Nonlinear dynamics of the topological helicity wave in a frustrated skyrmion string

A skyrmion in a frustrated magnetic system has the helicity degree of freedom. A skyrmion string is formed in a frustrated layered system, which is well described by the $XY$ model owing to the exchange coupling between adjacent layers. We consider a system where the interlayer exchange couplings are alternating, where the dimerized $XY$ model is materialized, whose linear limit is the Su-Schrieffer-Heeger model. We argue that it is a nonlinear topological system. We study the quench dynamics of the helicity wave under the initial condition that the helicity of the skyrmion in the bottommost layer is rotated. It yields a good signal to detect whether the system is topological or trivial. Our results show that the helicity dynamics of the skyrmion string have a rich physics in the modulated exchange-coupled system.

Figure 1

(a) Schematic of a Bloch-type skyrmion. (b) Schematic of a Néel-type skyrmion. (c) Illustration of a layered frustrated magnet. The alternating thicknesses of spacers are assumed to materialize alternating interlayer exchange couplings. A heavy-metal layer is underneath the layered frustrated magnet, in which a vertical spin current could be generated to drive the helicity dynamics of the skyrmion in the bottommost magnetic layer. (d) Illustration of a 3D Bloch-type skyrmion string.

Figure 2

(a) Simulated dipole-dipole energy of a static frustrated skyrmion as a function of the helicity $\eta$. The skyrmion energy is calculated by assuming a fixed skyrmion profile with a given helicity. The magnetic parameters are as follows: the nearest-neighbor exchange $J_1=30$ meV, next-nearest-neighbor exchange $J_2=-0.8$ (in units of $J_1=1$), next-next-nearest-neighbor exchange $J_3=-0.6$, and perpendicular magnetic anisotropy constant $K=0.01$. The simulated model is a square-lattice sample of $31\times 31$ spins with the lattice constant of 0.4 nm. More modeling details are given in Ref. [9]. Note that we do not apply an external magnetic field to stabilize the frustrated skyrmion since our system has the perpendicular magnetic anisotropy. (b) Snapshots showing the frustrated skyrmions with different $\eta$. The out-of-plane spin component is color coded: red is into the plane, blue is out of the plane, white is in-plane. The in-plane spin directions are indicated by arrows.

Figure 3

Illustration of a skyrmion string, where blue disks represent skyrmions, while red and green bonds represent the couplings $J_A$ and $J_B$. (a) The limit $\lambda =-1$, where the system is topological. (b) An intermediate state $\left|\lambda \right|<1$, where it is topological or trivial. (c) The limit $\lambda =1$, where it is trivial.

Figure 4

(a) The energy spectrum of the SSH model as a function of $\lambda$. Topological edge states are marked in red, while the bulk states are marked in cyan. The vertical axis is energy in units of $J$. (b) The spatial profile of the edge states $|{\psi }^{\text{b}}{|}^2+{\left|{\psi }^{\text{t}}\right|}^2$ as a function of $\lambda$. (c) $|{\psi }^{\text{b}}{|}^2+{\left|{\psi }^{\text{t}}\right|}^2$ as a function of $\lambda$. Red disk at $\lambda =1$ shows the peak amplitude owing to the dimer state. We have used a finite chain with length $L=50.$

Figure 5

Time evolution of $sin{\varphi }_i$ for $0\le t\le 50$. (a1), (a2), (c1), and (c2) Topological phase with $\lambda =-0.5$. (b1), (b2), (d1), and (d2) Trivial with $\lambda =0.5$. (a1), (a2), (b1), and (b2) $\xi =0.1$. (c1), (c2), (d1), and (d2) $\xi =0.5$. We have set $m=1$, $J=1$, and $V=0.1$. We have used a finite chain with length $L=50$.

Figure 6

Phase indicator $\mathrm{\Phi }$ as a function of the dimerization $\lambda$. (a) $\xi =0.1$, (b) $\xi =0.25$, (c) $\xi =0.5$, and (d) $\xi =0.75$. Black curves indicate $V=0$, magenta curves indicate $V=0.1$, and cyan curves indicate $V=0.2$. We have set $m=1$, $J=1$, $L=50$, and $T=100$.

Figure 7

Phase indicator $\mathrm{\Phi }$ in the $\lambda \text{−}\xi$ plane. (a) Bird's eye view together with the color palette indicating the amplitude. We have set $V=0.2$, $m=1$, $J=1$, $L=50$, and $T=100$. (b) Schematic illustration of the phase diagram.

Figure 8

Disorder effects of the phase indicator $\mathrm{\Phi }$ as a function of the dimerization $\lambda$. (a) $\xi =0.1$, (b) $\xi =0.25$, (c) $\xi =0.5$, and (d) $\xi =0.75$. Cyan curves indicate $R=0$, magenta curves indicate $R=0.1$, magenta curves indicate $R=0.2$, orange curves indicate $R=0.3$, and green curves indicate $R=0.4$. We have set $m=1$, $J=1$, $V=0.2$, $L=50$, and $T=100$.