386
X. S. Wang and X. R. Wang
where M α = K α + B + 3J , d(k) = i D
i (−1)
i e
ik·b i , f (k) =
J +
F
2
j e
ik·a j ,
g 1 (k) =
F
2
j e
2iθ j e
ik·a j , g 2 (k) =
F
2
j e
−2iθ j e
ik·a j (α = A, B; j = 1, 2, 3; i =
1 ∼ 6). θ j is the angle between a j and e x and η i is the angle between b i and e x .
The solutions come in pairs ±λ, obeying a particle-hole symmetry, so it is enough to
consider only the positive bands. Although H is not Hermitian, H can be expressed
as S H 1 where H 1 is Hermitian and S = σ 0
σ 3 (with σ 0 being the 2 × 2 identity
matrix and σ 3 the Pauli matrix), and all the eigenvalues of H are real as long as the
ferromagnetic state is a stable equilibrium state. H is pseudo-Hermitian with metric
S, expressed as [159],
H
†
= S H S
−1
.
(14.48)
The metric S comes from the bosonic nature of system in the presence of the pseudodipolar term [160]. Thus, for finite F, unlike previous studies [142, 143], this
model is exclusive in bosonic systems and has no counterparty in electronic systems.
The spin wave dispersion relation ω(k) is shown in Fig. 14.14c for J = 0.1,
F = 0.5, = D = 0, and M ≡ 3J + K + B = 1.3 (in units of μ 0 μ
2
/a
3 ). The band
gap at K and K
points is g = M −
M 2 − 9F 2 /4, as shown in the right panel of
Fig. 14.14b. When F = 0, the bands linearly cross each other at K and K
to form
Dirac cones, similar to electron Dirac cones in graphene [161]. Thus, the pseudodipolar NN exchange interaction plays a crucial role in the band gap opening.
We consider now a long strip of width 100a with zigzag edges along x direction
as shown in Fig. 14.14a. The density plot of the spectral function on the top edge is
shown in Fig. 14.15a, for the same parameters as those in Fig. 14.14b. The negative
slope of the dispersion curve (the bright line) of the top edge states in the band gap
proves that these states propagate to the left. Similarly, the states on the bottom edge
propagate unidirectionally to the right. The edge channels connect the upper and
lower bands and linearly cross each other in the momentum space. Figure 14.15b
shows spatial distribution of the edge spin wave eigenstate of ω = 1.2 (γμ/a
3 ). The
symbol shapes and sizes are the spin precession trajectory and the precession radius
at each site, respectively. The azimuthal angle φ of m at t = 0 is encoded by the
colors with the color ring shown in the inset. It is evident that the spin wave is mainly
localized on the outermost sites. Here we use a zigzag edge as an example, and the
existence of the topologically protected unidirectional edge states does not depend
on edge types.
To understand the Berry phase of a spin wave, we consider the time evolution of a
state of ϕ k (t) with momentum k in an infinite sample. Without breaking the translational symmetry, k is a good quantum number that can be used to label eigenstates.
ϕ k (t) must be the Bloch state in the nth band n (k) multiplying a time dependent
phase, and satisfies following time-dependent linearized LLG equation, similar to a
Schrödinger equation,
i
μ 0 μ
γ
∂ϕ k
∂t
= H ϕ k .
(14.49)
Note that the same expression can be derived from quantum-mechanical model by
using Bogoliubov transformation, given that
μ 0 μ
γ
= s where s is the total spin of each
X. S. Wang and X. R. Wang
where M α = K α + B + 3J , d(k) = i D
i (−1)
i e
ik·b i , f (k) =
J +
F
2
j e
ik·a j ,
g 1 (k) =
F
2
j e
2iθ j e
ik·a j , g 2 (k) =
F
2
j e
−2iθ j e
ik·a j (α = A, B; j = 1, 2, 3; i =
1 ∼ 6). θ j is the angle between a j and e x and η i is the angle between b i and e x .
The solutions come in pairs ±λ, obeying a particle-hole symmetry, so it is enough to
consider only the positive bands. Although H is not Hermitian, H can be expressed
as S H 1 where H 1 is Hermitian and S = σ 0
σ 3 (with σ 0 being the 2 × 2 identity
matrix and σ 3 the Pauli matrix), and all the eigenvalues of H are real as long as the
ferromagnetic state is a stable equilibrium state. H is pseudo-Hermitian with metric
S, expressed as [159],
H
†
= S H S
−1
.
(14.48)
The metric S comes from the bosonic nature of system in the presence of the pseudodipolar term [160]. Thus, for finite F, unlike previous studies [142, 143], this
model is exclusive in bosonic systems and has no counterparty in electronic systems.
The spin wave dispersion relation ω(k) is shown in Fig. 14.14c for J = 0.1,
F = 0.5, = D = 0, and M ≡ 3J + K + B = 1.3 (in units of μ 0 μ
2
/a
3 ). The band
gap at K and K
points is g = M −
M 2 − 9F 2 /4, as shown in the right panel of
Fig. 14.14b. When F = 0, the bands linearly cross each other at K and K
to form
Dirac cones, similar to electron Dirac cones in graphene [161]. Thus, the pseudodipolar NN exchange interaction plays a crucial role in the band gap opening.
We consider now a long strip of width 100a with zigzag edges along x direction
as shown in Fig. 14.14a. The density plot of the spectral function on the top edge is
shown in Fig. 14.15a, for the same parameters as those in Fig. 14.14b. The negative
slope of the dispersion curve (the bright line) of the top edge states in the band gap
proves that these states propagate to the left. Similarly, the states on the bottom edge
propagate unidirectionally to the right. The edge channels connect the upper and
lower bands and linearly cross each other in the momentum space. Figure 14.15b
shows spatial distribution of the edge spin wave eigenstate of ω = 1.2 (γμ/a
3 ). The
symbol shapes and sizes are the spin precession trajectory and the precession radius
at each site, respectively. The azimuthal angle φ of m at t = 0 is encoded by the
colors with the color ring shown in the inset. It is evident that the spin wave is mainly
localized on the outermost sites. Here we use a zigzag edge as an example, and the
existence of the topologically protected unidirectional edge states does not depend
on edge types.
To understand the Berry phase of a spin wave, we consider the time evolution of a
state of ϕ k (t) with momentum k in an infinite sample. Without breaking the translational symmetry, k is a good quantum number that can be used to label eigenstates.
ϕ k (t) must be the Bloch state in the nth band n (k) multiplying a time dependent
phase, and satisfies following time-dependent linearized LLG equation, similar to a
Schrödinger equation,
i
μ 0 μ
γ
∂ϕ k
∂t
= H ϕ k .
(14.49)
Note that the same expression can be derived from quantum-mechanical model by
using Bogoliubov transformation, given that
μ 0 μ
γ
= s where s is the total spin of each