15 Topological Dynamics of Spin Texture Based Metamaterials
427
Q 0 = ω 0 −
ξ 2
1 + 2ξ 2
2
2ω K
,
Q 1 = −
ξ 1 ξ 2
2ω K
exp(i
2π
3
)
exp[ik · (a 2 − a 1 )] + exp(−ik · a 1 )
−
ξ 2
2
2ω K
exp(i
2π
3
),
Q 2 = ζ 1 exp[ik · (a 2 − a 1 )],
Q 3 = −
ξ 1 ξ 2
2ω K
exp(−i
2π
3
)
exp[ik · (a 2 − a 1 )] + exp(ik · a 2 )
−
ξ 2
2
2ω K
exp(−i
2π
3
),
Q 4 = −
ξ 1 ξ 2
2ω K
exp(i
2π
3
)
exp[ik · (a 2 − a 1 )] + exp(ik · a 2 )
−
ξ 2
2
2ω K
exp(i
2π
3
),
Q 5 = −
ξ 1 ξ 2
2ω K
exp(−i
2π
3
)[exp(ik · a 2 ) + exp(ik · a 1 )] −
ξ 2
2
2ω K
exp(−i
2π
3
),
Q 6 = −
ξ 1 ξ 2
2ω K
exp(i
2π
3
)[exp(ik · a 2 ) + exp(ik · a 1 )] −
ξ 2
2
2ω K
exp(i
2π
3
),
Q 7 = −
ξ 1 ξ 2
2ω K
exp(i
2π
3
)
exp[ik · (a 1 − a 2 )] + exp(ik · a 1 )
−
ξ 2
2
2ω K
exp(i
2π
3
).
(15.13)
Topological invariants can be used to distinguish different phases. For any insulator with translational symmetry, the gauge-invariant Chern number of bulk bands
[58, 138]
C =
i
2π
BZ
dk x dk y Tr
P(
∂ P
∂k x
∂ P
∂k y
−
∂ P
∂k y
∂ P
∂k x
)
(15.14)
is often adopted for determining the FOTI phase, where P is projection matrix
P(k) = φ(k)φ(k)
† , with φ(k) being the normalized eigenstate (column vector) of
Hamiltonian, and the integral is over the first Brillouin zone. However, to determine
whether the system allows the HOTI phase, another different topological invariant
should be considered.
In the presence of six-fold rotational (C 6 ) symmetry, a proper topological invariant
is the Z 6 Berry phase [48–52]:
θ =
L 1
Tr[A(k)] · dk (mod 2π),
(15.15)
where A(k) is the Berry connection:
A(k) = i
†
(k)
∂
∂k
(k).
(15.16)
Here, (k) = [φ 1 (k),φ 2 (k),φ 3 (k)] is the 6 × 3 matrix composed of the eigenvectors
of (15.12) for the lowest three bands. L 1 is an integral path in momentum space
G
→ → K
; see the green line segment in Fig. 15.19a. The Wilson-loop approach
is adopted for evaluating the Berry phase θ to avoid the difficulty of the gauge choice
[23, 24]. It is worth mentioning that the six high-symmetry points G, K , G
, K
, G
,