Soliton dynamics and elastic collisions in a spin chain with an external time-dependent magnetic field

Abstract

In this paper, we investigate the nonlinear dynamics of a Heisenberg spin chain with an external time-oscillating magnetic field. Such dynamics can be described by a Landau–Lifshitz-type equation. We apply the Darboux transformation method to the linear eigenvalue problem associated with this equation, and obtain the multi-soliton solution with a purely algebraic iterative procedure. Through the analytical analysis and graphical illustrations for the solutions obtained, we discuss in detail the effects of an external magnetic field on the nonlinear wave. Under the action of an external field, although the amplitude, width and depth of soliton vary periodically with time and its symmetry property is changeable, the soliton can also propagate stably and it possesses particle-like behavior.

Introduction

Over the last decades, the nonlinear magnetization dynamics of the classical Heisenberg spin chain has attracted a great deal of interest both in soliton theory and condensed-matter physics [1], [2], [3], [4], [5], [6], [7]. Much effort has been put into understanding the nonlinear mechanisms and rich variety of spin excitations in different spin systems [2], [3], [4], [5], [6]. The Landau–Lifshitz equation for a spin chain in the form

$$S_t=S\times S_{xx}+S\times \mathrm{JS}\text{,}\left|S\right|=1\text{,}$$

where $S$ represents a chain spin, and $J=\mathrm{diag}(J_1,J_2,J_3)$ describes the magnetic property of chain, has usually been used to model the precessional motion of the magnetization in studying the dispersive theory of magnetization of ferromagnets [1]. The integrable aspects and wide applications of Eq. (1) have been studied extensively in much literature and many monographs [5], [6], [7], [8], [9].

The relevant nonlinear waves are of current interest for magnetic systems under the action of an external magnetic field [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23]. Regarding the motion of a continuum Heisenberg spin chain, people have paid attention towards the new effects of an external magnetic field on the nonlinear wave. The dynamical behaviors of the magnetic chain have been observationally indicated and theoretically addressed for isotropic and anisotropic spin chains with an external magnetic field depending on time [10], [11], [12], [13], [14], [15], [20], [21]. In particular, investigation of the partial-like excitation described by the soliton solution in a spin system is of great interest, to study the peculiarities of slow neutron scattering on magnets, dynamic structures from factors, and so on [16], [17], [18]. In addition, the influence of an external magnetic field with various forms of time dependence, such as linear, exponential and sinusoidal time dependence [20], [21], [22], [23], is different for a magnetic spin system, and the nonlinear dynamics exhibits rich phenomena and properties such as coherent and chaotic structures.

Motivated by the above physical interest, in this paper, we further extend the theoretical investigation on the nonlinear dynamics of a ferromagnetic spin chain with an external time-dependent magnetic field. We will explore the new effects of nonlinear dynamics and discuss the possibilities for future applications of a spin chain under the action of a time-varying magnetic field.

In a ferromagnetic spin chain, under the influence of a time-oscillating magnetic field with an arbitrary direction, the Heisenberg Hamiltonian takes the form [19]

$$\hat{H}=-J\sum _{〈n,n^{\prime }〉}{\hat{S}}_n\cdot {\hat{S}}_{n^{\prime }}-g{\mu }_{B}B\left(t\right)\cdot \sum _n{\hat{S}}_n\text{,}$$

where ${\hat{S}}_n=({\hat{S}}_n^x,{\hat{S}}_n^y,{\hat{S}}_n^z)(n=1,2\dots ,N)$ represents the classical three-component spin vector, $g$ and ${\mu }_B$, respectively, are the Lande factor and the Bohr magneton, $J>0$ is the pair interaction parameter, and $B\left(t\right)=Bcos\left(\omega t\right)e$ is the external magnetic field with $e=(sin\theta ,0,cos\theta )$ as the unit vector in the field direction. The angle $\theta$ between the direction of the magnetic field and the $z$-axis is arbitrary. Through considering the spin as a classical vector at low temperatures, the equation of motion corresponding to the spin Hamiltonian (2) in a continuum spin chain with a time-dependent magnetic field becomes the following Landau–Lifshitz-type equation [19]:

$$S_t=S\times S_{xx}+S\times \epsilon \text{,}$$

where $\epsilon =g{\mu }_{B}B\left(t\right)/\left(2J\right)$, $S(x,t)=(S^x,S^y,S^z)$. The dimensional time $t$ and coordinate $x$ are scaled in units $1/\left(2J\right)$ and the lattice constant $d$, respectively. By virtue of the inverse scattering transformation, [19] has obtained the soliton trains of Eq. (3) and discussed the fascinating shape-changing inelastic collisions of solitons.

In the following sections, we adopt the Darboux transformation method, which is more straightforward and effective than the method used in [19], to construct the multi-soliton solutions of Eq. (3). Through the consideration of soliton solutions, the effects of the external magnetic field on the width, depth, amplitude and symmetry property of solitons will be clearly explored. The collisions between two solitons with equal or different amplitudes in the presence of an external magnetic field for different values of the angle between the direction of the magnetic field and the $z$-axis will also be discussed.