394
X. S. Wang and X. R. Wang
167] in electronic systems, similar to Weyl points in photonic systems [168]. Weyl
magnons have also been identified in pyrochlore magnets [156, 169, 170]. For F = 0
and a finite D, the spin wave spectrum can be analytically solved. Two positive bands
are ω(k) = ω 0 (k) ±
β=x,y,z h
2
β (k), where ω 0 (k) = 3J + K + 2J
(1 − cos kz),
h x (k) = −J
3
i=1 cos(k · a i ), h y (k) = −J
3
i=1 sin(k · a i ), and h z (k) = 2D
3
i=1
sin(k · b i ) + δK + 2δ J (1 − cos k z ) (the frequency is in units of γ J/μ as before).
The negative bands are related to positive ones by ω(−k) = −ω(k) and they actually
represent the same modes as the positive ones.
The two bands cross each other only when h β (k) = 0. The crossing points locate at
k
±
η = (−
4π
3
√
3
, 0, ± cos
−1 f η ), where η = 1, 2 and f η =
δK
2δ J
+ 1 + (−1)
η 3
√
3D
2δ J
. The
requirement of
f η
≤ 1 yields the phase boundaries of δ J = ±3
√
3D and 4δ J +
δK = ±3
√
3D. Figure 14.21b shows the phase diagram in δK /D − δ J/D plane.
In the green region, two bands do not cross each other, and each band has a nontrivial Chern number. This is the weak topological insulator phase and topologically
protected surface modes exist on the side surfaces. The band is gapped (the dark
green) when |δ J | < 3
√
3D − J
(the dashed lines). This constraint plays a role only
when J
> 3
√
3D/2. Otherwise the whole green region is gapped. The white regions
are topologically trivial. The rest of the regions belong to three different phases. In
the pink (yellow) regions there is one pair of band crossing points at k
±
1 (k
±
2 ). In the
blue regions, there are two pairs of band crossing points at k
±
1,2 .
To see whether the band crossing points are Weyl nodes (WNs), we calculate the
Berry curvatures of the magnon bands. We fix the model parameters to D = J
=
0.2J , K = 60J , and choose (δK , δ J ) = (0.2J, 0.5J ) and (−1.8J, 0.9J ) (the black
and the white dots in Fig. 14.21b). Energy bands in the k x − k z plane for fixed k y = 0
(the blue plane in Fig. 14.19b) of the WMs are shown in Fig. 14.22a, b in which one
and two pairs of WNs appear. The red and blue dots denote chirality ±1 of the WNs.
The corresponding Berry curvatures are shown in Figs. 14.22c, d, respectively, in
which the black arrows encode the direction of Berry curvatures projected onto the
k x − k z plane and the background color represents the divergence of Berry curvature
∇ k · k with red for positive and blue for negative. Thus, the red and blue spots in
Figs. 14.22c, d are indeed monopoles of Berry curvature and correspond to the WNs
in Figs. 14.22a, b. The spectral functions on the front (100) surface along ¯
Z ¯
¯
Z of
the first (100) surface BZ (see Fig. 14.19b) are shown in Figs. 14.22e, f, respectively.
The surface states with high density (red color) on the front surface between WNs
can be clearly seen. Near the energy of WNs, these surface states form magnon arcs
(an analog to the Fermi arcs) on sample surfaces.
The Weyl magnons demonstrated above are type-I, in analogy to the type-I Weyl
electrons [171]. Two spin wave bands cross at WNs. When J
> δ J , the system
can also be type-II Weyl semimetals. Two bands of type-II semimetals intersect
with each other with a line called nodal-lines [171] so the constant-energy surface
passing through WN is a line. In an inhomogeneous electric field, the magnons can
also form Landau levels [157, 172]. Chiral anomaly for magnons results in the linear
X. S. Wang and X. R. Wang
167] in electronic systems, similar to Weyl points in photonic systems [168]. Weyl
magnons have also been identified in pyrochlore magnets [156, 169, 170]. For F = 0
and a finite D, the spin wave spectrum can be analytically solved. Two positive bands
are ω(k) = ω 0 (k) ±
β=x,y,z h
2
β (k), where ω 0 (k) = 3J + K + 2J
(1 − cos kz),
h x (k) = −J
3
i=1 cos(k · a i ), h y (k) = −J
3
i=1 sin(k · a i ), and h z (k) = 2D
3
i=1
sin(k · b i ) + δK + 2δ J (1 − cos k z ) (the frequency is in units of γ J/μ as before).
The negative bands are related to positive ones by ω(−k) = −ω(k) and they actually
represent the same modes as the positive ones.
The two bands cross each other only when h β (k) = 0. The crossing points locate at
k
±
η = (−
4π
3
√
3
, 0, ± cos
−1 f η ), where η = 1, 2 and f η =
δK
2δ J
+ 1 + (−1)
η 3
√
3D
2δ J
. The
requirement of
f η
≤ 1 yields the phase boundaries of δ J = ±3
√
3D and 4δ J +
δK = ±3
√
3D. Figure 14.21b shows the phase diagram in δK /D − δ J/D plane.
In the green region, two bands do not cross each other, and each band has a nontrivial Chern number. This is the weak topological insulator phase and topologically
protected surface modes exist on the side surfaces. The band is gapped (the dark
green) when |δ J | < 3
√
3D − J
(the dashed lines). This constraint plays a role only
when J
> 3
√
3D/2. Otherwise the whole green region is gapped. The white regions
are topologically trivial. The rest of the regions belong to three different phases. In
the pink (yellow) regions there is one pair of band crossing points at k
±
1 (k
±
2 ). In the
blue regions, there are two pairs of band crossing points at k
±
1,2 .
To see whether the band crossing points are Weyl nodes (WNs), we calculate the
Berry curvatures of the magnon bands. We fix the model parameters to D = J
=
0.2J , K = 60J , and choose (δK , δ J ) = (0.2J, 0.5J ) and (−1.8J, 0.9J ) (the black
and the white dots in Fig. 14.21b). Energy bands in the k x − k z plane for fixed k y = 0
(the blue plane in Fig. 14.19b) of the WMs are shown in Fig. 14.22a, b in which one
and two pairs of WNs appear. The red and blue dots denote chirality ±1 of the WNs.
The corresponding Berry curvatures are shown in Figs. 14.22c, d, respectively, in
which the black arrows encode the direction of Berry curvatures projected onto the
k x − k z plane and the background color represents the divergence of Berry curvature
∇ k · k with red for positive and blue for negative. Thus, the red and blue spots in
Figs. 14.22c, d are indeed monopoles of Berry curvature and correspond to the WNs
in Figs. 14.22a, b. The spectral functions on the front (100) surface along ¯
Z ¯
¯
Z of
the first (100) surface BZ (see Fig. 14.19b) are shown in Figs. 14.22e, f, respectively.
The surface states with high density (red color) on the front surface between WNs
can be clearly seen. Near the energy of WNs, these surface states form magnon arcs
(an analog to the Fermi arcs) on sample surfaces.
The Weyl magnons demonstrated above are type-I, in analogy to the type-I Weyl
electrons [171]. Two spin wave bands cross at WNs. When J
> δ J , the system
can also be type-II Weyl semimetals. Two bands of type-II semimetals intersect
with each other with a line called nodal-lines [171] so the constant-energy surface
passing through WN is a line. In an inhomogeneous electric field, the magnons can
also form Landau levels [157, 172]. Chiral anomaly for magnons results in the linear