Damping-like effects in Heisenberg spin chain caused by the site-dependent bilinear interaction
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Yu-Juan Zhang,1,∗, Dun Zhao2, Zai-Dong Li31School of Mathematics and Statistics, Xidian University, Xi’an 710126, China 2School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China 3Department of Applied Physics, Hebei University of Technology, Tianjin 300401, China
First author contact:∗ Author to whom any correspondence should be addressed. Received:2020-05-20Revised:2020-11-3Accepted:2020-11-4Online:2021-01-07
Abstract We investigate a continuous Heisenberg spin chain equation which models the local magnetization in ferromagnet with time- and site-dependent inhomogeneous bilinear interaction and time-dependent spin-transfer torque. By establishing the gauge equivalence between the spin chain equation and an integrable generalized nonlinear Schrödinger equation, we present explicitly a novel nonautonomous magnetic soliton solution for the spin chain equation. The results display how the dynamics of the magnetic soliton can be controlled by the bilinear interaction and spin-polarized current. Especially, we find that the site-dependent bilinear interaction may break some conserved quantity, and give rise to damping-like effect in the spin evolution. Keywords：Heisenberg spin chain;site-dependent bilinear interaction;spin-transfer torque;magnetic soliton;damping effect
PDF (998KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Yu-Juan Zhang, Dun Zhao, Zai-Dong Li. Damping-like effects in Heisenberg spin chain caused by the site-dependent bilinear interaction. Communications in Theoretical Physics, 2021, 73(1): 015105- doi:10.1088/1572-9494/abc7ab
1. Introduction
Nonlinear excitations [1–3] are universal phenomena in magnetic ordered materials. During the past several decades there has been great progress for magnetization dynamics in magnetic nanostructures, many efforts have been devoted to the investigation about the dynamics of three-types magnetic excitation states in the ferromagnetic nanowires, namely spin wave, domain wall and dynamic soliton [1, 2]. In particular, the dynamic soliton can describe the localized excited states of magnetization, and the motion of dynamic soliton is of topic research in confined ferromagnetic materials, especially with the generation and detection of magnons excitation in a magnetic multilayer.
In statistical physics, various magnetic properties of low-dimensional materials can be depicted by a Heisenberg spin chain, which is a discrete one-dimensional model to describe ferromagnetism with spin–spin interactions. In the continuum limit of pair spin with nearest neighbor Heisenberg interaction, where the infinite spins lie dense on a line, one can get the continuous Heisenberg spin chain model [4], which was first derived phenomenologically by Landau and Lifshitz [5]. In solid physics, this model has successfully explained the existence of ferromagnetism and antiferromagnetism below the Curie temperature, so it can be considered as the starting point for understanding the complex magnetic structures. After the first observation of soliton solution by Nakamura and Sasada [6], the soliton dynamics of the one-dimensional continuous Heisenberg spin chain equation has received much attention, see, for examples [7–21], and the references therein.
In general, the continuous Heisenberg spin chain equation (Landau–Lifshitz equation) takes the form ${\vec{S}}_{t}=\vec{S}\times {\vec{H}}_{\mathrm{eff}}$, where $\vec{S}=\vec{S}(x,t)=\{{S}_{1}(x,t),{S}_{2}(x,t),{S}_{3}(x,t)\}$ denotes the magnetization (spin) density vector, ${\vec{S}}^{2}=1$, and ${\vec{H}}_{\mathrm{eff}}$ denotes the effective fields including the external field, the anisotropy field, the demagnetization field, and the exchange field. Based on various choices of ${\vec{H}}_{\mathrm{eff}}$, this equation has been studied by many physicists and mathematicians. For example, in the sense of the homogeneous bilinear interaction, ${\vec{H}}_{\mathrm{eff}}={\vec{S}}_{{xx}}$, Lakshmanan [8] obtained the soliton solution by mapping the spin chain onto a moving helical curve in the Euclidean space. Takhtajan [9] proved its complete integrability by associating it with a Lax pair representation. Zakharov and Takhtajan [10] showed that it is equivalent to the standard focusing cubic nonlinear Schrödinger equation. In the case of the site-dependent inhomogeneous bilinear interaction ${\vec{H}}_{\mathrm{eff}}={\left(f{\vec{S}}_{x}\right)}_{x}\,=f{\vec{S}}_{{xx}}+{f}_{x}{\vec{S}}_{x}$, where f=f(x) is the coupling function of the interaction, Balakrishnan [14] has proved that the spin equation is equivalent to the following inhomogeneous nonlinear Schrödinger equation$\begin{eqnarray}{\rm{i}}{q}_{t}+{\left({fq}\right)}_{{xx}}+2f| q{| }^{2}q+2q{\int }_{-\infty }^{x}{f}_{x}{q}^{2}{\rm{d}}x=0,\end{eqnarray}$where the spin vector $\vec{S}$ has been set as the unit tangent of a moving space curve, and q=q(x, t) in the nonlinear Schrödinger equation has been associated with the curvature and torsion of this curve by the Hasimoto transformation. Equation (1) can be associated with an Ablowitz–Kaup–Newell–Segur formalism [15, 22], which means that it is the compatibility condition of a pair of two linear partial differential equations. However, even until recently [20], the N-soliton solutions, Bäcklund transformation and conservation laws for equation (1) have been discussed for f(x)=α1x+α2, where α1 and α2 are constants. On the other hand, when a spin-polarized current injects into the ferromagnet, the dynamics of localized magnetization is described by the modified equation [23–25] with spin-transfer torque$\begin{eqnarray}{\vec{S}}_{t}=\vec{S}\times {\vec{H}}_{\mathrm{eff}}+{T}_{b},\end{eqnarray}$where ${T}_{b}=\gamma {\vec{S}}_{x}$ describes the spin-transfer torque resulted from the spin-polarized current. Recently, a lot of theoretical and experimental research demonstrates that the spin-polarized current can arouse many significant phenomena [26, 27], such as spin-wave excitation [28, 29], magnetization switching [30] and reversal [31, 32] in magnetic multilayers. Some significant results are reported for the dynamics of magnetization associated with spin-polarized current in layered materials [30, 33]. Nowadays, spin polarized currents are commonly used to create, manipulate, and control nanoscale magnetic excitations such as domain walls and magnetic soliton. Owing to the considerable interest of magnetoelectronics in potential technological applications, it is important to find various dynamical soliton in ferromagnet, and consider the manipulation of magnetic solitons.
In this paper, we will devote to the continuous Heisenberg spin chain equation (2) with time- and site-dependent bilinear interaction and time-dependent spin-transfer torque. Concretely speaking, we will take ${\vec{H}}_{\mathrm{eff}}={(h(x,t){\vec{S}}_{x})}_{x}$, where h(x, t)=α(t)+β(t)x, and γ=γ(t), and thus consider the following generalized inhomogeneous spin chain equation$\begin{eqnarray}{\vec{S}}_{t}=h(x,t)\vec{S}\times {\vec{S}}_{{xx}}+\beta (t)\vec{S}\times {\vec{S}}_{x}+\gamma (t){\vec{S}}_{x}.\end{eqnarray}$We will present explicitly a novel nonautonomous magnetic soliton solution for equation (3) and show how the dynamics of the magnetic soliton can be controlled by the bilinear interaction and spin-polarized current. Especially, we find that in such a model, the parameter β(t) will play a very interesting role, it may break some conserved quantity, and may cause the damping-like effect, which, to our knowledge, has not been reported in literatures.
To deal with the spin chain equation (3), we will establish the equivalence to the generalized inhomogeneous nonlinear Schrödinger equation$\begin{eqnarray}\begin{array}{l}{\rm{i}}{q}_{t}+h(x,t)({q}_{{xx}}+2| q{| }^{2}q)\\ \quad -\,{\rm{i}}\gamma (t){q}_{x}+2\beta (t)\left({q}_{x}+q{\displaystyle \int }_{-\infty }^{x}| q{| }^{2}{\rm{d}}x\right)=0,\end{array}\end{eqnarray}$which is integrable, and thus get the explicit solutions of (3) from the solutions of (4). We point out that here we use the gauge equivalence to establish the equivalence of the spin chain equation (3) and the generalized inhomogeneous nonlinear Schrödinger equation (4), the procedure is a generalization of the gauge equivalence between the Heisenberg ferromagnetic equation and the classical cubic nonlinear Schrödinger equation [10], which is different from the concern of Lakshmanan and Balakrishnan in [12, 14], where they presented the equivalence from the geometric point of view.
This paper is organized as follows. In section 2, the relation between the solutions of the equations (4) and (3) is established through the gauge equivalence. In section 3, by using the Darboux transformation, explicit solutions of equation (4) are presented, and the corresponding spin vector of equation (3) are displayed. Section 4 is devoted to a brief summary. Finally, we add two appendix to present the calculation details.
2. Gauge equivalence of the generalized nonlinear Schrödinger equation and the spin chain equation
Denote σi the Pauli matrices, i.e.$\begin{eqnarray}{\sigma }_{1}=\left(\begin{array}{cc}0 & 1\\ 1 & 0\end{array}\right),\quad {\sigma }_{2}=\left(\begin{array}{cc}0 & -{\rm{i}}\\ {\rm{i}} & 0\end{array}\right),\quad {\sigma }_{3}=\left(\begin{array}{cc}1 & 0\\ 0 & -1\end{array}\right),\end{eqnarray}$and set$\begin{eqnarray}{U}_{0}=\left(\begin{array}{cc}0 & q\\ -r & 0\end{array}\right).\end{eqnarray}$Let $\vec{S}=\{{S}_{1}(x,t),{S}_{2}(x,t),{S}_{3}(x,t)\}$ be the spin vector governed by equation (3) and$\begin{eqnarray}\tilde{S}={S}_{1}{\sigma }_{1}+{S}_{2}{\sigma }_{2}+{S}_{3}{\sigma }_{3}=\left(\begin{array}{cc}{S}_{3} & {S}_{1}-{{\rm{i}}{S}}_{2}\\ {S}_{1}+{{\rm{i}}{S}}_{2} & -{S}_{3}\end{array}\right),\end{eqnarray}$then a simple computation shows that equation (3) is equivalent to$\begin{eqnarray}{\tilde{S}}_{t}=\displaystyle \frac{1}{2{\rm{i}}}h(x,t)[\tilde{S},{\tilde{S}}_{{xx}}]+\displaystyle \frac{1}{2{\rm{i}}}\beta (t)[\tilde{S},{\tilde{S}}_{x}]+\gamma (t){\tilde{S}}_{x},\end{eqnarray}$where [·,·] denotes the Lie bracket of the matrices.
Furthermore, set$\begin{eqnarray}\begin{array}{rcl}{U}_{1} & = & -{\sigma }_{3},\quad {V}_{1}=-\gamma (t){\sigma }_{3}-2{\rm{i}}\ h(x,t){U}_{0},\\ {V}_{2} & = & 2{\rm{i}}h(x,t){\sigma }_{3},\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{rcl}{V}_{0} & = & {\rm{i}}\left[h(x,t){qr}+\beta (t){\displaystyle \int }_{-\infty }^{x}{qr}{\rm{d}}x\right]{\sigma }_{3}\\ & & +{\rm{i}}\displaystyle \frac{\partial }{\partial x}[h(x,t){\sigma }_{3}{U}_{0}]+\gamma (t){U}_{0},\end{array}\end{eqnarray}$it is known that equation (4) has the Lax representation [34]$\begin{eqnarray}\displaystyle \frac{\partial \phi }{\partial x}=U\phi ,\quad \displaystyle \frac{\partial \phi }{\partial t}=V\phi ,\end{eqnarray}$where$\begin{eqnarray}U={U}_{0}+{U}_{1}\eta ,\quad V={V}_{0}+{V}_{1}\eta +{V}_{2}{\eta }^{2},\end{eqnarray}$and η=η(t) is the spectral parameter that satisfies $\tfrac{{\rm{d}}\eta (t)}{{\rm{d}}t}\,=-2{\rm{i}}\beta (t)\eta {\left(t\right)}^{2}$, i.e.$\begin{eqnarray}\eta (t)=1/(2{\rm{i}}{\int }_{0}^{t}\beta (t){\rm{d}}t+{\rm{\Omega }}),\end{eqnarray}$with Ω an arbitrary complex number, and the generalized nonlinear Schrödinger equation (4) can be yielded from the so-called zero curvature condition Ut−Vx+[U,V]=0.
Denote$\begin{eqnarray}{\rm{\Phi }}(x,t,\eta )=\left({\phi }_{1}(x,t,\eta ),{\phi }_{2}(x,t,\eta )\right),\end{eqnarray}$where φ1(x, t, η) and φ2(x, t, η) are two linear independent eigenfunctions of (11) for spectral parameter η(t) such that Φ(x,t,η) is an invertible 2×2 matrix. Let g(x,t)= Φ(x,t,η)∣η = 0 [10] and set$\begin{eqnarray}\begin{array}{rcl}\tilde{{\rm{\Phi }}}(x,t) & = & g{\left(x,t\right)}^{-1}{\rm{\Phi }}(x,t),\\ \tilde{S} & = & -g{\left(x,t\right)}^{-1}{\sigma }_{3}g(x,t),\quad \tilde{U}=\eta \tilde{S},\end{array}\end{eqnarray}$$\begin{eqnarray}\tilde{V}=-2{\rm{i}}h(x,t){\eta }^{2}\tilde{S}+\gamma (t)\eta \tilde{S}-{\rm{i}}h(x,t)\eta \tilde{S}{\tilde{S}}_{x},\end{eqnarray}$we can confirm that (see appendix A for details)$\begin{eqnarray}\displaystyle \frac{\partial \tilde{{\rm{\Phi }}}}{\partial x}=\tilde{U}\tilde{{\rm{\Phi }}},\quad \displaystyle \frac{\partial \tilde{{\rm{\Phi }}}}{\partial t}=\tilde{V}\tilde{{\rm{\Phi }}}.\end{eqnarray}$is a Lax representation of equation (8), so the generalized inhomogeneous nonlinear Schrödinger equation (4) and the generalized inhomogeneous spin chain equation (3) are gauge equivalent.
From the above gauge transformation we know that if a nonzero solution q(x, t) of the equation (4) is known, then the eigenfunctions are determined explicitly from the Lax representation (11), such that the invertible matrix Φ(x, t, η) is in hand. Set g(x,t)=Φ(x,t,η)∣η = 0, then $\tilde{S}=-g{(x,t)}^{-1}{\sigma }_{3}g(x,t)$ is a solution of (8). By the definition of $\tilde{S}$, we can solve the spin components S1,S2,S3 and thus get $\vec{S}$, the solution of (3).
We remark that the above scheme is ready for giving the n-soliton solution for equation (3). As the explicit expression of the n-soliton solution is too complex, we only present the one-soliton solution in this paper.
3. Explicit magnetic soliton solutions
Let ${\phi }^{[0]}{(\eta )=({\phi }_{1}^{[0]}(\eta ),{\phi }_{2}^{[0]}(\eta ))}^{{\rm{T}}}$ be the eigenfunction of the Lax pair (11) corresponding to the zero solution of the nonlinear Schrödinger equation (4), for spectral parameter η(t)=η1(t), we denote ${h}_{11}={\phi }_{1}^{[0]}({\eta }_{1}),{h}_{12}={\phi }_{2}^{[0]}({\eta }_{1})$. By building the Darboux transformation via the Lax pair (11), from the zero solution we get the one-soliton solution of the generalized inhomogeneous nonlinear Schrödinger equation (4) as (see appendix B for details)$\begin{eqnarray}{q}^{[1]}=\displaystyle \frac{2\left({\eta }_{1}+{\bar{\eta }}_{1}\right){h}_{11}{\bar{h}}_{12}}{-| {h}_{11}{| }^{2}-| {h}_{12}{| }^{2}},\end{eqnarray}$the corresponding eigenfunction$\begin{eqnarray}{\phi }^{[1]}(\eta )={\left({\phi }_{1}^{[1]}(\eta ),{\phi }_{1}^{[1]}(\eta )\right)}^{{\rm{T}}}\end{eqnarray}$for η=0 is given by$\begin{eqnarray}\left\{\begin{array}{l}{\phi }_{1}^{[1]}(\eta ){| }_{\eta =0}=\tfrac{{\eta }_{1}| {h}_{11}{| }^{2}-{\bar{\eta }}_{1}| {h}_{12}{| }^{2}+({\eta }_{1}+{\bar{\eta }}_{1}){h}_{11}{\bar{h}}_{12}}{-| {h}_{11}{| }^{2}-| {h}_{12}{| }^{2}},\\ {\phi }_{2}^{[1]}(\eta ){| }_{\eta =0}=\tfrac{{\eta }_{1}| {h}_{12}{| }^{2}-{\bar{\eta }}_{1}| {h}_{11}{| }^{2}+({\eta }_{1}+{\bar{\eta }}_{1}){h}_{12}{\bar{h}}_{11}}{-| {h}_{11}{| }^{2}-| {h}_{12}{| }^{2}}.\end{array}\right.\end{eqnarray}$For convenience, write$\begin{eqnarray}{\phi }_{1}^{[1]}={\phi }_{1}^{[1]}(\eta ){| }_{\eta =0},\qquad {\phi }_{2}^{[1]}={\phi }_{2}^{[1]}(\eta ){| }_{\eta =0},\end{eqnarray}$according to the process mentioned above, we get the spin components of equation (3), which reads$\begin{eqnarray}\left\{\begin{array}{l}{S}_{1}=2({\mathfrak{R}}{\phi }_{1}^{[1]}{\mathfrak{R}}{\phi }_{2}^{[1]}-{\mathfrak{I}}{\phi }_{1}^{[1]}{\mathfrak{I}}{\phi }_{2}^{[1]}),\\ {S}_{2}=2({\mathfrak{R}}{\phi }_{1}^{[1]}{\mathfrak{I}}{\phi }_{2}^{[1]}+{\mathfrak{I}}{\phi }_{1}^{[1]}{\mathfrak{R}}{\phi }_{2}^{[1]}),\\ {S}_{3}=| {\phi }_{1}^{[1]}{| }^{2}-| {\phi }_{2}^{[1]}{| }^{2}.\end{array}\right.\end{eqnarray}$Here ${\mathfrak{R}}$ and ${\mathfrak{I}}$ represent the real part and the imaginary part of a complex function, respectively.
Let Ω=ω1+iω2 be an arbitrary complex number in the spectral parameter η(t) defined in (13), write ${\eta }_{1}(t)\,=\eta (t){| }_{{\rm{\Omega }}={\omega }_{1}+{\rm{i}}{\omega }_{2}}$, and ${\eta }_{2}(t)=-{\bar{\eta }}_{1}(t)$, let$\begin{eqnarray}A(t)=-\displaystyle \frac{2{\omega }_{1}}{{\omega }_{1}^{2}+{\left({\omega }_{2}+2{\int }_{0}^{t}\beta (t){\rm{d}}t\right)}^{2}}=-2{\mathfrak{R}}{\eta }_{1}(t),\end{eqnarray}$$\begin{eqnarray}B(t)=\displaystyle \frac{2({\omega }_{2}+2{\int }_{0}^{t}\beta (t){\rm{d}}t)}{{\omega }_{1}^{2}+{\left({\omega }_{2}+2{\int }_{0}^{t}\beta (t){\rm{d}}t\right)}^{2}}=2{\mathfrak{I}}{\bar{\eta }}_{1}(t),\end{eqnarray}$then from (18), we obtain the one soliton solution of the corresponding generalized nonlinear Schrödinger equation (4) which reads$\begin{eqnarray}{q}^{[1]}(x,t)=A(t){\rm{sech}} (\mu (x,t)){{\rm{e}}}^{{\rm{i}}\nu (x,t)},\end{eqnarray}$where$\begin{eqnarray}\mu (x,t)=2{\int }_{0}^{t}A(t)B(t)\alpha (t){\rm{d}}t-A(t)x+{\int }_{0}^{t}A(t)\gamma (t){\rm{d}}t,\end{eqnarray}$$\begin{eqnarray}\begin{array}{rcl}\nu (x,t) & = & {\int }_{0}^{t}(A{\left(t\right)}^{2}-B{\left(t\right)}^{2})\alpha (t){\rm{d}}t+B(t)x\\ & & -{\int }_{0}^{t}B(t)\gamma (t){\rm{d}}t,\end{array}\end{eqnarray}$and the spin components of the corresponding spin chain equation (3) read:$\begin{eqnarray}{S}_{1}=1+{\omega }_{1}A(t){{\rm{sech}} }^{2}(\mu (x,t)),\end{eqnarray}$$\begin{eqnarray}\begin{array}{rcl}{S}_{2} & = & {\omega }_{1}(B(t)\cosh (\mu (x,t))\cos (\nu (x,t))\\ & & +A(t)\sinh (\mu (x,t))\sin (\nu (x,t))){{\rm{sech}} }^{2}(\mu (x,t)),\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{rcl}{S}_{3} & = & {\omega }_{1}(B(t)\cosh (\mu (x,t))\sin (\nu (x,t))\\ & & -A(t)\sinh (\mu (x,t))\cos (\nu (x,t))){{\rm{sech}} }^{2}(\mu (x,t)).\end{array}\end{eqnarray}$
Note that when ω1=0, the solutions display some interesting special cases. If ω1=0 but ${\omega }_{2}+2{\int }_{0}^{t}\beta (t){\rm{d}}t\ne 0$ for any t, then A(t)=0 and ${q}^{[1]}(x,t)=0,\vec{S}=(1,0,0)$, this is a plain solution; however, if ${\omega }_{2}+2{\int }_{0}^{t}\beta (t){\rm{d}}t$ has zero, the situation becomes more complicated. For example, let $\beta (t)=\cos (t)$, then ${\int }_{0}^{t}\beta (t){\rm{d}}t=\sin (t)$, and thus if ∣ω2∣ <2, for times such that ${\omega }_{2}+2\sin t\ne 0$, q[1](x,t)=0, $\vec{S}=(1,0,0);$ but for some certain times such that ${\omega }_{2}+2\sin t=0$, one has A(t)=∞ and B(t)=0, so at these times, both μ(x, t) and ν(x, t) may diverge. Due to this reason, in what follows we only consider the case ω1≠0 such that the above solution gives a magnetic soliton whose propagation is determined by the time- and site-dependent inhomogeneous bilinear interaction, we will discuss the solution in detail at follows.
We see that the components S2 and S3 precess around the direction of S1. It is clear that if ${\int }_{0}^{t}\beta (t){\rm{d}}t\to \infty $ (for example, β (t)=1 or β (t)=t when t→∞), then A(t)→0, B(t)→0, which leads to $\vec{S}\to (1,0,0)$. In such a situation, β (t) acts just like a damping term. However, a periodic β (t) could cause the oscillation of A(t), and gives rise to oscillatory motion of the soliton. Figure 1 shows the damping and oscillation effects of the spin caused by the parameter β(t), this can also be confirmed by the evolution of the spin components as shown in figure 2.
Figure 1.
New window|Download| PPT slide Figure 1.Damping and oscillation of the spin caused by the parameter β(t). Both in (a) and (b), the black, red and blue curves on the Bloch sphere describe the spin state of the spin chain at t=0.5, t=1 and t=2, respectively. In (a), β(t)=1, it is shown that with the time increasing, the trajectory curves shrink gradually; in (b), $\beta (t)=2\sin (t)$, it is shown that with the time increasing, the trajectory curves oscillate around the direction of S1. The other parameters used are ω1=1, ω2=0, α(t)=1, γ(t)=0.
Figure 2.
New window|Download| PPT slide Figure 2.Damping-like effects and oscillation of the spin components caused by β(t). In (a)–(c), β(t)=t, it is shown that with the time increasing, S1→1 and S2→0, S3→0 (the case of β(t)=1 is similar), here β(t) gives rise to a damping-like effect. In (d)–(f), $\beta (t)=2\sin (t)$, it is clear that the spin components oscillate with β(t). The other parameters used are ω1=1, ω2=0, α(t)=1, γ(t)=0.
Furthermore, we point out that in the case of β(t)=0, solution (30) gives a conserved quantity which is independent of the choice of α(t) and γ(t):$\begin{eqnarray}{\int }_{-\infty }^{+\infty }(| {S}_{2}(x,t){| }^{2}+| {S}_{3}(x,t){| }^{2}){\rm{d}}x=\displaystyle \frac{4}{3}\,\displaystyle \frac{{\omega }_{1}({\omega }_{1}^{2}+3{\omega }_{2}^{2})}{{\omega }_{1}^{2}+{\omega }_{2}^{2}};\end{eqnarray}$but when β(t)≠0, this conservation could be broken. For example, take α(t)=1, γ(t)=0 and β(t)=1, we have$\begin{eqnarray}\begin{array}{l}{\displaystyle \int }_{-\infty }^{+\infty }(| {S}_{2}(x,t){| }^{2}+| {S}_{3}(x,t){| }^{2}){\rm{d}}x\\ \quad =\,\displaystyle \frac{4}{3}\,\displaystyle \frac{{\omega }_{1}({\omega }_{1}^{2}+3{\omega }_{2}^{2}+12{t}^{2}+12t{\omega }_{2})}{{\omega }_{1}^{2}+{\omega }_{2}^{2}+4{t}^{2}+4t{\omega }_{2}},\end{array}\end{eqnarray}$which is time-dependent.
By virtue of (30), we also see clearly the influence of the parameters α(t), β(t) and γ(t) to the propagation of the magnetic soliton. Compared with the standard dark soliton for S1 component given by ω1=1, ω2=1, α(t)=1 and β(t)=γ(t)=0, figure 3 shows how β(t) and γ(t) affect the propagation of the soliton. We also point out that in such a situation, the influence of α(t) is very similar to that of γ(t).
Figure 3.
New window|Download| PPT slide Figure 3.Influences of the parameter β(t) and γ(t) to the dynamics of the magnetic soliton given by equation (30). (a) β(t)=1; (b) β(t)=2t; (c) $\beta (t)=2\sin (t);$ (d) γ(t)=1; (e) γ(t)=2t; (f) $\gamma (t)=2\sin (t)$. In (a)–(c), γ(t)=0; and in (d)–(f), β(t)=0. The other parameters are ω1=1, ω2=1, α(t)=1, x∈[−10,10], t∈[−5,5].
As $| \vec{S}| =1$, the components of the vector $\vec{S}$ can be described as$\begin{eqnarray}{S}_{1}=\cos \theta ,\quad {S}_{2}+{\rm{i}}{S}_{3}=\sin \theta {{\rm{e}}}^{{\rm{i}}\phi },\end{eqnarray}$where θ and φ are the polar and azimuthal angles, respectively. From (30) we have$\begin{eqnarray}\theta =\arccos ({S}_{1})=\arccos (1+{\omega }_{1}A(t){{\rm{sech}} }^{2}(\mu (x,t))),\end{eqnarray}$and$\begin{eqnarray}\begin{array}{rcl}\phi & = & \arctan \left(\displaystyle \frac{{S}_{3}}{{S}_{2}}\right)\\ & = & \left\{\begin{array}{l}\nu (x,t)+\arctan \left(\tfrac{{\omega }_{1}}{{\omega }_{2}+2{\displaystyle \int }_{0}^{t}\beta (t){\rm{d}}t}\tanh (\mu (x,t))\right),\quad \mathrm{if}\quad {\omega }_{2}^{2}+{\beta }^{2}(t)\ne 0,\\ -\arctan \left(\cot \left(\tfrac{4}{{\omega }_{1}^{2}}{\displaystyle \int }_{0}^{t}\alpha (t){\rm{d}}t\right)\right.,\quad \mathrm{if}\quad {\omega }_{2}=\beta (t)=0.\end{array}\right.\end{array}\end{eqnarray}$When β(t)=0, θ and φ can be rewritten more clearly$\begin{eqnarray}\theta =\arccos \left(\displaystyle \frac{1}{1+{\rm{sech}} (2\xi )}-\displaystyle \frac{3{\omega }_{1}^{2}-{\omega }_{2}^{2}}{2({\omega }_{1}^{2}+{\omega }_{2}^{2})}{{\rm{sech}} }^{2}(\xi )\right),\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}\phi =\displaystyle \frac{2{\omega }_{2}\left(x+{\displaystyle \int }_{0}^{t}\gamma (t){\rm{d}}t\right)}{{\omega }_{1}^{2}+{\omega }_{2}^{2}}+\displaystyle \frac{4({\omega }_{1}^{2}-{\omega }_{2}^{2}){\displaystyle \int }_{0}^{t}\alpha (t){\rm{d}}t}{{\left({\omega }_{1}^{2}+{\omega }_{2}^{2}\right)}^{2}}\\ +\,\arctan \left(\displaystyle \frac{{\omega }_{1}}{{\omega }_{2}}\tanh \left(\displaystyle \frac{2{\omega }_{1}\left(x+{\displaystyle \int }_{0}^{t}\gamma (t){\rm{d}}t\right)}{{\omega }_{1}^{2}+{\omega }_{2}^{2}}\right.\right.\\ \quad \left.\left.-\,\displaystyle \frac{8{\omega }_{1}{\omega }_{2}{\displaystyle \int }_{0}^{t}\alpha (t){\rm{d}}t}{{\left({\omega }_{1}^{2}+{\omega }_{2}^{2}\right)}^{2}}\right)\right),\end{array}\end{eqnarray}$where$\begin{eqnarray}\xi =\displaystyle \frac{2{\omega }_{1}\left(\left({{\omega }_{1}}^{2}+{{\omega }_{2}}^{2}\right)\left(x+{\int }_{0}^{t}\gamma (t){\rm{d}}t\right)-4{\omega }_{2}{\int }_{0}^{t}\alpha (t){\rm{d}}t\right)}{{\left({{\omega }_{1}}^{2}+{{\omega }_{2}}^{2}\right)}^{2}}.\end{eqnarray}$Equations (34) and (35) give us all the information how the parameters ω1, ω2, α(t), β(t) and γ(t) affect the dynamics of the spin determined by (30). Via the evolution of θ(x, t), figure 4 displays the damping and oscillation effects caused by β(t), which agree with the conclusion as shown in figures 1 and 2.
Figure 4.
New window|Download| PPT slide Figure 4.Evolution of the polar angle θ(x, t) manipulated by the parameter β(t). (a) β(t)=0; (b) β(t)=1; (c) $\beta (t)=2\sin (t)$. The other parameters are ω1=1, ω2=1, α(t)=1, γ(t)=0, x∈[−10, 10], t∈[−5, 5].
4. Summary
This paper investigates the dynamics of magnetization in ferromagnet governed by the continuous Heisenberg spin chain equation with time-dependent inhomogeneous bilinear interaction and spin-transfer torque. By virtue of a gauge equivalence between the spin chain equation and an integrable generalized nonlinear Schrödinger equation, we get a novel nonautonomous magnetic soliton solution, which shows the possibility to control the dynamics of the spin chain through time- and site-dependent bilinear interaction and the spin-polarized current, an interesting phenomenon we found is that although the system we discussed has no damping term, the site-dependent bilinear interaction may break some conserved quantity, and give rise to damping or oscillation in the spin evolution. These results are beneficial to understand the related experiments.
Acknowledgments
The work was supported in part by NSFC under the grants No. 12075102, No. 61807025, and No. 61774001; Natural Science Foundation of Shannxi under the grant No. 2018JQ1065.
Appendix A
From the representation (14), we know that Φ(x, t, η) satisfies the Lax representation (11) as well as the eigenfunction φ(x,t,η). Then from the definition g(x,t)=Φ(x,t,η)∣η = 0 and the matrix representation of equation (11), we obtain$\begin{eqnarray}{g}_{x}={U}_{0}g,\quad {g}_{t}={V}_{0}g.\end{eqnarray}$The zero curvature condition U0t−V0x+[U0,V0]=0 also yields (4).
From (11), by the transform$\begin{eqnarray}{\rm{\Phi }}(x,t)=g(x,t)\tilde{{\rm{\Phi }}}(x,t),\end{eqnarray}$we obtain$\begin{eqnarray}\displaystyle \frac{\partial \tilde{{\rm{\Phi }}}}{\partial x}=\tilde{U}\tilde{{\rm{\Phi }}},\quad \displaystyle \frac{\partial \tilde{{\rm{\Phi }}}}{\partial t}=\tilde{V}\tilde{{\rm{\Phi }}}.\end{eqnarray}$where$\begin{eqnarray}\tilde{U}={g}^{-1}{Ug}-{g}^{-1}{g}_{x}={g}^{-1}(U-{U}_{0})g={g}^{-1}{U}_{1}g\eta ,\end{eqnarray}$$\begin{eqnarray}\begin{array}{rcl}\tilde{V} & = & {g}^{-1}{Vg}-{g}^{-1}{g}_{t}={g}^{-1}(V-{V}_{0})g\\ & = & {g}^{-1}{V}_{1}g\eta +{g}^{-1}{V}_{2}g{\eta }^{2}.\end{array}\end{eqnarray}$
On the other hand, from the definition of $\tilde{S}$, together by using the condition (A1), we obtain$\begin{eqnarray}\tilde{S}{\tilde{S}}_{x}=-{g}^{-1}{\sigma }_{3}{U}_{0}{\sigma }_{3}g+{g}^{-1}{\sigma }_{3}^{2}{U}_{0}g=2{g}^{-1}{U}_{0}g,\end{eqnarray}$here we have used the fact that ${\sigma }_{3}^{2}=I,{\sigma }_{3}{U}_{0}{\sigma }_{3}=-{U}_{0}.$ Substitute the matrix representation U1,V1,V2 into (A4) and (A5), together by using the condition (A6), we obtain$\begin{eqnarray}\tilde{U}=\eta \tilde{S},\quad \tilde{V}=-2{\rm{i}}h(x,t){\eta }^{2}\tilde{S}+\gamma (t)\eta \tilde{S}-{\rm{i}}h(x,t)\eta \tilde{S}{\tilde{S}}_{x}.\end{eqnarray}$The compatibility condition ${\tilde{U}}_{t}-{\tilde{V}}_{x}+[\tilde{U},\tilde{V}]=0$ yields$\begin{eqnarray}\begin{array}{l}{\rm{i}}h(x,t)({\tilde{S}}_{x}+\tilde{S}{\tilde{S}}_{x}\tilde{S})=0,\\ {\tilde{S}}_{t}=-{\rm{i}}h(x,t)({\tilde{S}}_{x}^{2}+\tilde{S}{\tilde{S}}_{{xx}})-{\rm{i}}\beta (t)\tilde{S}{\tilde{S}}_{x}+\gamma (t){\tilde{S}}_{x}.\end{array}\end{eqnarray}$Notice that ${\tilde{S}}^{2}=I$, differentiate it from both sides we obtain$\begin{eqnarray}\tilde{S}{\tilde{S}}_{x}+{\tilde{S}}_{x}\tilde{S}=0,\end{eqnarray}$thus $\tilde{S}{\tilde{S}}_{x}=-{\tilde{S}}_{x}\tilde{S}$, $\tilde{S}{\tilde{S}}_{x}\tilde{S}=-{\tilde{S}}_{x}$, such that (A8) is automatically satisfied. Differentiate (A9) again, we obtain ${\tilde{S}}_{x}^{2}\,=-\tfrac{1}{2}\left(\tilde{S}{\tilde{S}}_{{xx}}+{\tilde{S}}_{{xx}}\tilde{S}\right)$, substitute it into (A8), we obtain equation (8), which is equivalent to the spin chain equation (3). Thus we proved the gauge equivalence of the nonlinear evolution equation (4) and the corresponding generalized inhomogeneous spin chain equation (3).
Appendix B
Following the procedure presented in [35], let us construct the Darboux transformation of the system (11). For convenience, at this beginning, we assume η(t)=ηi(t), i=1, 2 are two spectral parameters, denote ${h}_{i1}={\phi }_{1}^{[0]}({\eta }_{i}),{h}_{i2}={\phi }_{2}^{[0]}({\eta }_{i}),i=1,2.$ Set$\begin{eqnarray}{H}_{1}=\left(\begin{array}{cc}{h}_{11} & {h}_{21}\\ {h}_{12} & {h}_{22}\end{array}\right),\quad {{\rm{\Lambda }}}_{1}=\left(\begin{array}{cc}{\eta }_{1} & 0\\ 0 & {\eta }_{2}\end{array}\right).\end{eqnarray}$If $\det {H}_{1}\ne 0$, set ${S}^{[1]}={H}_{1}{{\rm{\Lambda }}}_{1}{H}_{1}^{-1}$, then M[1]=η(t)I−S[1] is a Darboux matrix of (11), where I is the identity matrix. ${\phi }^{{\prime} }={M}^{[1]}\phi $ satisfies a system of the same form as (11), i.e.$\begin{eqnarray}\left\{\begin{array}{l}\tfrac{\partial {\phi }^{{\prime} }}{\partial x}={U}^{{\prime} }{\phi }^{{\prime} }=({U}_{0}^{{\prime} }-\eta {\sigma }_{3}){\phi }^{{\prime} },\\ \tfrac{\partial {\phi }^{{\prime} }}{\partial t}={V}^{{\prime} }{\phi }^{{\prime} }={\displaystyle \sum }_{j=0}^{2}{V}_{j}^{{\prime} }{\eta }^{j}{\phi }^{{\prime} }.\end{array}\right.\end{eqnarray}$Here ${U}_{0}^{{\prime} }$, ${V}_{j}^{{\prime} }$ are just replace q(x,t),r(x,t) of U0,Vj into ${q}^{{\prime} }(x,t),{r}^{{\prime} }(x,t)$.
The first order Darboux transformation is written as$\begin{eqnarray}{\rm{\Phi }}\to {{\rm{\Phi }}}^{[1]}={M}^{[1]}{\rm{\Phi }}=(\eta I-{S}^{[1]}){\rm{\Phi }},\end{eqnarray}$$\begin{eqnarray}q\to {q}^{[1]}=q-2{S}_{12}^{[1]},\quad r\to {r}^{[1]}=r+2{S}_{21}^{[1]},\end{eqnarray}$where ${S}^{[1]}={H}_{1}{{\rm{\Lambda }}}_{1}{H}_{1}^{-1}$, and ${S}_{{ij}}^{[1]},i,j=1,2$ denote the ith row jth column of the matrix S[1]. Thus$\begin{eqnarray}{q}^{[1]}=-2{S}_{12}^{[1]}=2\,\displaystyle \frac{{h}_{11}{h}_{21}\left({\eta }_{1}-{\eta }_{2}\right)}{{h}_{11}{h}_{22}-{h}_{21}{h}_{12}},\end{eqnarray}$$\begin{eqnarray}\left\{\begin{array}{l}{\phi }_{1}^{[1]}(\eta )=\left(\tfrac{{\eta }_{2}{h}_{12}{h}_{21}-{\eta }_{1}{h}_{11}{h}_{22}}{{h}_{11}{h}_{22}-{h}_{21}{h}_{12}}+\eta \right){\phi }_{1}^{[0]}(\eta )+\tfrac{({\eta }_{1}-{\eta }_{2}){h}_{11}{h}_{21}}{{h}_{11}{h}_{22}-{h}_{21}{h}_{12}}{\phi }_{2}^{[0]}(\eta ),\\ {\phi }_{2}^{[1]}(\eta )=\tfrac{({\eta }_{2}-{\eta }_{1}){h}_{12}{h}_{22}}{{h}_{11}{h}_{22}-{h}_{21}{h}_{12}}{\phi }_{1}^{[0]}(\eta )+\left(\tfrac{{\eta }_{1}{h}_{12}{h}_{21}-{\eta }_{2}{h}_{11}{h}_{22}}{{h}_{11}{h}_{22}-{h}_{21}{h}_{12}}+\eta \right){\phi }_{2}^{[0]}(\eta ).\end{array}\right.\end{eqnarray}$Here ${\phi }^{[1]}{(\eta )=({\phi }_{1}^{[1]}(\eta ),{\phi }_{2}^{[1]}(\eta ))}^{{\rm{T}}}$ is the eigenfunction of (11) corresponding to the one soliton solution (B5).
Recall that in the Lax system (11),$\begin{eqnarray}r=-\bar{q},\ \ A(-\bar{\eta }(t))=-\bar{A}(\eta (t)),\ \ B(-\bar{\eta }(t))=-\bar{C}(\eta (t)),\end{eqnarray}$where $\bar{q}$ denotes the complex conjugation of q. Such that if ${({\phi }_{1}(x,t),{\phi }_{2}(x,t))}^{{\rm{T}}}$ is a nontrivial solution of (11) for η(t)=η1(t), then ${({\bar{\phi }}_{2}(x,t),-{\bar{\phi }}_{1}(x,t))}^{{\rm{T}}}$ is a solution of (11) for $\eta (t)={\eta }_{2}(t)=-{\bar{\eta }}_{1}(t)$. Thus we set$\begin{eqnarray}{h}_{21}={\bar{h}}_{12},\quad {h}_{22}=-{\bar{h}}_{11},\end{eqnarray}$substitute it into (B5), we obtain (18).
On the other hand, from the initial zero solution, we get the eigenfunction as$\begin{eqnarray}\left\{\begin{array}{l}{\phi }_{1}^{[0]}(\eta )={{\rm{e}}}^{\displaystyle \int 2{\rm{i}}{\eta }^{2}\alpha (t)-\eta \gamma (t){\rm{d}}t-\eta x},\\ {\phi }_{2}^{[0]}(\eta )={{\rm{e}}}^{-\left(\displaystyle \int 2{\rm{i}}{\eta }^{2}\alpha (t)-\eta \gamma (t){\rm{d}}t-\eta x\right)}.\end{array}\right.\end{eqnarray}$It is easy to check that ${\phi }_{1}^{[0]}(\eta ){| }_{\eta =0}={\phi }_{2}^{[0]}(\eta ){| }_{\eta =0}=1$, such that from (B6) we have$\begin{eqnarray}\left\{\begin{array}{l}{\phi }_{1}^{[1]}(\eta ){| }_{\eta =0}=\tfrac{{\eta }_{2}{h}_{12}{h}_{21}-{\eta }_{1}{h}_{11}{h}_{22}+({\eta }_{1}-{\eta }_{2}){h}_{11}{h}_{21}}{{h}_{11}{h}_{22}-{h}_{21}{h}_{12}},\\ {\phi }_{2}^{[1]}(\eta ){| }_{\eta =0}=\tfrac{{\eta }_{1}{h}_{12}{h}_{21}-{\eta }_{2}{h}_{11}{h}_{22}+({\eta }_{2}-{\eta }_{1}){h}_{12}{h}_{22}}{{h}_{11}{h}_{22}-{h}_{21}{h}_{12}}.\end{array}\right.\end{eqnarray}$Substitute the condition (B8) into (B10), we obtain (20).
Denote$\begin{eqnarray}{{\rm{\Phi }}}^{[1]}=\left(\begin{array}{cc}{\phi }_{1}^{[1]}(\eta ) & {\bar{\phi }}_{2}^{[1]}(\eta )\\ {\phi }_{2}^{[1]}(\eta ) & -{\bar{\phi }}_{1}^{[1]}(\eta )\end{array}\right),\end{eqnarray}$as the invertible matrix of eigenfunction, and set g(x,t)= Φ[1]∣η = 0. Then $\tilde{S}=-g{(x,t)}^{-1}{\sigma }_{3}g(x,t)$ is the solution of (8) corresponding to the one-soliton solution (B5) of the nonlinear Schrödinger equation (4). The spin components read ${S}_{1}\,={\mathfrak{R}}({\tilde{S}}_{12}),{S}_{2}=-{\mathfrak{I}}({\tilde{S}}_{12}),{S}_{3}={\tilde{S}}_{11}$, which is the expression (22).
,1,∗, Dun Zhao2, Zai-Dong Li31School of Mathematics and Statistics, 
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