Magnetic Hopfions: A Review

Abstract

Recent advances in the research area of 3D magnetic topological solitons (hopfions) in restricted geometries are reviewed. The description of the magnetic solitons is based on a macroscopic micromagnetic approach and the Landau–Lifshitz equation of the magnetization motion. The concepts of the gauge emergent vector potential and emergent magnetic field are widely used to calculate the 3D topological charge (the Hopf index) of magnetic textures. The relation of the magnetic hopfions with classical field theory is demonstrated, and a special role of the curvilinear toroidal coordinates in the description of the hopfions is underlined. The hopfion stability and dynamics in ferromagnetic films and dots are considered. A critical discussion of calculations of the magnetization emergent magnetic field and the Hopf index of the toroidal magnetic hopfions in restricted geometries is presented.

1. Introduction

Non-linear configurations of the continuous classical field (solitons) play an important role in the different fields of modern physics such as classical field theory, optics, condensed matter physics, etc. The solitons are localized, particle-like objects, and, in many cases, the solitons can be classified by integer numbers related to their topology (topological charges) [1].

Let us assume that a physical system can be described by a classical 3D vector field $\mathit{n}\left(\mathit{r}\right)$ of unit length ${\mathit{n}}^2=1$ (the order-parameter space is a unit sphere $S_2$ in 3D space) depending on three spatial coordinates, represented by the vector $\mathit{r}$. For instance, in the area of ferromagnetism, such a field is the unit magnetization field [2]. The vector field in optics is the Stokes vector field [3,4], whereas in liquid crystals, it is a director field [5,6], etc. There are topologically non-trivial configurations of the vector field $\mathit{n}\left(\mathit{r}\right)$, which describes an order parameter in the media. Such field configurations can be classified by using a mapping $\mathit{r}\to \mathit{n}\left(\mathit{r}\right)$ from the coordinate space (r) to the order parameter space $\mathit{n}\left(\mathit{r}\right)$ [7].

The macroscopic equation to describe a magnetization equilibrium configuration and magnetization dynamics in ordered magnetic media (the Landau–Lifshitz equation) is essentially nonlinear and allows for several soliton-type solutions including topological magnetic solitons [2]. Topologically non-trivial magnetization configurations in ferromagnets and ferri- and antiferromagnets, such as domain walls, vortices, skyrmions, and hopfions, etc., are currently a focus of researchers working in the area of solid-state magnetism. However, the existence of the different stable 3D magnetization configurations M(r) and the role of 3D (Hopf index) and 2D (skyrmion number) topological charges in the magnetization dynamics are still subject of intensive research. Nowadays, complicated 3D magnetization configurations in ferromagnetic media can be observed experimentally using electron holography or X-ray magnetic imaging [8,9].

The magnetization field M(r) is the vector order parameter of a ferromagnet. The magnetization configuration M(r) in 3D space is represented by the unit field vector m(r) = M(r)⁄|M(r)| depending on three spatial coordinates $\mathit{r} \left(x_1,x_2,x_3\right)$. There are particular cases when the magnetization configurations depend only on one spatial coordinate (domain walls) or two spatial coordinates (magnetic vortices and skyrmions in thin magnetic films or in flat magnetic dots). For stable magnetization configurations $\mathit{m}\left(\mathit{r}\right)$, it is possible to calculate topological charges, which describe the degrees of mappings (homotopy invariants) of a 1D- ($R^1$) or 2D coordinate space ($R^2$) to the unit sphere m² = 1 in the magnetization space S₂(m), i.e., $R^1$ → S₂(m), $R^2$ → S₂(m). The mapping of the 3D coordinate space $R^3$(r) to the magnetization unit sphere m² = 1 (S₂) is more complicated and is related to the Hopf index (3D topological charge). The corresponding magnetization configurations with a non-zero Hopf charge are called magnetic hopfions. The 3D magnetization textures are topologically equivalent if they have the same degree of mapping (the Hopf index). The magnetic energy of the given field configuration $E\left[\mathit{m}\left(\mathit{r}\right)\right]$ can be calculated as a functional of the unit magnetization field m(r) and spatial derivatives of the vector m(r). Such energy includes the exchange, magnetostatic, magnetic anisotropy, etc., energy contributions. The minimization of the energy functional $E\left[\mathit{m}\left(\mathit{r}\right)\right]$ yields some equilibrium (stable or metastable) magnetization field configurations $\mathit{m}\left(\mathit{r}\right)$. Then, the magnetization dynamics of these equilibrium configurations can be considered on the base of the Landau–Lifshits equation of the magnetization motion, and any parameters, including the topological charges, can be calculated.

In this review, I consider 3D magnetic topological solitons—hopfions and Bloch points. In particular, toroidal hopfions, which were introduced in field theory, are analyzed in detail. Such magnetic solitons reveal intriguing and novel physical properties due to their topologically non-trivial 3D magnetization configurations $\mathit{m}\left(\mathit{r}\right)$. The article is organized as follows. The basic properties of the hopfions in classical field theory are considered in Section 2. The concept of the emergent electromagnetic field and definition of 3D topological charges of the magnetic textures are presented in Section 3. The static and slow dynamics behaviors of the magnetic hopfions are considered in Section 4. The fast linear and nonlinear hopfion dynamics are considered in Section 5. The review is concluded by a summary in Section 6.

2. Hopfions in the Classical Field Theory

Topological three-dimensional solitons with a non-zero Hopf charge, named hopfions, were first introduced in classical field theory in connection with the non-linear $\sigma$-model [7,10,11,12]. The explicit form of the mapping of a 3D space ($R^3$) to the surface of the unit sphere ${\mathit{n}}^2=1$ of the classical field $\mathit{n}\left(\mathit{r}\right)$, $R^3\to S_2$, was introduced by Hopf in 1931 [13]. Later, it was found that the Hopf charge (a degree of the mapping $R^3\to S_2\left(\mathit{n}\right)$) can be expressed as some integral of the function composed by a continuous classical field $\mathit{n}\left(\mathit{r}\right)$ (for instance, the magnetization field $\mathit{m}\left(\mathit{r}\right)$) and spatial derivatives of the field $\mathit{n}\left(\mathit{r}\right)$ [14].

Faddeev [15] suggested a new Lagrangian to describe the field $\mathit{n}\left(\mathit{r}\right)$ (sometimes referred to as the Faddeev’s or Faddeev–Skyrme´s Lagrangian), which has stable soliton solutions for the three-component classical field $\mathit{n}\left(\mathit{r}\right)$ in 3D coordinate space. The stable solutions conserve a 3D topological charge (the Hopf charge). The Faddeev–Skyrme´s Lagrangian is a linear combination of two invariants of the rotation group O(3) represented by the spatial derivatives of the vector field $\mathit{n}\left(\mathit{r}\right)$. Then, de Vega [16] explicitly found the field $\mathit{n}\left(\mathit{r}\right)$ components (calling the field configurations “closed vortices”) for the simplest non-trivial unit Hopf charge in the toroidal coordinates. Such hopfions are now called toroidal hopfions or torus-like vortex rings. Nicole [17] suggested an analytical form of the Hopf mapping of 3D-coordinate space $R^3$ to the surface of the unit sphere ${\mathit{n}}^2=1$, $R^3\to S_2\left(\mathit{n}\right)$, which allowed for the explicit introduction of analytic equations for the soliton field configuration with the unit Hopf index. Later, motivated by the excellent paper by Faddeev and Niemi [7], a series of papers on the toroidal hopfions in classical field theory were published [10,11,12,18]. Gladikowski et al. [10] introduced the field configurations $\mathit{n}\left(\mathit{r}\right)$ of the toroidal hopfions with an arbitrary Hopf index and presented explicit expressions for the emergent magnetic field and vector potential components in the toroidal coordinates. The toroidal coordinates play a special role in the theory of the hopfions because the analytical equations are essentially simplified in these curvilinear coordinates. The toroidal hopfions attracted considerable attention from researchers because such topological solitons are special stable solutions of the Faddeev–Skyrme´s model [15]. A recent review can be found in Ref. [1].

For calculations of the soliton topological charges, it is important to distinguish localized and non-localized solitons. The field $\mathit{n}\left(\mathit{r}\right)$ approaches a constant vector value ${\mathit{n}}_0$ at infinity $\left|\mathit{r}\right|\to \infty$ for the localized solitons. The field is inhomogeneous at infinity $\left|\mathit{r}\right|\to \infty$ for non-localized solitons. The condition $\mathit{n}\left(\mathit{r}\right)\to {\mathit{n}}_0$ means that the Hopf index, which characterizes the different homotopy classes for the mapping $R^3\to S_2\left(\mathit{n}\right)$, is an integer number ${\pi }_3\left(S_2\right)=Z$ in infinite media [7]. Therefore, the toroidal hopfions, which are described by an integer Hopf index $Q_H=0, \pm 1,\pm 2 \dots$, represent a class of the hopfions in ordered media. The Hopf index of the toroidal hopfions can be represented as a product of two winding numbers [10], the planar winding (the azimuthal vorticity) and the twisting of the field configuration $\mathit{n}\left(\mathit{r}\right)$ along the hopfion tube (the poloidal vorticity), respectively. Due to the property that the vector field is asymptotically trivial, $\mathit{n}\left(\mathit{r}\right)\to {\mathit{n}}_0$ at $\left|\mathit{r}\right|\to \infty$, it is possible to compactify the three-dimensional coordinate space to the four-dimensional unit sphere surface, $R^3\to S_3$. The toroidal hopfions as localized solitons of a classical three-dimensional vector field resemble particle-like objects.

To illustrate the toroidal hopfion field $\mathit{n}\left(\mathit{r}\right)$, we introduce the toroidal coordinates $\mathit{r}\left(\eta ,\beta ,\phi \right)$. The connection between the cylindrical $\left(\rho ,\phi ,z\right)$ and toroidal $\left(\eta ,\beta ,\phi \right)$ coordinates is $\rho =a{\tau }^{-1}sinh\left(\eta \right)$, $z=a{\tau }^{-1}sin\left(\beta \right)$, $\phi =\phi ,$ and $\tau =cosh\left(\eta \right)-cos\left(\beta \right)$, where the toroidal parameter $\eta$ varies from 0 to $\infty$, the poloidal angle $\beta$ varies from $-\pi$ to $\pi$, the azimuthal angle $\phi$ varies from 0 to $2\pi$, and a is a scale parameter (the hopfion radius in the plane $z=0$). The toroidal hopfion field components are [19]

$$\begin{gathered} n_z\left(\eta \right)=p{\displaystyle \frac{1-{cosh}^{2m}\left(\eta \right){tanh}^{2n}\left(\eta \right)}{1+{cosh}^{2m}\left(\eta \right){tanh}^{2n}\left(\eta \right)}}, \\ {n}_x\left(\mathit{r}\right)+i{n}_y\left(\mathit{r}\right)=\sqrt{1-n_z^2\left(\eta \right)}exp\left[i\left(n\phi +m\beta \right)\right], \end{gathered}$$

(1)

where $p=n_z\left(\eta =0\right)$ is the hopfion polarity, $p=\pm 1$, and the integer numbers m, n are the hopfion poloidal and azimuthal vorticities, respectively.

The profiles of the out-of-plane field component, $n_z\left(\rho \right)$, of the toroidal hopfions are shown in Figure 1 for the different hopfion vorticities $\left(m,n\right)$. The profiles can be plotted by substituting the poloidal coordinate $\eta \left(\rho ,z\right)=atanh\left(2a\rho /\left({\rho }^2+z^2+a^2\right)\right)$ to Equation (1). The component $n_z\left(0\right)=n_z\left(\infty \right)$ = 1 and $n_z\left(a\right)=-1$ at the hopfion radius $\rho =a$ in the basal plane $z=0$ if the hopfion polarity $p=+1$.

Nowadays, different kinds of hopfions are investigated in the field of condensed matter physics (magnetic media [20], liquid crystals and colloids [5], ferroelectrics [21]) and in electromagnetism and gravitation [22] photonics [3], optics [4], etc. There is a growing interest in 3D inhomogeneous magnetization textures classified by a linking number of the preimages of two different points on an S₂(m) unit sphere in the 3D coordinate space ($R^3$), i.e., by the non-zero Hopf index. For a given magnetization texture $\mathit{m}\left(\mathit{r}\right)$, it is possible to plot preimages, the 3D curves in real space, $\mathit{r}\left({\mathit{m}}_1\right)$ and $\mathit{r}\left({\mathit{m}}_2\right)$ of two points ${\mathit{m}}_1$ and ${\mathit{m}}_2$ on the magnetization unit sphere ${\mathit{m}}^2=1$ and visually check the number of their crossings (a linking number of the preimages). The linking number introduced solely for the toroidal hopfions is an integer number by definition and is equal to the Hopf index of a magnetization texture $\mathit{m}\left(\mathit{r}\right)$. However, aside from the toroidal hopfions, there are other kinds of hopfions, and the Hopf index is, in general, not integer and cannot be calculated as the linking number of preimages. Although, 3D localized topological magnetic solitons were introduced by Dzyaloshinskii et al. [23] a long time ago, and the reincarnation of interest to such magnetization field 3D textures started after the publication of the papers by Sutcliffe [24,25] relatively recently.

The simplest magnetic toroidal hopfions with the Hopf index $\left|Q_H\right|=1$were considered in infinite ferromagnetic films [26,27] and cylindrical dots [25,28,29,30] using the vector field toroidal hopfion ansatz [10,11]. It was shown numerically that the toroidal hopfions with $\left|Q_H\right|=1$ can be the ground state of circular ferromagnetic nanodots [29] assuming a strong surface uniaxial magnetic anisotropy. The first experimental observation of the 3D magnetization configurations interpreted as magnetic hopfions was carried out in the Ir/Co/Pt multilayer films [31]. It is shown in Ref. [32] that aside from the toroidal magnetic hopfions, there is, in soft magnetic materials, another class of the magnetization textures with a non-zero Hopf index—the Bloch points. The skyrmion-like winding magnetic hopfions are considered in Section 5.

For the simplest magnetic hopfion with the vorticities $m=1, n=1$, $\left|Q_H\right|=1$, the hopfion magnetization field components (1) in the cylindrical coordinates are [11]

$$m_z=1-{\displaystyle \frac{8{\rho }^2{a}^2}{{\left(a^2+{\rho }^2+z^2\right)}^2}}, m_x+i{m}_y={\displaystyle \frac{4a\rho exp\left(i\left(\phi +{\phi }_0\right)\right)\left(2za+i\left({\rho }^2+z^2-a^2\right)\right)}{{\left({a^2+\rho }^2+z^2\right)}^2}}.$$

(2)

It is evident from Equation (2) that the toroidal hopfion magnetization field $\mathit{m}\left(\mathit{r}\right)$ approaches a constant vector at infinity $\left|\mathit{r}\right|\to \infty$, $\mathit{m}\left(\infty \right)=\left(0,0,1\right)$, as it should be for the localized solitons. The constant angle ${\phi }_0$ describes a freedom of choosing the directions of the $Ox$ and $Oy$ coordinate axes in the basal $xOy$ plane. According to Ref. [27], we can distinguish Bloch hopfions (${\phi }_0=0,\pi$) and Neel hopfions ${\phi }_0=\pm \pi /2$ analogously to the Bloch and Neel skyrmions [33]. The magnetization rotates in the $\widehat{\mathit{\phi }}\widehat{\mathit{z}}$ plane ($m_{\rho }=0$) for the Bloch hopfions or in the $\widehat{\mathit{\rho }}\widehat{\mathit{z}}$ plane ($m_{\phi }=0$) for the Neel hopfions when one goes from the coordinate system origin $\mathit{r}=0$ to the hopfion radius $\rho =a$ along the radial direction $\widehat{\mathit{\rho }}$ in the $xOy$ plane.

The in-plane magnetization configurations ($z=0$) of the toroidal Bloch and Neel hopfions plotted according to Equation (2) are shown in Figure 2. The spatial distribution of the magnetization components $m_x,m_z$ of the toroidal Bloch hopfion $\left({\phi }_0=0\right)$with the vorticities $\left(m,n\right)=\left(1,1\right)$ plotted in the $xz$ plane is shown in Figure 3.

3. Emergent Electromagnetic Field and the Hopf Index

An inhomogeneous and moving magnetization texture $\mathit{m}\left(\mathit{r},t\right)$ results in the appearance of effective magnetic and electric fields, which act on other subsystems. These so-called emergent magnetic and electric fields are related to the spatial and time derivatives of the magnetization field $\mathit{m}\left(\mathit{r},t\right)$. Initially, this effect was calculated by Korenman et al. [34] and Volovik [35] as a result of the exchange interaction of the local spins with the spins of itinerant electrons in ferromagnetic metals. Then, it was shown [36] that this is a more general, pure geometrical effect related to the choice of the local moving coordinate frame with the Oz axis parallel to the local instant magnetization $\mathit{m}\left(\mathit{r},t\right)$.

The emergent electromagnetic field tensor can be written as

$$F_{\mu \nu }\left(\mathit{m}\right)=\mathit{m}·\left({\partial }_{\mu }\mathit{m}\times {\partial }_{\nu }\mathit{m}\right),$$

(3)

where ${\partial }_{\mu }=\partial /\partial x_{\mu }$ denotes spatial and time derivatives. The indices $\mu , \nu =0, 1, 2 ,3$, where $x_0=ct$ and $\mathit{r}=\left(x_1,x_2,x_3\right)$, correspond to the components of a 3D radius–vector r in an orthogonal coordinate system. The field tensor (represented in the units of ${\mathsf{\Phi }}_0/4\pi$, where ${\mathsf{\Phi }}_0=h/e$ is the magnetic flux quantum) is related to the emergent field four-component vector potential $\mathcal{A}=\left(A_0,A_1,A_2,A_3\right)$ as $F_{\mu \nu }\left(\mathit{m}\right)={\partial }_{\mu }{\mathcal{A}}_{\nu }\left(\mathit{m}\right)-{\partial }_{\nu }{\mathcal{A}}_{\mu }\left(\mathit{m}\right)$. The emergent magnetic field (sometimes called gyrocoupling density) $\mathit{B}=\nabla \times \mathit{A}$, $\mathit{A}=\left(A_1,A_2,A_3\right)$ can be defined as in classical electrodynamics $B_i\left(\mathit{m}\right)={\epsilon }_{ijk}{F}_{jk}\left(\mathit{m}\right)/2$ [36], where the indices i, j, k mark the spatial coordinates $x_i$. It is important that the emergent field $\mathit{B}$ is a divergence-free vector field, $\nabla ·\mathit{B}=0$. The flux of the emergent magnetic field $\mathit{B}=\nabla \times \mathit{A}$ through a closed surface defines a 2D topological charge (skyrmion number).

The emergent magnetic field introduced above differs from the real magnetic field defined in standard electrodynamics and is “fictious” in some sense. Nevertheless, the emergent field results in some experimentally measurable effects. Prominent examples of such effects are the topological Hall effect (influence of the emergent magnetic field on trajectories of the conductivity electrons) and skyrmion Hall effect (appearance of the magnetic soliton velocity component transverse to the driving force direction due to the gyroforce). A theoretical approach to exploiting the topological Hall resistance to electrically detect the magnetic hopfion 3D magnetization textures was recently suggested by Göbel et. al. [30]. The flux of the emergent magnetic field trough the plane z = const in 2D ferromagnets determines the gyrovector, an important parameter to describe the motion of 2D topological solitons, magnetic vortices, and skyrmions. The consequences of the non-zero gyrovector in 2D nanostructures have been investigated many times experimentally. A prominent example is the vortex/skyrmion gyrotropic excitation mode immediately related to the gyrovector (see a recent review [33]).

The dot product $\mathit{A}·\mathit{B}$ of the emergent field vector potential $\mathit{A}$ and the emergent magnetic field $\mathit{B}$ defines a 3D topological charge or the Hopf index [37]

$$Q_H={\displaystyle \frac{1}{{\left(4\pi \right)}^2}}\int d^3\mathit{r}\mathit{A}\left(\mathit{m}\right)·\mathit{B}\left(\mathit{m}\right)$$

(4)

of a 3D magnetization texture $\mathit{m}\left(\mathit{r}\right)$. There is a gauge freedom in choosing the vector potential $\mathit{A}$. To have a physical sense, the Hopf index should be gauge invariant. The necessary condition for infinite samples is either the nullification of the emergent magnetic field at the sample borders S, ${\mathit{B}}_S=0$, or the field $\mathit{B}$ being tangential to the sample surface, $\mathit{B}·\mathit{\sigma }=0$, where $\mathit{\sigma }$ is the external normal to the surface. The condition ${\mathit{B}}_S=0$ is satisfied for the localized solitons (toroidal hopfions, for instance) in infinite samples. The condition $\mathit{B}·\mathit{\sigma }=0$, in general, is not satisfied at the sample surface. It was shown in Ref. [19] for the toroidal hopfions that only a particular choice of $\mathit{A}=-2\nabla \gamma +{\mathit{A}}^e$, ${\mathit{A}}^e=\left(1-cos\mathsf{\Theta }\right)\nabla \mathsf{\Phi }$, related to the hopfion helicity $\gamma$ leads to the integer and invariant values of the Hopf indices $Q_H$ in infinite samples, allowing for considering them as the degrees of mapping (the linking numbers or integer numbers of crossings of the magnetization configuration $\mathit{m}\left(\mathit{r}\right)$ preimages). Here, $\mathsf{\Theta }\left(\mathit{r}\right)=\mathsf{\Theta }\left(\rho ,z\right)$, $\mathsf{\Phi }\left(\mathit{r}\right)=n\phi +\gamma \left(\rho ,z\right)$, are the spherical angles $\mathsf{\Theta },\mathsf{\Phi }$ of the magnetization, and the hopfion helicity is represented by the poloidal angle $\beta$ as $\gamma \left(\rho ,z\right)=m\beta \left(\rho ,z\right)$, $\beta \left(\rho ,z\right)=atan\left(2az/\left({\rho }^2+z^2-a^2\right)\right)$ [19]. The gauge invariance should be considered with respect to this particular choice of the vector potential $\mathit{A}=-2\nabla \gamma +{\mathit{A}}^e$. In the case of finite samples and magnetization textures different from the toroidal hopfions, the situation is more complicated. It is difficult to choose a proper form of the emergent field vector potential and prove the gauge invariance of the Hopf index. The important question about an integer Hopf index and its gauge invariance should be carefully investigated for each magnetization texture in a restricted geometry.

Recently, Zheng et al. [38] reported on a direct experimental observation of the magnetic toroidal hopfions forming coupled states with skyrmion tube strings in the submicron FeGe plates. They also provided a theoretical interpretation of the observed hopfions (hopfion rings). Zheng et al. [38] used the hopfion ring magnetization to calculate the Hopf invariant (Hopf index) by applying the concept of the emergent magnetic field. The hopfion topological charge ($Q_H$) was calculated for confined samples. However, the large values of $Q_H=5, 6, 10, 12$ are very strange from the point of view of the theory of 3D magnetization configurations (including the magnetic hopfions [7,19,27]). The calculation method resulted in such large integer values of $Q_H$, and gauge invariance of the calculated values of the Hopf index $Q_H$ should be carefully analyzed.

Obviously, the magnetization configurations considered by Zheng et al. are not the so-called toroidal hopfions (torus-like vortex rings) introduced in field theory for infinite media; see Refs. [19,27] and references therein. It was proved that the Hopf index is an integer only for the toroidal hopfions in infinite space (see Section 2). In [38], Zheng et al. considered neither toroidal magnetic hopfions nor infinite samples. Therefore, there is no ground for the speculations about the integer Hopf index and its equivalence to the linking number of their 3D magnetization textures. The Hopf index can be calculated by Equation (4) and is, in general, non-integer for arbitrary 3D magnetization configurations in a finite domain like the ones considered in the paper by Zheng et al. [38].

The strong statement about the integer Hopf index of an arbitrary magnetization configuration in a finite sample was made by Zheng et al. without proof or a proper reference. In modern hopfion theory, such a statement is only valid for the toroidal hopfions in infinite media. The simulations of the equilibrium magnetization configurations $\mathit{m}\left(\mathit{r}\right)$ and the corresponding Hopf indices were conducted by Zheng et al. for finite samples: 0.5 μm diameter circular dot with a thickness of 180 nm; the square plate, 180 × 1000 × 1000 nm or “bulk” sample 700 × 350 × 350 nm. None of the samples used for the simulations in Ref. [38] is an infinite sample.

The Hopf index can be calculated only if an explicit expression for the vector potential A of the emergent magnetic field is known. The authors of Ref. [38] suggested a heuristic equation to calculate the vector potential A. However, they did not discuss the important problem of the vector potential gauge and the gauge invariance of the Hopf index. It was shown in Ref. [27] that to make the Hopf index (4) gauge invariant, the emergent field B(r) components should decay fast enough when approaching zero and increasing the absolute value of the radius vector r up to infinity $\left|\mathit{r}\right|\to \infty$. This is possible to implement only in an infinite sample. In other words, the Hopf index is invariant and does not depend on the vector potential gauge in infinite samples. Even if one introduces the artificial boundary condition for the magnetization $\mathit{m}$ in a finite sample at the sample borders ($\mathit{m}={\mathit{m}}_0=const$) as Zheng et al. [38] suggested, such boundary conditions do not guarantee that the emergent magnetic field B goes to zero at the borders because this field is defined via the magnetization space derivatives, not via the magnetization $\mathit{m}$ itself. There is a specific case of the axially symmetric magnetization configuration $\mathit{m}\left(\mathit{r}\right)=\mathit{m}\left(\rho ,z\right)$ in a cylindrical sample [19,27] ($\mathit{r}=\left(\rho ,\phi ,z\right)$ are the cylindrical coordinates), for which $\mathit{B}=0$ at the sample surface if $\mathit{m}={\mathit{m}}_0=\widehat{\mathit{z}}$ ($\widehat{\mathit{z}}$ is a unit vector in the out-of-plane direction) at the borders. None of the magnetization configurations simulated by Zheng et al. is axially symmetric. Moreover, in many cases, the square plates or “bulk” rectangular samples were used for the simulations. The boundary condition $\mathit{m}=\widehat{\mathit{z}}$ used in Ref. [38] means a strong surface uniaxial magnetic anisotropy. It has no physical sense for such magnetic material as FeGe with a cubical crystal structure [39]. Therefore, we can exclude this particular case of axial symmetry from the consideration. It is important that if $\mathit{B}\ne 0$ at the sample borders, then the Hopf index is not gauge invariant [27] and cannot be used to characterize a magnetization texture $\mathit{m}\left(\mathit{r}\right)$. Even if one can suggest an artificial expression for the vector potential like the equation used by Zheng et al. to satisfy the equation $\mathit{B}=\nabla \times \mathit{A}$, this does not mean that the introduced vector potential is correct.

An expression, alternative to Equation (4) for the Hopf index, is suggested in Refs. [40,41]. This expression is based on the double volume integral from the emergent magnetic field B(r) components and can be derived as a result of the application of the Helmholtz theorem [42] (the fundamental theorem of vector calculus) to the field B(r). An important consequence of the Helmholtz theorem used in Refs. [40,41] is the integral representation of the vector potential A via the emergent field B, namely

$$\mathit{A}\left(\mathit{r}\right)=-{\displaystyle \frac{1}{4\pi }}\int d^3{\mathit{r}}^{\prime }{\displaystyle \frac{\mathit{R}\times \mathit{B}\left({\mathit{r}}^{\prime }\right)}{R^3}}, \mathit{R}=\mathit{r}-{\mathit{r}}^{\prime }.$$

(5)

This expression for $\mathit{A}\left(\mathit{r}\right)$ (5) is different from the expression for the emergent field vector potential given by Zheng et al. [38]. Therefore, the heuristic vector potential introduced by Zheng et al. contradicts the Helmholtz theorem [42] and the integral expressions for the Hopf index used in Refs. [40,41]. The similar wrong heuristic expressions for the vector potential (represented as one-fold integral from the emergent magnetic field B) were used in Refs. [20,43] without any justification.

An expression for the vector potential was used by Liu et al. [28] for calculations of the Hopf index of the circular magnetic dots in the momentum representation. Liu et al. calculated the Hopf index to be 0.96 for their dot parameters. However, Liu et al. [26,28] used the Coulomb gauge for the emergent field vector potential, $\nabla ·\mathit{A}=0$, which is not compatible with Equation (5) for the potential $\mathit{A}\left(\mathit{r}\right)$ given above. The gauge $\nabla ·\mathit{A}=0$ is unphysical for the particular case of the toroidal hopfions. This can be checked if one uses the toroidal vector potential components calculated explicitly in Ref. [19].

The value of the Hopf index represented by Equation (4) depends essentially on the choice of the vector potential $\mathit{A}$ of the emergent magnetic field. Recently, it was shown [19] that a definite gauge of the vector potential $\mathit{A}$ should be chosen for the toroidal hopfions in infinite samples to guarantee the integer values of the Hopf index $Q_H$. The heuristic vector potential given by Zheng et al. [38] can guarantee neither the integer Hopf indices nor their stability with respect to the vector potential gauge change in the finite samples. It seems that the integer values of $Q_H$ simulated in the paper by Zheng et al. [38] are some arbitrary and ungrounded numbers resulting in a confusion of the readers who are not specialists in the theory of 3D magnetic textures (hopfions, in particular). Therefore, the methods and calculations used in Ref. [38] should be reconsidered to bring them in accordance with the hopfion theory.

4. Static Properties of Magnetic Hopfions

Recently, the stability of the toroidal hopfions was studied numerically in chiral ferromagnets FeGe [28,29] and in confined ferroelectric nanoparticles (nanospheres of PbZrTi oxide) [21]. Based on the previous theoretical predictions [28,29], the first experimental observation of magnetic toroidal hopfions was carried out in the Ir/Co/Pt multilayer systems [31] with ultrathin Co layers. A restricted sample cylindrical geometry was used for the chiral ferromagnets FeGe and Ir/Co/Pt with a strong Dzyaloshinskyi-Moriya interaction and out-of-plane uniaxial magnetic anisotropy on the sample faces. The calculated Hopf index $Q_H$ for chiral ferromagnets was equal to 1 [29,31] or close to 1 [28]. It was simulated in Ref. [28] that the toroidal hopfions are metastable states of a chiral nanodisk existing at a large-enough disk thickness and radius. Their magnetic energy $E\left[\mathit{m}\left(\mathit{r}\right)\right]$ is higher than the energy of the monopole–antimonopole pair 3D magnetization configuration.

The hopfions were also considered in ferroelectrics. The Hopf index of the polarization field $\mathit{P}$ was defined using a non-standard equation in Ref. [21]. If one assumes that the polarization field P(r) for ferroelectrics is similar to the magnetization field M(r) for ferromagnets, then the definition of the Hopf index should be the emergent magnetic field B(r), not the field M(r) (or P(r)). Therefore, the parameter H defined in Ref. [21] as a volume integral from the dot product $\mathit{P}·\mathit{A}$ is not a Hopf index, although the spatial configuration of the field P(r) is similar to the one for the toroidal hopfion of the magnetization field M(r). It is not a surprise that the Hopf index H is not an integer for the considered spherical samples. The calculated values of the Hopf index are not listed in Ref. [21].

The magnetization configurations of the toroidal magnetic hopfions in infinite media with the integer Hopf index $Q_H=mn$ and the arbitrary poloidal and azimuthal hopfion vorticities m, n were explicitly calculated in Ref. [19] using the toroidal and cylindrical coordinates. The calculation method was based on the Hopf mapping’s explicit definition and the concept of the emergent magnetic field defined via spatial derivatives of the magnetization field $\mathit{m}\left(\mathit{r}\right)$. It was shown that the Hopf index density can be represented as a Jacobian of the transformation from the toroidal to the cylindrical coordinates [19]. The calculated components of the emergent magnetic field and emergent field vector potential can be used, in particular, for explanations of the topological and skyrmion Hall effects of the toroidal magnetic hopfions.

The separate problem is the stability of the different 3D magnetization textures $\mathit{m}\left(\mathit{r}\right)$. It was proven within field theory [44,45] that for any physical systems with a squared gradient field term in the Lagrangian, there are no stable, stationary, localized solutions in the 3D case for any form of the potential. This statement is known as the Hobart–Derrick theorem. However, stable localized solutions (localized solitons) may exist if any energy contributions linear with respect to spatial derivatives or with higher-order spatial derivatives of the magnetization field are present in the Lagrangian [46,47]. In the theory of magnetism, there are some specific energy terms with the first-order derivatives, so-called Lifshitz invariants, accounting for the Dzyaloshinskii–Moriya interactions (DMI) in ferro-magnetic materials with broken inversion symmetry. Another opportunity to obtain a stable 3D magnetization field configurations is accounting for the higher-order spatial derivatives in the Lagrangian. The term quartic in spatial derivatives was introduced by Skyrme [48] and Faddeev [15] a long time ago within classical field theory. It was shown that the Faddeev–Skyrme Lagrangian

$$\mathcal{L}={\displaystyle \frac{1}{2}}\int d^3\mathit{r}\left[{\left({\partial }_{\mu }\mathit{n}\right)}^2+g{\left(F_{\mu \nu }\left(\mathit{n}\right)\right)}^2\right]$$

(6)

has stable 3D localized soliton field solutions in the form of toroidal hopfions [7,10,11,12]. Here, $g$ is the coupling constant.

The toroidal hopfions belong to a class of the localized topological solitons and are characterized by integer values of a 3D topological charge (Hopf index) [13,14]. It was shown recently [20] that the classical Heisenberg model with competing long-range exchange interactions results in quadratic terms in the second spatial derivatives of the magnetization field. Although such Heisenberg model is beyond the standard theory of micromagnetism, it may lead to the stabilization of the toroidal magnetic hopfions. The question is this: is it possible to stabilize the toroidal magnetic hopfions in a ferromagnet with the standard micromagnetic exchange energy (avoiding exotic exchange interactions) due to non-zero DMI terms and/or magnetostatic energy? Such micromagnetic energy contributions are beyond the field theory frameworks and, therefore, the applicability of the Hobart–Derrick theory to the evaluation of the stability of the magnetization field configurations should be reconsidered. The simple scaling analysis accounting for the DMI energy terms was conducted in Ref. [49]. However, this scaling analysis ignored the finite sample sizes and the magnetostatic interaction (which is unavoidable in real ferromagnetic samples). The complicated non-local magnetostatic energy is usually not accounted for in the theory of magnetic skyrmions and hopfions or accounted for in a simplified form. The skyrmions are usually considered in the bulk ferromagnetic crystals without inversion symmetry or in ultrathin magnetic films. In both cases, the magnetostatic energy is reduced to a local form and accounted for as an extra contribution to the magnetic anisotropy energy. Accounting for the magnetostatic energy in relatively thick ferromagnetic dots [50] allows us to stabilize quasi-2D magnetic skyrmions without the presence of any DMI terms if a small out-of-plane magnetic uniaxial anisotropy is included in the energy functional. The magnetostatic energy was not included in the energy functional in Refs. [25,29,30] describing the toroidal magnetic hopfions in thick cylindrical dots. The hopfion’s constant phase angle ${\phi }_0$ introduced in Section 2 does not contribute to the exchange energy. However, the magnetostatic and DMI energies depend on the particular value of ${\phi }_0$ and are different for the Bloch and Neel toroidal hopfions. We note that the magnetostatic energy can also lead to the stabilization of other kinds of 3D magnetization textures: the Bloch point hopfions with non-zero Hopf charge or half-hedgehog (3D quasi-Neel skyrmion) magnetization textures even in soft magnetic materials with no DMI [32,51,52].

It is demonstrated in Ref. [53] that the Bloch toroidal hopfion magnetization texture is a metastable state of a thick cylindrical dot or a long cylindrical wire of a finite radius R. The existence of this metastable state is a result of a competition of the exchange and magnetostatic energies. The Dzyaloshinskyi–Moriya exchange energy and magnetic uniaxial anisotropy are of the second importance for the toroidal hopfion stabilization. The important role of the magnetostatic interaction in the toroidal hopfion stabilization in a cylindrical dot was confirmed by simulations [43].

5. Dynamics of Magnetic Hopfions

A numerical study of the spin excitation spectra of the toroidal magnetic hopfions was performed in recent papers [54,55]. Very recently, theoretical papers on the magnetic hopfion dynamics [26,43,56,57] were published.

The spin-polarized electrical current-induced 3D dynamics of the toroidal magnetic hopfions were studied both analytically (the collective coordinates approach) and numerically in Ref. [26] in an infinite frustrated ferromagnet (different signs of the exchange integrals for the nearest neighbor spins). The hopfions exhibit complicated dynamics including a longitudinal motion along the current direction, a transverse motion with respect to the current direction, a rotational motion, and dilation [26]. The skyrmion Hall effect (due to non-zero hopfion gyrovector) is clearly seen in the solutions, although the gyrovector was not explicitly defined. The calculated hopfion velocity is quite small, <10 m/s, for the typical values of the current intensity. The results by Liu et al. [26] are in some contradiction with the hopfion dynamics calculated by Wang et al. in Ref. [27], where the hopfion’s internal structure dynamics were ignored. Wang et al. did not observe any skyrmion Hall effect in the course of the toroidal hopfion motion in magnetic stripes. The hopfions moved under the drive of the spin-transfer torque or spin Hall torque along the nanostripe with velocities up to 20 m/s. The origin of this discrepancy is still unclear. The Hopf index was $\left|Q_H\right|=1$ in both cases.

The hopfion dynamics depend essentially on the phase angle ${\phi }_0$ defining the toroidal hopfion static magnetization (2). The spin wave excitation spectra of the Bloch and Neel toroidal hopfions in FeGe nanodisks were investigated numerically in Ref. [43]. It was shown that the Bloch hopfions reveal very dense eigenfrequencies in the low-frequency region (<5 GHz). Only a few eigenfrequencies were detected for the Neel hopfions. The eigenmode spatial distributions were found for some selected spin excitation modes. The modes were mainly localized at the disk edge for the Bloch hopfions or near the dot center for Neel hopfions [43]. The eigenmode classification according to their symmetry or number of the nodes in different spatial directions were not presented. It seems that the Bloch hopfion frequency spectra are numerical artifacts and further careful analysis is necessary. The Hopf index of the Neel hopfions calculated using the concept of the emergent electromagnetic is above 0.9. It is unclear why the Neel hopfions transform from a circularly symmetric shape (azimuthal symmetry) to a square shape, increasing the axial magnetic field [43]. There is no reason to break the azimuthal symmetry of the static magnetization configurations.

The spin excitation modes of the Bloch toroidal hopfions in nanodisks (the radius R is 100 nm and thickness L is 70 nm) were simulated in Ref. [54] in a wide frequency range of 0–20 GHz. The spin eigenmodes were classified according to their symmetry, number of the nodes in the azimuthal and out-of-plane directions, and degree of the mode localization. The dot edge, dot middle, and dot center localized modes were identified. No simple rule establishing a connection of the mode localization with its frequency was found. The edge and middle localized modes had large azimuthal indices. Such complicated inhomogeneous mode patterns mean that the ferromagnetic resonance intensity (average mode volume magnetization) of the modes is small and it will be difficult to observe these modes by the standard ferromagnetic resonance spectroscopy technique. It was found numerically [55] that the toroidal hopfions have distinctly less resonance peaks in comparison with skyrmion tubes. It was also found in Ref. [55] that the hopfion breathing excitation modes (oscillations of the hopfion radius) and rotating spin modes (magnetization rotates in the hopfion basal plane $xOy$) hybridize, applying the oscillation magnetic field along the hopfion axis 0z.

As it is well known, the skyrmion Hall effect (existence of a gyroforce, which is perpendicular to the soliton velocity) for magnetic topological solitons is attributed to a non-zero gyrovector. The gyrovector, in the case of the magnetic vortices and skyrmions, determines their low-frequency dynamics in both 3D and 2D cases [58,59]. Therefore, the problem of the calculation of the magnitude and direction of the gyrovector of the magnetic hopfions is of practical importance. The components of the emergent magnetic field in the cylindrical coordinate system, defining the gyrovector $\mathit{G}$ of the toroidal magnetic hopfion, are presented in Ref. [27]. It is proven unambiguously [27] that the gyrovector out-of-plane component $G_z$ of an axially symmetric toroidal magnetic hopfion with the Hopf index $\left|Q_H\right|=1$ is equal to zero. Concerning the zero values of the two in-plane gyrovector components, only a plausible assumption is considered that they have to vanish due to the toroidal hopfion and system symmetry. Nevertheless, this assumption was accepted in a number of subsequent articles, where the equality to zero of all components of the toroidal hopfion gyrovector $\mathit{G}$ was mentioned as a proven fact [56,57,60]. Meanwhile, this issue needs to be clarified, since the possible non-zero values of the gyrovector components affect essentially both the three-dimensional current-induced hopfion translational motion [26] and the spin waves excited over the hopfion background [56] by the external magnetic field. Liu et al. [57] excluded the toroidal hopfion gyrovector from consideration and represented the hopfion dynamics solely in terms of the hopfion emergent magnetic field toroidal $\mathit{T}=1/2\int dV\left(\mathit{r}\times \mathit{B}\right)$ and octupole moments. Such an approach resulted in a specific toroidal hopfion dynamic presented in Ref. [57]. The static toroidal hopfion ansatz for the particular case of the Hopf index $\left|Q_H\right|=1$ in Ref. [26] was initially written incorrectly, and then it was corrected in a paper (Ref. [60]) by the same group.

The calculation approach suggested in Ref. [61] is based on the concept of the emergent magnetic field $\mathit{B}\left(\mathit{r}\right)$ introduced in Section 3 and a calculation of the emergent field components and their volume averages for the toroidal hopfion magnetization texture in the appropriate curvilinear coordinate systems. The simplest non-trivial toroidal hopfion with the Hopf index $\left|Q_H\right|=1$ in the cylindrical magnetic dot was considered, and the dependencies of the Hopfion gyrovector components on the dot sizes were calculated. It was demonstrated by analytical calculations that the magnetic hopfion gyrovector $\mathit{G}$ is not equal to zero and does not vanish even in the limit of an infinite sample. Namely, two components of the gyrovector in the curvilinear cylindrical coordinates, $G_z$ and $G_{\phi }$, are finite. The out of-plane z-component of the hopfion gyrovector ($G_z$) goes to zero, increasing the dot radius; however, the in-plane gyrovector φ-component ($G_{\phi }$) remains finite [61]. The calculated components of the hopfion emergent magnetic field and gyrovector can be used for calculations of the topological and skyrmion Hall effect of the toroidal magnetic hopfions, respectively. It was recently shown [62] that the toroidal hopfions reveal a Hall motion (skyrmion Hall effect) under the current pulses, while the skyrmionium moves only along the current direction. This is because the gyrovector (2D topological charge) of skyrmionium vanishes, whereas it is not equal to zero for the toroidal magnetic hopfions in accordance with Ref. [61]. The fractional Hopf index $Q_H=0.8$ was calculated in Ref. [62] due to the restricted geometry of FeGe samples. The current pulses were employed to drive the dynamics of the fractional hopfions. An asymmetric Hall motion of the hopfions with respect to the current direction was detected [62].

Space–time magnetic hopfions are suggested in Ref. [63]. They are treated as 2D magnetic textures (skyrmions) excited by an oscillating magnetic or electric field. The authors considered the coupled dynamics of the skyrmion radius and helicity using the model of the exchange interaction quadratic and quartic on the magnetization spatial derivatives. The authors of Ref. [63] believe that the introduced hopfion topological invariant, the space–time Hopf charge ($Q_H=\pm 1$), can be tuned by the applied electric field. The emergent magnetic field $B_i\left(\mathit{m}\right)={\epsilon }_{ijk}\mathit{m}·\left({\partial }_j\mathit{m}\times {\partial }_k\mathit{m}\right)$ (i, j, k = x, y, t) and gauge vector potential Awere introduced using the standard equations. However, these equations are valid only if the indices i, j, k used in their definition [63] mark the spatial coordinates, for instance, the Cartesian coordinates (x, y, z). However, one of the indices i, j, k corresponds to time (t) according to Knapman et al. [63]. If the index i = t, then the emergent field component $B_t$ has no physical sense because B= $\left(B_x,B_y,B_z\right)$ is a vector in three-dimensional coordinate space (x, y, z). If the indices j = t or k = t, then the vector B is not the emergent magnetic field, and the equation $\mathit{B}=\nabla \times \mathit{A}$ is not valid. In the case of j = t or k = t, the field B is an emergent electric field E proportional to the time derivative of the unit magnetization vector m. The relation of the emergent electric field defined as $E_i=F_{i0}$ ($i=x,y,z)$or $\mathit{E}=\nabla A_0-\partial \mathit{A}/c\partial t$ and the four-vector potential $\mathcal{A}=\left(A_0,\mathit{A}\right)$ is essentially different from the equation $\mathit{B}=\nabla \times \mathit{A}$ for the emergent magnetic field. Only the space Hopf index defined by Equation (4) via the dot product $\left(\mathit{A}·\mathit{B}\right)$ has physical sense. The time derivative of the magnetization m and emergent electric field $\mathit{E}~\partial \mathit{m}/\partial t$ cannot be used for the calculation of the Hopf index defined by Equation (4). Therefore, the equations defining the space–time Hopf index used by Knapman et al. [63] are wrong and should be corrected.

The hopfions considered by Knapman et al. are a particular case of 3D winding hopfions introduced by Kobayashi et al. [64] in the $R^2\times S_1$ coordinate space, where the third coordinate $S_1$ (z) is directed along the sample thickness, L. Knapman et al. [64] substituted the thickness coordinate (z) to time ($t$). There is the standard angular parameterization of the magnetization $\mathit{m}\left(\mathit{r}\right)$ components via the spherical angles $\mathsf{\Theta }\left(\mathit{r}\right)$, $\mathsf{\Phi }\left(\mathit{r}\right)$. The hopfion magnetization spherical angles should be written in an axially symmetric form, $\mathsf{\Theta }\left(\mathit{r}\right)=\mathsf{\Theta }\left(\rho ,z\right)$, $\mathsf{\Phi }\left(\mathit{r}\right)=n\phi +\gamma \left(\rho ,z\right)$, as it was performed for the azimuthally symmetric toroidal hopfions in Refs. [19,53]. If following Ref. [64] one introduces the hopfion helicity as $\gamma \left(\rho ,z\right)=-Pg\left(z\right)$, where the phase angle $g\left(z\right)$ satisfies the boundary condition $g\left(L\right)-g\left(0\right)=2\pi$ at the sample upper/down faces and P is an integer, then the calculated Hopf index of the winding hopfion is an integer. Kobayashi et al. assumed that the angle $\mathsf{\Theta }\left(\rho ,z\right)$ does not depend on the thickness coordinate $z$. Therefore, the dependence $\mathsf{\Theta }\left(\rho \right)$ describes a 2D skyrmion profile in the $xOy$ plane.

Following Kobayashi et al. [64], we introduce the winding hopfion magnetization components as

$${\displaystyle m_z\left(\rho \right)=cos\mathsf{\Theta }\left(\rho \right), m_x\left(\mathit{r}\right)+i{m}_y\left(\mathit{r}\right)=sin\mathsf{\Theta }\left(\rho \right)e^{i\left(n\phi -Pg\left(z\right)+{\phi }_0\right)},}$$

(7)

where $\mathsf{\Theta }\left(\rho \right)$ is the skyrmion magnetization polar angle satisfying the boundary conditions $\mathsf{\Theta }\left(0\right)=\pi$, $\mathsf{\Theta }\left(\infty \right)=0$, $n$ is the azimuthal vorticity, and ${\phi }_0$ is a constant phase angle.

To plot the winding hopfion magnetization, we take the skyrmion profile in the Belavin–Polyakov form [33] $cos\mathsf{\Theta }\left(\rho \right)=\left({\rho }^2-1\right)/\left({\rho }^2+1\right)$ ($\rho$ is taken in the units of the skyrmion radius ${\rho }_c$, $m_z\left({\rho }_c,\right)=0$) and the function $g\left(z\right)=2\pi \left(z/L\right)$ in the simplest form satisfying the conditions $g\left(L\right)-g\left(0\right)=2\pi$. The magnetization components of the winding hopfions are plotted in Figure 4. We note that the skyrmions at the film surfaces $z=0, L$ are of the Neel type at ${\phi }_0=0$ (Figure 4A) or Bloch type at ${\phi }_0=\pi /2$ (Figure 4B).

The calculation by Equation (4) yields $Q_H=QP$, where Q = n is the skyrmion number [36] (2D topological charge). However, in the finite samples, Q is not an integer and the condition $g\left(L\right)-g\left(0\right)=2\pi$, in general, is not satisfied. Therefore, the Hopf index of the winding hopfions can be an arbitrary number. The winding hopfions are very different from the toroidal magnetic hopfions, which have almost integer Hopf indices even in finite samples [28,29].

Very recently, the paper by Saji et al. [65] was published, where the authors rewrote the toroidal hopfion magnetization in the spherical coordinates regardless of whether the hopfion has a cylindrical (azimuthal) symmetry. The validity of the ansatz derived in Ref. [65] for the Hopf index $Q_H>1$ should be proven. It is also unclear why the emergent vector potential calculated by Saji et al. [65] differs from the vector potential presented in Ref. [59] calculated on the base of field theory [10,11]. In Refs. [26,60,65], the toroidal hopfion magnetization, emergent magnetic field, and vector potential are presented via an arbitrary smooth function $f\left(r\right)$ satisfying the boundary conditions $f\left(0\right)=0$, $f\left(\infty \right)=\pi$ to ensure an integer Hopf index $Q_H$ in an infinite sample. Therefore, these important local parameters $\mathit{m}\left(\mathit{r}\right)$, $\mathit{B}\left(\mathit{r}\right)$, $\mathit{A}\left(\mathit{r}\right)$ are not completely defined and cannot be used for calculations of any effects related to the toroidal magnetic hopfions. Using the function $f\left(r\right)$ is not necessary because the exact explicit expressions for $\mathit{m}\left(\mathit{r}\right)$, $\mathit{B}\left(\mathit{r}\right)$, $\mathit{A}\left(\mathit{r}\right)$ of the toroidal magnetic hopfions in infinite media were found in Ref. [19] for any values of the 3D topological charge $Q_H$.

The authors of Ref. [65] demonstrated that the spin waves excited over the hopfion background experience the emergent electromagnetic field generated by the toroidal hopfion magnetization texture $\mathit{m}\left(\mathit{r}\right)$. In particular, it was shown that the spin waves propagating along the toroidal hopfion symmetry axis (0z) are deflected by the hopfion magnetic texture. This effect using an analogy with the skyrmion topological Hall effect was named as “magnonic Hall effect”.

6. Summary

I briefly review above the current status of research on 3D magnetic topological solitons (hopfions) in restricted and infinite geometries. The hopfions are initially introduced, similarly to the skyrmions, in classical field theory. Then, these complicated 3D textures are calculated and measured in ferromagnetic media. It is shown that the concepts of the gauge emergent vector potential and emergent magnetic field are absolutely necessary to describe the magnetic hopfions. The toroidal hopfions with the integer Hopf indices found in the field theory occupy a special place among the magnetic hopfions. However, one should be careful using the equations developed for infinite media to calculate the hopfions in finite samples. Except for the toroidal hopfions, other hopfions with a non-zero and non-integer Hopf index are also possible (the Bloch points and winding hopfions, for instance). I note that the Hopf index (3D topological charge) can be used as an important parameter (invariant) to characterize a given 3D magnetization texture. The magnetization textures with the same values of the Hopf index are topologically equivalent. The Hopf index has less physical sense in comparison to a 2D topological charge (skyrmion number) because the latter defines the soliton gyrovector resulting, in particular, in the vortex/skyrmion gyrotropic excitation mode observed experimentally.

The main articles on the toroidal hopfion stability and dynamics in ferromagnetic films and dots are reviewed. Both topics are considered mainly theoretically (numerically) because it is difficult to observe stable hopfions and their dynamics experimentally. No applications of the magnetic hopfions have been realized to the moment. I believe that it can be achieved in the near future.

A critical discussion on the calculations and simulations of the magnetization, emergent magnetic field, and Hopf index of the toroidal magnetic hopfions and other kinds of magnetic hopfions in restricted geometries is presented. The critical point is using proper equations for the hopfion emergent magnetic field and vector potential and proof of gauge invariance of the calculated Hopf index.