J., Tai, J.-S. B., Dennis, M. R., and Smalyukh, I. I. (2019).  Two-dimensional skyrmion bags in liquid crystals and ferromagnets.  _Nature Physics_ 15\.  10.1038/s41567-019-0476-x
  * Graef (2001) Graef, M. D. (2001).  2\. lorentz microscopy: Theoretical basis and image simulations.  In _Experimental Methods in the Physical Sciences_ (Elsevier). 27–67.  10.1016/s1079-4042(01)80036-9
  * Henkelman and Jónsson (2000) Henkelman, G. and Jónsson, H. (2000).  Improved tangent estimate in the nudged elastic band method for finding minimum energy paths and saddle points.  _J. Chem. Phys._ 113, 9978.  https://doi.org/10.1063/1.1323224
  * Hopf (1931) Hopf, H. (1931).  Über die abbildungen der dreidimensionalen sphäre auf die kugelfläche.  _Mathematische Annalen_ 104, 637–665.  10.1007/bf01457962
  * Kent et al. (2021) Kent, N., Reynolds, N., Raftrey, D., Campbell, I. T. G., Virasawmy, S., Dhuey, S., et al. (2021).  Creation and observation of hopfions in magnetic multilayer systems.  _Nature Communications_ 12, 1562.  10.1038/s41467-021-21846-5
  * Kiselev (2016) Kiselev, N. S. (2016).  Magnoom software
URL: https://github.com/n-s-kiselev/magnoom

  * Kovács et al. (2017) Kovács, A., Caron, J., Savchenko, A. S., Kiselev, N. S., Shibata, K., Li, Z.-A., et al. (2017).  Mapping the magnetization fine structure of a lattice of bloch-type skyrmions in an FeGe thin film.  _Applied Physics Letters_ 111, 192410.  10.1063/1.5004394
  * Kuchkin et al. (2020) Kuchkin, V. M., Barton-Singer, B., Rybakov, F. N., Blügel, S., Schroers, B. J., and Kiselev, N. S. (2020).  Magnetic skyrmions, chiral kinks, and holomorphic functions.  _Phys. Rev. B_ 102, 144422.  10.1103/PhysRevB.102.144422
  * Liu et al. (2018) Liu, Y., Lake, R. K., and Zang, J. (2018).  Binding a hopfion in a chiral magnet nanodisk.  _Phys. Rev. B_ 98, 174437.  10.1103/PhysRevB.98.174437
  * Lobanov et al. (2021) Lobanov, I. S., Potkina, M. N., and Uzdin, V. M. (2021).  Stability and lifetimes of magnetic states of nano-and microstructures (brief review).  _JETP Letters_ 113, 801.  10.1134/S0021364021120109
  * Lobanov and Uzdin (2021) Lobanov, I. S. and Uzdin, V. M. (2021).  The lifetime of micron scale topological chiral magnetic states with atomic resolution.  _Comp. Phys. Commun._ 269, 108136.  10.1016/j.cpc.2021.108136
  * Lobanov and Uzdin (2023) Lobanov, I. S. and Uzdin, V. M. (2023).  Lifetime, collapse, and escape paths for hopfions in bulk magnets with competing exchange interactions.  _Phys. Rev. B_ 107, 104405.  10.1103/PhysRevB.107.104405
  * Mentink et al. (2010) Mentink, J. H., Tretyakov, M. V., Fasolino, A., Katsnelson, M. I., and Rasing, T. (2010).  Stable and fast semi-implicit integration of the stochastic landau–lifshitz equation.  _Journal of Physics: Condensed Matter_ 22, 176001.  10.1088/0953-8984/22/17/176001
  * Moriya (1960) Moriya, T. (1960).  Anisotropic superexchange interaction and weak ferromagnetism.  _Phys. Rev._ 120, 91–98.  10.1103/PhysRev.120.91
  * Müller et al. (2020) Müller, G. P., Rybakov, F. N., Jónsson, H., Blügel, S., and Kiselev, N. S. (2020).  Coupled quasimonopoles in chiral magnets.  _Physical Review B_ 101\.  10.1103/physrevb.101.184405
  * Park et al. (2014) Park, H. S., Yu, X., Aizawa, S., Tanigaki, T., Akashi, T., Takahashi, Y., et al. (2014).  Observation of the magnetic flux and three-dimensional structure of skyrmion lattices by electron holography.  _Nature Nanotechnology_ 9, 337–342.  10.1038/nnano.2014.52
  * Rybakov and Babaev (2020) Rybakov, F. N. and Babaev, E. (2020).  Excalibur software
URL: http://quantumandclassical.com/excalibur/

  * Rybakov et al. (2015) Rybakov, F. N., Borisov, A. B., Blügel, S., and Kiselev, N. S. (2015).  New type of stable particlelike states in chiral magnets.  _Phys. Rev. Lett._ 115, 117201.  10.1103/PhysRevLett.115.117201
  * Rybakov and Kiselev (2019) Rybakov, F. N. and Kiselev, N. S. (2019).  Chiral magnetic skyrmions with arbitrary topological charge.  _Phys. Rev. B_ 99, 064437.  10.1103/PhysRevB.99.064437
  * Rybakov et al. (2022) Rybakov, F. N., Kiselev, N. S., Borisov, A. B., Döring, L., Melcher, C., and Blügel, S. (2022).  Magnetic hopfions in solids.  _APL Materials_ 10, 111113.  10.1063/5.0099942
  * Rybakov (2021) Rybakov, P. (2021).  _Topological excitations in field theory models of superconductivity and magnetism_.  Ph.D. thesis, KTH Royal Institute of Technology, Stockholm
  * Sallermann et al. (2023) Sallermann, M., Jónsson, H., and Blügel, S. (2023).  Stability of hopfions in bulk magnets with competing exchange interactions.  _Phys. Rev. B_ 107, 104404.  10.1103/PhysRevB.107.104404
  * Savchenko et al. (2022) Savchenko, A. S., Kuchkin, V. M., Rybakov, F. N., Blügel, S., and Kiselev, N. S. (2022).  Chiral standing spin waves in skyrmion lattice.  _APL Materials_ 10, 071111.  10.1063/5.0097651
  * Shibata et al. (2013) Shibata, K., Yu, X. Z., Hara, T., Morikawa, D., Kanazawa, N., Kimoto, K., et al. (2013).  Towards control of the size and helicity of skyrmions in helimagnetic alloys by spin–orbit coupling.  _Nature Nanotechnology_ 8, 723–728.  10.1038/nnano.2013.174
  * Skyrme (1961) Skyrme, T. H. R. (1961).  A non-linear field theory.  _Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences_ 260, 127–138.  10.1098/rspa.1961.0018
  * Sutcliffe (2017) Sutcliffe, P. (2017).  Skyrmion knots in frustrated magnets.  _Physical Review Letters_ 118\.  10.1103/physrevlett.118.247203
  * Tai and Smalyukh (2019) Tai, J.-S. B. and Smalyukh, I. I. (2019).  Three-dimensional crystals of adaptive knots.  _Science_ 365, 1449–1453.  10.1126/science.aay1638
  * Tang et al. (2021) Tang, J., Wu, Y., Wang, W., Kong, L., Lv, B., Wei, W., et al. (2021).  Magnetic skyrmion bundles and their current-driven dynamics.  _Nature Nanotechnology_ 16, 1086–1091.  10.1038/s41565-021-00954-9
  * ter Haar (1965) ter Haar, D. (ed.) (1965).  _Collected Papers of L.D. Landau_ (Pergamon), chap. 18 - On the Theory of the Dispersion of Magnetic Permeability in Ferromagnetic bodies.  101–114.  https://doi.org/10.1016/B978-0-08-010586-4.50023-7
  * Tokura and Kanazawa (2020) Tokura, Y. and Kanazawa, N. (2020).  Magnetic skyrmion materials.  _Chemical Reviews_ 121, 2857–2897.  10.1021/acs.chemrev.0c00297
  * Vansteenkiste et al. (2014) Vansteenkiste, A., Leliaert, J., Dvornik, M., Helsen, M., Garcia-Sanchez, F., and Van Waeyenberge, B. (2014).  The design and verification of mumax3.  _AIP Advances_ 4, 107133.  10.1063/1.4899186
  * Voinescu et al. (2020) Voinescu, R., Tai, J.-S. B., and Smalyukh, I. I. (2020).  Hopf solitons in helical and conical backgrounds of chiral magnetic solids.  _Phys. Rev. Lett._ 125, 057201.  10.1103/PhysRevLett.125.057201
  * Whitehead (1947) Whitehead, J. H. C. (1947).  An expression of hopf's invariant as an integral.  _Proceedings of the National Academy of Sciences_ 33, 117–123.  10.1073/pnas.33.5.117
  * Yu et al. (2015) Yu, X., Kikkawa, A., Morikawa, D., Shibata, K., Tokunaga, Y., Taguchi, Y., et al. (2015).  Variation of skyrmion forms and their stability in mnsi thin plates.  _Phys. Rev. B_ 91, 054411.  10.1103/PhysRevB.91.054411
  * Yu et al. (2018) Yu, X., Morikawa, D., Yokouchi, T., Shibata, K., Kanazawa, N., Kagawa, F., et al. (2018).  Aggregation and collapse dynamics of skyrmions in a non-equilibrium state.  _Nature Physics_ 14, 832–836.  10.1038/s41567-018-0155-3
  * Yu et al. (2010a) Yu, X. Z., Kanazawa, N., Onose, Y., Kimoto, K., Zhang, W. Z., Ishiwata, S., et al. (2010a).  Near room-temperature formation of a skyrmion crystal in thin-films of the helimagnet FeGe.  _Nature Materials_ 10, 106–109.  10.1038/nmat2916
  * Yu et al. (2010b) Yu, X. Z., Onose, Y., Kanazawa, N., Park, J. H., Han, J. H., Matsui, Y., et al. (2010b).  Real-space observation of a two-dimensional skyrmion crystal.  _Nature_ 465, 901–904.  10.1038/nature09124
  * Zheng et al. (2023) Zheng, F., Kiselev, N. S., Rybakov, F. N., Yang, L., Blügel, S., and Dunin-Borkowski, R. E. (2023).  Hopfion rings in a cubic chiral magnet.  _Preprint at Research Square_ 10.21203/rs.3.rs-2681064/v1
  * Zheng et al. (2022) Zheng, F., Kiselev, N. S., Yang, L., Kuchkin, V. M., Rybakov, F. N., Blügel, S., et al. (2022).  Skyrmion–antiskyrmion pair creation and annihilation in a cubic chiral magnet.  _Nature Physics_ 18, 863–868.  10.1038/s41567-022-01638-4
  * Zheng et al. (2018) Zheng, F., Rybakov, F. N., Borisov, A. B., Song, D., Wang, S., Li, Z.-A., et al. (2018).  Experimental observation of chiral magnetic bobbers in b20-type FeGe.  _Nature Nanotechnology_ 13, 451–455.  10.1038/s41565-018-0093-3
  * Zheng et al. (2021) Zheng, F., Rybakov, F. N., Kiselev, N. S., Song, D., Kovács, A., Du, H., et al. (2021).  Magnetic skyrmion braids.  _Nature Communications_ 12\.  10.1038/s41467-021-25389-7

## Figure captions

Figure 1: The images illustrate the spin texture inside the simulated box of
the size $4L_{\mathrm{D}}\times 4L_{D}$ in the $xy$-plane and thickness of
$2L_{\mathrm{D}}$ along the $z$-axis. The first column of images correspond to
the hopfion anzatz (5). The second column represents the hopfion ansatz after
applying the spiralization (6) with the $\mathbf{k}$-vector of the spiral
parallel to the $y$-axis, $|\mathbf{k}|=2\pi/L_{\mathrm{D}}$. The third column
of images corresponds to the spin texture after the energy minimization
assuming periodical boundary conditions in the $xy$-plane and free boundaries
along the $z$-axis. The fourth column illustrates the relaxed spin texture
after applying despiralization. The first row of images shows the isosurfaces
$n_{\mathrm{z}}=0$. The images depicted in the second, third, and fourth rows
illustrate the magnetization field in the middle planes. For illustrative
purposes, these images are bounded by the blue box of the size
$0.64L_{\mathrm{D}}\times 0.64L_{\mathrm{D}}\times 2L_{\mathrm{D}}$ depicted
in the top left image. Figure 2: The diagram of stability for heliknoton in
extended film calculated with and without demagnetizing field. For
consistency, the external fields for both cases are given in reduced units
with respect to the saturation field of the cone phase,
$B_{\mathrm{c}}=B_{\mathrm{D}}+\mu_{0}M_{\mathrm{s}}$ and
$B_{\mathrm{c}}=B_{\mathrm{D}}$, for the case with and without demagnetizing
filed, respectively. For B20-type FeGe,
$B_{\mathrm{c}}=B_{\mathrm{D}}+\mu_{0}M_{\mathrm{s}}=0.682$ T and
$L_{\mathrm{D}}=70$ nm, see the top axis. Figure 3: (A) The theoretical
Lorentz TEM images of heliknoton at zero external field in the plate of
different thicknesses. The thicknesses in reduced units with respect to the
period of helical modulations $L_{\mathrm{D}}$ are indicated on the top of
each pair of over-focus and under-focus images. The defocus distance is 400
$\mu$m. (B) The over-focus and under-focus Lorentz TEM images of heliknoton in
the film of thickness $2L_{\mathrm{D}}$ calculated for different defocus
distances. Figure 4: (A) The minimum energy paths for two distinct mechanisms
for heliknoton collapse in the film: the escape through the free edges of the
plate (blue curve) and the decay inside the film via the nucleation of a pair
of Bloch points (red curve). The saddle points are indicated by hollow
symbols. The right inset illustrates the convergence of the energies of the
heliknoton, $E_{H}$, and the energy barrier for the decay, $\Delta E_{D}$, to
the micromagnetic limits with increasing mesh density, $L_{\mathrm{D}}/a$. (B)
The lifetime of heliknoton as a function of temperature. The escape mechanism
( blue line) is less probable than the decay mechanism for all temperatures.
Insets in (B) show the contributions of the exponent (activation barrier) in
Arrhenius law to the lifetime. (C) Snapshots of the system along the minimum
energy path for heliknoton decay. The spin texture after despiralization is
represented by the isosurfaces $m_{\mathrm{z}}=0$. The number above each image
corresponds to the reaction coordinate, $r$. Similar to Fig. 1, the images
show the volume confined by the blue box of the size $2.5L_{\mathrm{D}}\times
2.5L_{\mathrm{D}}\times 2L_{\mathrm{D}}$ – the quarter of the whole simulated
domain of the size $5L_{\mathrm{D}}\times 5L_{\mathrm{D}}\times
2L_{\mathrm{D}}$. (D) Snapshots of the system along the minimum energy path
for heliknoton escape through the free surface. The field of view and
notations are the same as in (C). In (C) and (D), the images in the middle
correspond to the saddle points of corresponding MEPs.  Figure 5: Snapshots of
the system in two different stochastic LLG simulations are shown. The set of
images in row (A) corresponds to the heliknoton collapse through the
nucleation of Bloch points, while images in row (B) correspond to its escaping
through the free surface of the plate. The shown box has size
$2L_{\mathrm{D}}$ in each dimension.