Controlling topological superconductivity by magnetization dynamics

We study theoretically a chain of precessing classical magnetic impurities in an $s$-wave superconductor. Utilizing a rotating wave description, we derive an effective Hamiltonian that describes the emergent Shiba band. We find that this Hamiltonian shows nontrivial topological properties, and we obtain the corresponding topological phase diagrams both numerically and analytically. We show that changing precession frequency offers control over topological phase transitions and the emergence of Majorana bound states. We propose driving the magnetic impurities or magnetic texture into precession by means of spin-transfer torque in a spin Hall setup, and manipulate it using spin superfluidity in the case of planar magnetic order.

Figure 1

Sketch of the precessing spin helix in a two-dimensional $s$-wave superconductor (the blue plane). The classical spins (red arrows) are separated by a distance $a$ and precess around the $z$ axis with a frequency ${\omega }_0$ at a polar angle $\theta$. The precession azimuthal angle ${\varphi }_j\left(t\right)={\omega }_{0}t+k_{h}ja$, with $k_h$ the step of the helix and $j$ the position of the spin in the chain. The local dynamical Shiba state wave functions which overlap to eventually give a dynamical Shiba band are shown in green.

Figure 2

The gap around quasienergy $E=0$ and the winding number of the Shiba band for the small and large coherence length regimes (first and second rows, respectively), plotted as functions of the driving frequency $B$ and the Fermi momentum $k_F$ (precession angle $\theta$) in the left (right) column. The continuous black lines separate regions with different winding numbers $\mathcal{W}$ which are well defined even for the gapless regions. The vertical red lines highlight the existence of localized Majorana end states in an open system (see Fig. 3 for details). We set $k_h=\pi /4$, $v_F=0.2$, ${\mathrm{\Delta }}_s=1$, $a=1$, $\alpha =0.9999$. The polar angle $\theta =\pi /3$ and $k_F=159$ in the left and right columns correspondingly.

Figure 3

The quasispectrum (normalized by the driving $B$) for an open Shiba chain, in the regime of small coherence length as a function of the driving frequency $B$. The horizontal red line stands for the zero quasienergy Majorana end mode, while $\mathcal{W}$ defines the bulk winding number (see main text). A region with two gaps, at $E=0$ and $E=B$, exists, but the latter is trivial as we find no Majoranas emerging. We set $k_F=159$, $k_h=\pi /4$, $v_F=0.2$, ${\mathrm{\Delta }}_s=1$, $a=1$, $\theta =\pi /3$, $\alpha =0.9999$.