416
Z. Li et al.
states with opposite chiralities coexist in a given soliton lattice. There is no analogue
in condensed matter system, to the best of our knowledge. In contrast, the propagation of ES3 is bidirectional, as shown in Fig. 15.10c. This non-chiral mode can
be simply explained in terms of the Tamm-Shockley mechanism [118, 119] which
predicts that the periodicity breaking of the crystal potential at the boundary can
lead to the formation of a conducting surface/edge state. Furthermore, it is rather
straightforward to show the propagation of the chiral modes is immune from the
defects and robust against the type of boundary, while the Tamm-Shockley mode is
not.
15.3.2 Theoretical Model
We adopt the generalized Thiele’s equation to theoretically understand the multiband
chiral skyrmionic edge states carrying opposite chiralities. First of all, we assume
that the steady-state magnetization m depends on not only the position of the guiding
center R(t) but also its velocity ˙
R(t) and acceleration ¨
R(t), and write m = m(r −
R(t), ˙
R(t), ¨
R(t)). After some algebra and by neglecting the damping term, one can
obtain the generalized Thiele’s equation including both a second-order inertial term
of skyrmion mass M [63, 92, 102] and a non-Newtonian third-order gyroscopic term
G 3 [120–122]:
G 3 ˆ
z ×
d
3 U j
dt 3 − M
d
2 U j
dt 2 + G ˆ
z ×
dU j
dt
+ F j = 0,
(15.5)
where U j and G share the same definition as (15.2). The conservative force can
be expressed as F j = −∂ W/∂U j . Here W is the total potential energy for the
sum of single disk and the interaction energy between nearest-neighbor disks:
W =
j K U
2
j /2 +
j =k U jk /2 with U jk = I U
j U
k − I ⊥ U
⊥
j U
⊥
k [83, 123, 124].
I and I ⊥ are the longitudinal and the transverse coupling constants, respectively. By
imposing U j = (u j , v j ) and defining ψ j = u j + iv j , we have:
ˆ
Dψ j = ω K ψ j +
k∈∈ j
(ζ ψ k + ξ e
i2θ jk ψ
∗
k ),
(15.6)
where the differential operator ˆ
D = iω 3
d
3
dt 3 − ω M
d
2
dt 2 + i
d
dt
, ω 3 = G 3 /|G|, ω M =
M/|G|, ω K = K /|G|, ζ = (I − I ⊥ )/2|G|, and ξ = (I + I ⊥ )/2|G|, θ jk is the angle
of the direction ˆ
e jk from x axis with ˆ
e jk = (R
0
k − R
0
j )/|R
0
k − R
0
j |, and j is the set
of the nearest neighbors of j (here Q = −1). We then expand the complex variable
to
ψ j = χ j (t) exp(−iω 0 t) + η j (t) exp(iω 0 t).
(15.7)
For counterclockwise (clockwise) skyrmion gyrations, one can justify |χ j | | |η j |
(|χ j | | |η j |). Substituting (15.7) into (15.6), one can obtain the following eigenvalue
equation:
Z. Li et al.
states with opposite chiralities coexist in a given soliton lattice. There is no analogue
in condensed matter system, to the best of our knowledge. In contrast, the propagation of ES3 is bidirectional, as shown in Fig. 15.10c. This non-chiral mode can
be simply explained in terms of the Tamm-Shockley mechanism [118, 119] which
predicts that the periodicity breaking of the crystal potential at the boundary can
lead to the formation of a conducting surface/edge state. Furthermore, it is rather
straightforward to show the propagation of the chiral modes is immune from the
defects and robust against the type of boundary, while the Tamm-Shockley mode is
not.
15.3.2 Theoretical Model
We adopt the generalized Thiele’s equation to theoretically understand the multiband
chiral skyrmionic edge states carrying opposite chiralities. First of all, we assume
that the steady-state magnetization m depends on not only the position of the guiding
center R(t) but also its velocity ˙
R(t) and acceleration ¨
R(t), and write m = m(r −
R(t), ˙
R(t), ¨
R(t)). After some algebra and by neglecting the damping term, one can
obtain the generalized Thiele’s equation including both a second-order inertial term
of skyrmion mass M [63, 92, 102] and a non-Newtonian third-order gyroscopic term
G 3 [120–122]:
G 3 ˆ
z ×
d
3 U j
dt 3 − M
d
2 U j
dt 2 + G ˆ
z ×
dU j
dt
+ F j = 0,
(15.5)
where U j and G share the same definition as (15.2). The conservative force can
be expressed as F j = −∂ W/∂U j . Here W is the total potential energy for the
sum of single disk and the interaction energy between nearest-neighbor disks:
W =
j K U
2
j /2 +
j =k U jk /2 with U jk = I U
j U
k − I ⊥ U
⊥
j U
⊥
k [83, 123, 124].
I and I ⊥ are the longitudinal and the transverse coupling constants, respectively. By
imposing U j = (u j , v j ) and defining ψ j = u j + iv j , we have:
ˆ
Dψ j = ω K ψ j +
k∈∈ j
(ζ ψ k + ξ e
i2θ jk ψ
∗
k ),
(15.6)
where the differential operator ˆ
D = iω 3
d
3
dt 3 − ω M
d
2
dt 2 + i
d
dt
, ω 3 = G 3 /|G|, ω M =
M/|G|, ω K = K /|G|, ζ = (I − I ⊥ )/2|G|, and ξ = (I + I ⊥ )/2|G|, θ jk is the angle
of the direction ˆ
e jk from x axis with ˆ
e jk = (R
0
k − R
0
j )/|R
0
k − R
0
j |, and j is the set
of the nearest neighbors of j (here Q = −1). We then expand the complex variable
to
ψ j = χ j (t) exp(−iω 0 t) + η j (t) exp(iω 0 t).
(15.7)
For counterclockwise (clockwise) skyrmion gyrations, one can justify |χ j | | |η j |
(|χ j | | |η j |). Substituting (15.7) into (15.6), one can obtain the following eigenvalue
equation: