Dynamical Toroidal Hopfions in a Ferromagnet with Easy-Axis Anisotropy


Three-dimensional toroidal precession solitons with a nonzero Hopf index, which uniformly move along the anisotropy axis in a uniaxial ferromagnet, have been found. The structure and existence region of the solitons have been numerically determined by solving the Landau–Lifshitz equation.

03.50.-k, 11.27.+d, 47.32.Cc, 75.10.Hk, 75.60.Ch, 94.05.Fg

Static and dynamical topological structures with a nonzero Hopf invariant (hopfions) in various models and media were discussed in bib:Faddeev1 (); bib:VM (); bib:Kamchatnov (); bib:DI (). In particular, the magnetization distribution in three-dimensional ferromagnetic materials is characterized by an integer Hopf index . The investigation of the dynamics of the hopfions in these media is not only of theoretical interest, but also important for various physical applications, particularly, in view of the prospect of the creation of fundamentally new memory elements.

Stable precession solitons with (magnon drops) in a uniaxial ferromagnet were found in bib:IvKos1 (); bib:IvKos2 (). However, precession uniformly moving toroidal solitons at were observed only in an isotropic ferromagnet bib:Cooper (). Their stability with respect to perturbations violating the axial symmetry was discussed in bib:Sut2007 (). The aim of this work is to analyze the existence region and structure of toroidal hopfions uniformly moving along the anisotropy axis in a uniaxial ferromagnet.

The dynamics of the magnetization vector is described by the Landau–Lifshitz equation; in the case of negligible relaxation, it has the form


where - is the gyromagnetic ratio (). The energy of the ferromagnet is the sum of the exchange energy


and the energy of the uniaxial magnetic anisotropy


Many magnetic media are characterized by a high quality factor and the contribution of the energy of the magnetic dipole interactions is insignificant for them.

At each point of the space with a Cartesian coordinate system , the orientation of vector is specified by a point on the two-dimensional sphere in terms of the angular variables and :


Let us consider localized solutions (1), for which


where , - is the polar angle of the cylindrical coordinate system , and at . Such solutions describe three-dimensional precession solitons with a stationary profile, which propagate along the anisotropy axis. For simplicity, we consider the configuration of a unit vector field


in the coordinate system moving along the axis (with the velocity ).

The desired vector field specifies the mapping and is characterized by an integer Hopf topological index . If , the solution given by (5) corresponds to a toroidal hopfion bib:KUR (); bib:Glad () with the index


Let us represent the energy as a functional of the vector field :


In addition to energy, equation (1) has two integrals of motion: the number of spin deviations (magnons),


and the projection of the momentum of the magnetization field bib:PT () on the anisotropy axis,


To determine the structure of three-dimensional solitons, we used the same method for minimizing energy functional (8) with constraint (10) as in bib:BorRyb1 (), but constraint (11) was taken into account through an additional additive square penalty function, and an initial field configuration was specified by smooth functions and , corresponding to the Hopf bundle with the index :


where and are the toroidal coordinates specified by the relations:


The parameters and are determined by the integrals of motion and :


where and .

For comparison, the calculations were performed by the same method for the case of nontopological solitons with , which are stationary bib:IvKos1 () and moving bib:Sut2001 () magnon drops. The initial field configuration in these cases was specified by the different functions


where , and the parameters and are expressed in terms of the integrals of motion and :


The calculations of the nontopological solitons are in agreement with the data reported in bib:IvKos2 ()bib:Sut2001 ().

The desired field ensures an extremum of the functional


Using the necessary condition of the extremum, we arrive at a pair of equations


Solving this system with respect to and , we obtain the convenient formulas:




After several thousands of iterations, minimized energy functional (8) reaches a minimum. To test the resulting field configurations, we calculated and by equations (24) and, then, the discrepancy for Landau– Lifshitz differential equations (1). Let us discuss the results.

Figure 1: Contours of the angles parameterizing the unit vector , for the (a) stationary hopfion and (b) hopfion moving with the velocity with the same index and . The solid and dashed curves correspond to and , respectively.
Figure 2: Velocity dependence of the energy for nontopological () and topological () solitons with the same precession frequency . The points are the results of the numerical calculation.
Figure 3: Coordinate dependence of the normalized energy density for hopfions of the (a) lower and (b) upper energy branches for , , .

Fig.1 shows the contours of the angles parameterizing vector . The coordinates are normalized to the characteristic length


It is seen that the radius of the central vortex ring corresponding to the value , i.e., to the southern pole of the sphere, is larger for the moving soliton. The contours are not constructed near small values, because this numerical method determining the vectors does not provide an accurate calculation of the azimuth angle when . However, the shapes of these lines are significantly different for stationary and moving hopfions. The localization region of the moving soliton is somewhat larger, but is about as in the stationary case.

Figure 4: Velocity dependence of energy for the hopfion with for two values of the precession frequency.

Fig.2 shows the reduced energy as a function of the ratio , where . The plot manifests an important revealed property, the existence of two types of moving hopfions with the same , and values at least in a certain velocity interval. Fig.3 presents typical distributions of the normalized energy density . A high energy density along the wall of the toroidal surface is characteristic of hopfions of the lower energy branch with low velocities, whereas the energy density increases from the wall to the center of the toroid for the hopfions of the upper energy branch with high energies. It is also seen in Figs. 2 and 4 that the velocity of topological solitons is limited. The limiting velocity of a hopfion decreases with decreasing precession frequency, whereas its energy increases. The frequency is normalized to the frequency of the homogeneous ferromagnetic resonance:


The general notions on the structure of the stationary hopfion with cannot allow for a comparison of its energy with the energy of the corresponding nontopological soliton bib:kniga1 (); bib:KBK (). The features of the structure of the class of objects under investigation prevent a numerical analysis of the case with and  bib:BorRyb1 (). However, the extrapolation of the dependence to the region of small values in Fig.2 for certainly indicates that the energy of the stationary precession topological soliton is higher than that for the nontopological soliton at the same frequency at least in a certain precession frequency range.


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