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Z. Li et al.
edge participate in the oscillation, as shown in Fig. 15.22e. Figure 15.22b shows the
bulk state with oscillations spreading over the whole lattice except the boundaries.
To judge whether these edge and corner states are topologically protected or not,
moderate defects and disorder are introduced into the system and we evaluate the
change of the spectrum. It can be found that the eigenfrequency of “zero-energy”
corner state 3 at the obtuse-angled corner (see Fig. 15.22f) is well confined around
927 MHz, which means that this corner state is suitably immune from external frustrations. This feature is due to the topological protection from the generalized chiral
symmetry. However, the frequencies of other corner modes (Fig. 15.22c, d) have
obvious shifts, revealing that these crystalline-symmetry-induced modes are sensitive to disorder. The origin of the edge state (Fig. 15.22e) is again attributed to the
so-called Tamm-Shockley mechanism [118, 119].
To figure out why the chiral symmetry-protected (CSP) corner modes emerge at
only obtuse-angled corners instead of acute-angled corners, the topological index
N = |N + − N − | is introduced, which captures the interplay between the topology
of the bulk Hamiltonian and the defect structure [31, 141]. Here, N counts the
number of topologically stable modes bound to corners, and N ± are integers counting
the number of eigenstates of the chiral symmetry operator ˆ
with eigenvalues +1
and −1, respectively. In the zero-correlation length limit d 2 → ∞, the breathing
honeycomb lattice is then reduced to isolated dimers (see Fig. 15.23b). As long as
the gap is not closed, the symmetry and the Berry phase remain, as evidenced by the
topological invariant θ . When the system is in the HOTI phase, N + = N − = 1 in
each edge unit cell and N + = N − = 2 for acute-angled corners, such that N = 0,
indicating that there may exist non-CSP modes at acute-angled corners or edges.
However, a similar analysis results in totally different outcome for obtuse-angled
corners: N + = 1 and N − = 2 or N + = 2 and N − = 1, which leads to N = 1. Thus,
CSP or “zero-energy” modes must exist in each obtuse-angled corner. It is worth
noting that the “zero-energy” corner state appears at acute-angled corners rather
than obtuse-angled corners if the edges of lattice change to zigzag type. This result
also can be fully explained in terms of the topological index N (see Fig. 15.23d, e).
On the other hand, in the limit d 1 → ∞ (see Fig. 15.23c, f), we find that there are
no uncoupled vortices. The six corners of the isolated hexagon are equivalent, with
no special edge or corner states.
15.5.3 Micromagnetic Simulations
The micromagnetic simulations are implemented to verify theoretical predictions.
Here, the parallelogram-shaped breathing honeycomb lattice of magnetic vortices
with an armchair edge is considered, as shown in Fig. 15.20b. All material parameters are the same as those for theoretical calculations in Fig. 15.21d. The numerical
package MUMAX3 [117] is used to simulate the collective dynamics of vortex lattice.
Figure 15.24a shows the temporal Fourier spectra of the vortex oscillations at
different positions, with the black, red, blue, and green curves indicating the positions
Z. Li et al.
edge participate in the oscillation, as shown in Fig. 15.22e. Figure 15.22b shows the
bulk state with oscillations spreading over the whole lattice except the boundaries.
To judge whether these edge and corner states are topologically protected or not,
moderate defects and disorder are introduced into the system and we evaluate the
change of the spectrum. It can be found that the eigenfrequency of “zero-energy”
corner state 3 at the obtuse-angled corner (see Fig. 15.22f) is well confined around
927 MHz, which means that this corner state is suitably immune from external frustrations. This feature is due to the topological protection from the generalized chiral
symmetry. However, the frequencies of other corner modes (Fig. 15.22c, d) have
obvious shifts, revealing that these crystalline-symmetry-induced modes are sensitive to disorder. The origin of the edge state (Fig. 15.22e) is again attributed to the
so-called Tamm-Shockley mechanism [118, 119].
To figure out why the chiral symmetry-protected (CSP) corner modes emerge at
only obtuse-angled corners instead of acute-angled corners, the topological index
N = |N + − N − | is introduced, which captures the interplay between the topology
of the bulk Hamiltonian and the defect structure [31, 141]. Here, N counts the
number of topologically stable modes bound to corners, and N ± are integers counting
the number of eigenstates of the chiral symmetry operator ˆ
with eigenvalues +1
and −1, respectively. In the zero-correlation length limit d 2 → ∞, the breathing
honeycomb lattice is then reduced to isolated dimers (see Fig. 15.23b). As long as
the gap is not closed, the symmetry and the Berry phase remain, as evidenced by the
topological invariant θ . When the system is in the HOTI phase, N + = N − = 1 in
each edge unit cell and N + = N − = 2 for acute-angled corners, such that N = 0,
indicating that there may exist non-CSP modes at acute-angled corners or edges.
However, a similar analysis results in totally different outcome for obtuse-angled
corners: N + = 1 and N − = 2 or N + = 2 and N − = 1, which leads to N = 1. Thus,
CSP or “zero-energy” modes must exist in each obtuse-angled corner. It is worth
noting that the “zero-energy” corner state appears at acute-angled corners rather
than obtuse-angled corners if the edges of lattice change to zigzag type. This result
also can be fully explained in terms of the topological index N (see Fig. 15.23d, e).
On the other hand, in the limit d 1 → ∞ (see Fig. 15.23c, f), we find that there are
no uncoupled vortices. The six corners of the isolated hexagon are equivalent, with
no special edge or corner states.
15.5.3 Micromagnetic Simulations
The micromagnetic simulations are implemented to verify theoretical predictions.
Here, the parallelogram-shaped breathing honeycomb lattice of magnetic vortices
with an armchair edge is considered, as shown in Fig. 15.20b. All material parameters are the same as those for theoretical calculations in Fig. 15.21d. The numerical
package MUMAX3 [117] is used to simulate the collective dynamics of vortex lattice.
Figure 15.24a shows the temporal Fourier spectra of the vortex oscillations at
different positions, with the black, red, blue, and green curves indicating the positions