14 Topology in Magnetism
391
transverse electric voltage or current by a longitudinal electric field or thermal gradient in an electronic system. It is natural to ask whether there is a similar effect
for magnons. Magnons are charge neutral quasiparticles and do not have the usual
Lorentz force. However, moving magnons experience gyroscopic forces because of
nonzero Berry curvature of a magnetic system [130–134]. When the transverse components of the currents due to opposite Berry curvatures do not cancel each other,
a net transverse magnon current is generated under a longitudinal force such as a
thermal gradient in the absence of a magnetic field, leading to the anomalous magnon
Nernst effect (AMNE) [155].
14.3.2 3D Topological Spin Waves
In the previous subsection, we have shown that a honeycomb ferromagnetic layer
supports topologically protected edge states. Since pure 2D system is difficult to
realize, it is useful to see whether a stacked 3D system can preserve the gapped
bulk band and TPESWs. We discuss now a 3D ferromagnet made from a directly
AA-stacking of honeycomb layers, as shown in Fig. 14.19a.
For simplicity and without loss of generality, we consider intralayer NN distance
a and interlayer distance d = a. The generic Hamiltonian of the system is
(b)
Г
A
K
H
K
H’
A
H
H’
k x
k y
k z
Г
Z
Z
–
–
–
U
–
U
–
U
–
U
–
A
B
(a)
a 3
b 1
b 2
b 3
x
y
z
a 1
a 2
Fig. 14.19 a AA-stacked honeycomb ferromagnet. b The corresponding Brillouin Zone