Precessing ball solitons in kinetics of a spin-flop phase transition

The fundamentals of precessing ball solitons (PBS), arising as a result of the energy fluctuations during spin-flop phase transition induced by a magnetic field in antiferromagnets with uniaxial anisotropy, are presented. The PBS conditions exist within a wide range of amplitudes and energies, including negative energies relative to an initial condition. For each value of the magnetic field, there exists a precession frequency for which a curve of PBS energy passes through a zero value (bifurcation point), and hence, in the vicinity of this point the PBS originate with the highest probability. The characteristics of PBS, including the time dependences of configuration, energy, and precession frequency, are considered. As a result of dissipation, the PBS transform into the macroscopic domains of a new phase.


Introduction
Magnetic solitons with spherical symmetry, which can arise in crystals with magnetic ordering during the phase transitions induced by a magnetic field, were considered in some papers [1][2][3][4][5][6][7][8]. The cases when it is possible to confine oneself to uniaxial symmetry [4][5][6][7][8] are of particular interest. For such a crystal, in addition to the amplitude (with corresponding configuration) and pulse, the solitons have the third parameter: the frequency of precession. In these cases, each value of an external field relates to a continuous spectrum of solitons by frequency and corresponding energy. Near boundaries of metastability, for each value of a field, the spectrum of precessing ball solitons (PBS) has the bifurcation point where the probability of PBS spontaneous origin is increasing abnormally. The frequency dependencies of energy and PBS configuration, as well as the process of PBS spontaneous origin at spinflop transition in antiferromagnets have been discussed in [8]. However, the analysis of time evolution connected with dissipation of energy in PBS is not correct in this article.
In the present article, it is shown that dissipation of energy is accompanied not only by the change of configuration of solitons but also by the change of the precession frequency. Taking into account this time dependency, we can carry out a more comprehensive analysis of the quantities of PBS and to consider the whole process of transformation of PBS into macroscopic domains of a new phase.
In the first chapter, we represent a more correct analysis of equations and expressions for PBS than in [7,8]. In the second one, the character of the time change of PBS that connected with dissipation is shown. Then, in the third chapter, we analyze the so-called "equilibrium PBS". In the fourth chapter, the PBS in overcritical range of a field, i.e. outside the metastability region, are considered. In the fifth chapter, the influence of the movement on the form of PBS is discussed. Finally, in the sixth chapter, we estimate the influence of the demagnetizing fields.

Equations for precessing ball solitons (PBS)
To analyze magnetic solitons in an antiferromagnet with uniaxial anisotropy, we used the following expression for the macroscopic energy (as in [8] (2) Here m and l are non-dimensional ferromagnetism and antiferromagnetism vectors; ; the absolute value of the vector l at H=0 equals 1, M 0 is the magnetization of each sublattice, X K x . xy The equations of motion with dissipative terms (in the Gilbert form) for the l and m vectors, taking into account the energy dissipation, are The solutions of equations (2) and (3) can be presented in the form: For the simplicity, it is supposed that the excitations advance along the x-axis. The time dependencies of the ω and k values are necessary for analyzing the time evolution of PBS.
From Eqs. (2)-(5), we have the following system of equations: where In (6)-(13), the differentiation is carried out with respect to the dimensionless time First of all, take notice that inserting Eqs. To analyze the approximate soliton solutions of (6)-(11) system, it is convenient to transform the system of (6)-(11) equations into one equation. First of all, we will receive expression for magnetization. The relaxation time of ferromagnetic moment is less for some orders than the relaxation time for the antiferromagnetism vector. Therefore, for z m component we use its quasi-equilibrium value, which can be obtained from 0 m is the ratio of two magnetic susceptibilities.
Using the (15) correlation, from (7) we obtain the following equation for PBS: Let's confine ourselves to low temperature approximation, i.e. suppose that ( ) 0 and the equation for PBS is as follows: This equation should be supplemented by the following expression obtained from (9), (14), and describing the evolution of PBS: Only in the case of immovable PBS, i.e. at 0 ≡ k , the (19) equation has the solutions with a spherical symmetry. For such case, the equation has the following form: (taking into account that ). In this equation, the frequency ω , as well as q values, depends on time. Thus we make the replacement , which is possible considering a rather slow change of the precession frequency in comparison with the change of precession phase. Note that in the (21) equation, the addition connected with the dissipation is negligibly small, 1 ). Therefore, in further calculations we will neglect this addition.
Transforming Eq. (21) with respect to l z parameter, we can obtain the following equation, that can be used to analyze the PBS originated during the reverse phase transition, at decrease of the field: Using Eqs. (14) and (15) and 1 << k 1 , from (2) we obtain the following expression for the energy of PBS: The last term in this integral corresponds to the kinetic energy of PBS.

Spontaneous origin of PBS and their evolution into domains of a new phase
As in [8], we divide the processes relating to PBS into two stages. At the first stage, the PBS originate spontaneously because of the thermal fluctuations, or by any different way, in the 0 = τ moment, and further evolution of PBS is carried out at the second stage. Equations   The same frequency dependencies as in Fig.1 for the reverse spin-flop transition, i.e. at the decreasing magnetic field, if 91 0. h = .
Probability of the spontaneous creation of PBS related to the fluctuations of energy at nonzero temperatures is proportional to probability of such fluctuations. We can use the following expression for the probability of PBS creation with the energy s E near the bifurcation point (see [7,8]): In conformity with Eq.(25), the character of PBS change is determined by a sign of the precession frequency and by its amplitude.
In Fig.4 -Fig.7    (see in Fig.4). The dotted line shows the frequency value corresponding to zero energy.  is illustrated in Fig.13.        In Fig.19, the field dependencies of the energy, amplitude and frequency for the equilibrium PBS are shown. In Fig.20, the configurations of equilibrium solitons are presented. Fig.19 The field dependencies of the energy, amplitude, and frequency for the equilibrium PBS.

Overcritical PBS
The PBS states also exist when the initial phase state is absolutely unstable [6][7][8].
Therefore, creation of PBS is possible at disintegration of the initial phase, i.e. at 1 h > (and for all PBS at 1 h > the precession frequencies are negative. . The curve of amplitude is shown for 001 (see Fig.21).
In Fig.24, the time dependencies of energy, amplitude, and precession frequency at are shown, when the PBS are transforming into the domains of the high-field phase.

The change of PBS during their movement
Now, we consider the influence of movement on soliton form. Using Eqs. (9), (11), and (12) for the 0 Q = ω case, we obtain the following expression: (here s r is the radial coordinate in the system of moving soliton) describes the deformation of a soliton because of its movement along the x-axis. For our solitons, the derivative 0 r q s < ∂ ∂ .
Consequently, the PBS frontal side is decreasing, i.e. it becomes steeper, but the back side is increasing, i.e. it becomes more slope in the same extent. Thereby, "the centre of gravity" of soliton is displaced in a direction opposite to a direction of the main movement. Thus, the form of moving soliton differs from spherical, but if 0 = Q its energy does not change.
However, the dissipation ( 0 ≠ Q ) results not only in change of frequency, but also in decrease of the velocity and, accordingly, of kinetic energy: If to us the expression (32) for the magnetization in Eq.(31), we receive:

5.
At spatial movement of PBS, its form is deformed, but it does not change the size and the amplitude. The dissipation results not only in change of precession frequency, but also in reduction of the velocity of movement. As a result of the velocity reduction, the shape of PBS is approaching to spherical.