15 Topological Dynamics of Spin Texture Based Metamaterials
437
of acute-angled corner, obtuse-angled corner, edge, and bulk bands, respectively.
It can be seen that around the frequency of 944 MHz (948 MHz), the spectra for
acute-angled corner (obtuse-angled corner) have an obvious peak, which does not
happen for the spectra for edge and bulk bands. Therefore, these two peaks denote
two different corner states that are located at acute-angled or obtuse-angled corners.
Similarly, the frequency range for bulk and edge states is identified. Further, to
visualize the spatial distribution of the vortex oscillations for different modes, four
representative frequencies are chosen and are marked by green, blue, black, and red
dots: 872 MHz for the bulk state, 934 MHz for the edge state, 944 MHz for the acuteangled corner state, and 948 MHz for the obtuse-angled corner state, respectively. We
then stimulate their dynamics by applying a sinusoidal field to the whole system. The
10 ns gyration paths of all vortices are plotted in Fig. 15.24b–e when the excitation
field drives a steady-state vortex dynamics. The spatial distribution of vortices motion
for the bulk and edge states are shown in Fig. 15.24b, c, respectively. We observe
type I corner state with vortex oscillation localized at the acute-angled corner in
Fig. 15.24d, which is in good agreement with the theoretical result. Interestingly, we
note a strong hybridization between the type II and type III corner states, as shown
in Fig. 15.22f, which is because their frequencies are very close to each other and
their wavefunctions have a large overlap (see Fig. 15.22).
15.6 Conclusion and Outlook
We have introduced the concept of topological insulator (both first-order and higherorder) based on spin texture metamaterials. The emerging multiband chiral edge
modes possessing different handness in skyrmion lattice should be appealing for
designing future skyrmionic topological devices. Besides, the predicted second-order
insulating phase in vortex lattice can facilitate designing new functional devices
based on magnetic solitons. For instance, we can realize a magnetic imaging system
by designing one vortex lattice of the desired shape in the HOTI phase surrounded
by another vortex lattice in the trivial phase [142, 143]. It is noted that the unique
localization property of vortex gyrations strongly depends on the working frequency,
which would motivate us to devise magnetic nano-oscillators with high spatiotemporal resolution. Furthermore, the multiband nature of corner modes (spectrum ranging
from sub GHz to dozens of GHz) is very useful to design broad-band topological
devices. In the present model, we have considered nanodisks with identical radius.
When the translational symmetry is broken, for instance, by introducing Kekulé distortions to disk sizes, one may realize novel devices supporting robust Majorana-like
zero modes localized in the device’s geometric center [144]. From an experimental
point of view, the fabrication of artificial vortex or skyrmion lattices is readily within
reach of current technology, e.g., electron-beam lithography [78, 79, 145] or X-ray
illumination [146]. By tracking the nanometer-scale vortex orbits using the recently
developed ultrafast Lorentz microscopy technique in a time-resolved manner [147],
one can directly observe the second-order topological corner states.
437
of acute-angled corner, obtuse-angled corner, edge, and bulk bands, respectively.
It can be seen that around the frequency of 944 MHz (948 MHz), the spectra for
acute-angled corner (obtuse-angled corner) have an obvious peak, which does not
happen for the spectra for edge and bulk bands. Therefore, these two peaks denote
two different corner states that are located at acute-angled or obtuse-angled corners.
Similarly, the frequency range for bulk and edge states is identified. Further, to
visualize the spatial distribution of the vortex oscillations for different modes, four
representative frequencies are chosen and are marked by green, blue, black, and red
dots: 872 MHz for the bulk state, 934 MHz for the edge state, 944 MHz for the acuteangled corner state, and 948 MHz for the obtuse-angled corner state, respectively. We
then stimulate their dynamics by applying a sinusoidal field to the whole system. The
10 ns gyration paths of all vortices are plotted in Fig. 15.24b–e when the excitation
field drives a steady-state vortex dynamics. The spatial distribution of vortices motion
for the bulk and edge states are shown in Fig. 15.24b, c, respectively. We observe
type I corner state with vortex oscillation localized at the acute-angled corner in
Fig. 15.24d, which is in good agreement with the theoretical result. Interestingly, we
note a strong hybridization between the type II and type III corner states, as shown
in Fig. 15.22f, which is because their frequencies are very close to each other and
their wavefunctions have a large overlap (see Fig. 15.22).
15.6 Conclusion and Outlook
We have introduced the concept of topological insulator (both first-order and higherorder) based on spin texture metamaterials. The emerging multiband chiral edge
modes possessing different handness in skyrmion lattice should be appealing for
designing future skyrmionic topological devices. Besides, the predicted second-order
insulating phase in vortex lattice can facilitate designing new functional devices
based on magnetic solitons. For instance, we can realize a magnetic imaging system
by designing one vortex lattice of the desired shape in the HOTI phase surrounded
by another vortex lattice in the trivial phase [142, 143]. It is noted that the unique
localization property of vortex gyrations strongly depends on the working frequency,
which would motivate us to devise magnetic nano-oscillators with high spatiotemporal resolution. Furthermore, the multiband nature of corner modes (spectrum ranging
from sub GHz to dozens of GHz) is very useful to design broad-band topological
devices. In the present model, we have considered nanodisks with identical radius.
When the translational symmetry is broken, for instance, by introducing Kekulé distortions to disk sizes, one may realize novel devices supporting robust Majorana-like
zero modes localized in the device’s geometric center [144]. From an experimental
point of view, the fabrication of artificial vortex or skyrmion lattices is readily within
reach of current technology, e.g., electron-beam lithography [78, 79, 145] or X-ray
illumination [146]. By tracking the nanometer-scale vortex orbits using the recently
developed ultrafast Lorentz microscopy technique in a time-resolved manner [147],
one can directly observe the second-order topological corner states.