418
Z. Li et al.
zigzag termination at y = 0 and y = W 2 . The average vertical position of the modes
y ≡
j R
0
j,y |U j |
2
/
j |U j |
2 are also shown in Fig. 15.11, where R
0
j,y is the equilibrium position of the skyrmion projected onto the y axis, represented by different
colors: closer to magenta indicating more localized at the upper edge. It is interesting
to note that the chirality of ES4 is opposite to those of ES1 and ES2. An interpretation is as following: The direction reversal of the skyrmion gyration generates a π
phase accumulation in the next-nearest-neighbor hopping term of (15.8). The Chern
numbers of two neighbouring bulk bands then switch their signs, so that the chirality
of the edge state in between reverses.
15.4 Corner States in a Breathing Kagome Lattice of
Vortices
In previous sections, we have discussed the topological insulating phases in magnetic
system. All these phases, however, are first order by nature. In this section, we will
discuss the higher-order topological edge states (corner states) in magnetic system,
and demonstrate their features in breathing kagome lattice of magnetic vortices.
15.4.1 The Theoretical Results and Discussions
A breathing kagome lattice of magnetic nanodisks with vortex states is considered.
Figure 15.12a plots the lattice structure with alternate distance parameters d 1 and
d 2 . We start with the generalized Thiele’s equation (15.6) to describe the collective
dynamics of vortex lattice:
ˆ
Dψ j = ω K ψ j +
k∈∈ j,l
(ζ l ψ k + ξ l e
i2θ jk ψ
∗
k ),
(15.9)
where the differential operator ˆ
D = iω 3
d
3
dt 3 − ω M
d
2
dt 2 − i
d
dt
, ζ l = (I ,l − I ⊥,l )/2|G|,
and ξ l = (I ,l + I ⊥,l )/2|G|, with l = 1 (or l = 2) representing the distance d 1 (or d 2 )
between the nearest neighbor vortices, here the topological charge Q = 1/2.
The coupling strengths I and I ⊥ are strongly depend on the parameter d (d = d
/r
with d
the distance between two vortices and r being the radius of nanodisk) [127–
129]. The analytical expression of I (d) and I ⊥ (d) are essential for evaluating the
spectrum and the phase diagram of vortex gyrations. With the help of micromagnetic simulations for two-vortex system with different combinations of vortex polarities, one can obtain the best fit of the numerical data: I = μ 0 M
2
s r (−1.72064 ×
10
−4
+ 4.13166 × 10
−2
/d
3
− 0.24639/d
5
+ 1.21066/d
7
− 1.81836/d
9 ) and I ⊥ =
μ 0 M
2
s r (5.43158 × 10
−4
− 4.34685 × 10
−2
/d
3
+ 1.23778/d
5
− 6.48907/d
7
+
Z. Li et al.
zigzag termination at y = 0 and y = W 2 . The average vertical position of the modes
y ≡
j R
0
j,y |U j |
2
/
j |U j |
2 are also shown in Fig. 15.11, where R
0
j,y is the equilibrium position of the skyrmion projected onto the y axis, represented by different
colors: closer to magenta indicating more localized at the upper edge. It is interesting
to note that the chirality of ES4 is opposite to those of ES1 and ES2. An interpretation is as following: The direction reversal of the skyrmion gyration generates a π
phase accumulation in the next-nearest-neighbor hopping term of (15.8). The Chern
numbers of two neighbouring bulk bands then switch their signs, so that the chirality
of the edge state in between reverses.
15.4 Corner States in a Breathing Kagome Lattice of
Vortices
In previous sections, we have discussed the topological insulating phases in magnetic
system. All these phases, however, are first order by nature. In this section, we will
discuss the higher-order topological edge states (corner states) in magnetic system,
and demonstrate their features in breathing kagome lattice of magnetic vortices.
15.4.1 The Theoretical Results and Discussions
A breathing kagome lattice of magnetic nanodisks with vortex states is considered.
Figure 15.12a plots the lattice structure with alternate distance parameters d 1 and
d 2 . We start with the generalized Thiele’s equation (15.6) to describe the collective
dynamics of vortex lattice:
ˆ
Dψ j = ω K ψ j +
k∈∈ j,l
(ζ l ψ k + ξ l e
i2θ jk ψ
∗
k ),
(15.9)
where the differential operator ˆ
D = iω 3
d
3
dt 3 − ω M
d
2
dt 2 − i
d
dt
, ζ l = (I ,l − I ⊥,l )/2|G|,
and ξ l = (I ,l + I ⊥,l )/2|G|, with l = 1 (or l = 2) representing the distance d 1 (or d 2 )
between the nearest neighbor vortices, here the topological charge Q = 1/2.
The coupling strengths I and I ⊥ are strongly depend on the parameter d (d = d
/r
with d
the distance between two vortices and r being the radius of nanodisk) [127–
129]. The analytical expression of I (d) and I ⊥ (d) are essential for evaluating the
spectrum and the phase diagram of vortex gyrations. With the help of micromagnetic simulations for two-vortex system with different combinations of vortex polarities, one can obtain the best fit of the numerical data: I = μ 0 M
2
s r (−1.72064 ×
10
−4
+ 4.13166 × 10
−2
/d
3
− 0.24639/d
5
+ 1.21066/d
7
− 1.81836/d
9 ) and I ⊥ =
μ 0 M
2
s r (5.43158 × 10
−4
− 4.34685 × 10
−2
/d
3
+ 1.23778/d
5
− 6.48907/d
7
+