14 Topology in Magnetism
383
on topological magnetic states in recent years [130–146]. Magnons are quanta of
spin waves, which are the propagation of spin fluctuation. Magnons are useful particles/excitations due to their low energy consumption and long coherence distance, as
well as a control knob of magnetization dynamics [31, 33, 39]. Generation, detection,
and manipulation of magnons is an emerging subfield called magnonics in condensed
matter physics [147–150], and topologically protected unidirectional surface spin
waves should be very useful in magnonics because they are highly non-susceptible
to external perturbations.
In this section, we consider a magnetic thin film whose bulk spin waves are gapped,
but its edge spin waves are chiral and gapless. This is a magnonic counterparty of
quantum Hall systems or topological insulators [126, 127]. The film consists of a
set of perpendicularly magnetized ferromagnetic spins on a honeycomb lattice with
a nearest-neighbour (NN) pseudodipolar interaction and/or next-nearest-neighbour
(NNN) Dzyaloshinskii-Moriya interaction (DMI) as well as the usual NN Heisenberg
ferromagnetic interaction. Although the magnons are bosons, the bulk spin wave
bands carry nontrivial Chern numbers so that one knows for sure the existence of
topologically-protected unidirectional (chiral) gapless spin waves at edges inside the
bulk gap according to the bulk-boundary correspondence [151, 152]. A staggered
sublattice anisotropy can change the system from a topologically nontrivial phase
to a trivial phse, and vice versa. It is also possible to tune the topological phase by
changing DMI. In 14.3.1, we show how to calculate the Chern number of the bosonic
spin wave bands, nontrivial spin wave bands of a finite sample and topologically
protected edge spin waves [153–155]. In 14.3.2, we go a step further to consider
3D magnets that are either magnonic weak topological insulators or magnonic Weyl
semimetals [156, 157].
14.3.1 Topologically Protected Edge Spin Waves
Various magnetic systems has been proposed for realizing nontrivial topology of spin
wave bands, such as kagome ferromagnet with NN DMI [132, 135, 137], dipolar
magnonic crystals [144, 145], honeycomb ferromaget with NNN DMI [142, 143],
and so on. We call them topological magnonic systems. Here, we provide the basic
ingredients and a theoretical framework for topological magnonic materials.
As a representative, we consider ferromagnetic spins on a honeycomb lattice as
shown in Fig. 14.14a. a 1(2,3) are three neighbouring site-vectors of length a. The
blue and red arrows represent the magnetic moments μ i = μm i of magnitude μ and
direction m i at site i of sublattices A and B, respectively. The system is described
by a Hamiltonian,
383
on topological magnetic states in recent years [130–146]. Magnons are quanta of
spin waves, which are the propagation of spin fluctuation. Magnons are useful particles/excitations due to their low energy consumption and long coherence distance, as
well as a control knob of magnetization dynamics [31, 33, 39]. Generation, detection,
and manipulation of magnons is an emerging subfield called magnonics in condensed
matter physics [147–150], and topologically protected unidirectional surface spin
waves should be very useful in magnonics because they are highly non-susceptible
to external perturbations.
In this section, we consider a magnetic thin film whose bulk spin waves are gapped,
but its edge spin waves are chiral and gapless. This is a magnonic counterparty of
quantum Hall systems or topological insulators [126, 127]. The film consists of a
set of perpendicularly magnetized ferromagnetic spins on a honeycomb lattice with
a nearest-neighbour (NN) pseudodipolar interaction and/or next-nearest-neighbour
(NNN) Dzyaloshinskii-Moriya interaction (DMI) as well as the usual NN Heisenberg
ferromagnetic interaction. Although the magnons are bosons, the bulk spin wave
bands carry nontrivial Chern numbers so that one knows for sure the existence of
topologically-protected unidirectional (chiral) gapless spin waves at edges inside the
bulk gap according to the bulk-boundary correspondence [151, 152]. A staggered
sublattice anisotropy can change the system from a topologically nontrivial phase
to a trivial phse, and vice versa. It is also possible to tune the topological phase by
changing DMI. In 14.3.1, we show how to calculate the Chern number of the bosonic
spin wave bands, nontrivial spin wave bands of a finite sample and topologically
protected edge spin waves [153–155]. In 14.3.2, we go a step further to consider
3D magnets that are either magnonic weak topological insulators or magnonic Weyl
semimetals [156, 157].
14.3.1 Topologically Protected Edge Spin Waves
Various magnetic systems has been proposed for realizing nontrivial topology of spin
wave bands, such as kagome ferromagnet with NN DMI [132, 135, 137], dipolar
magnonic crystals [144, 145], honeycomb ferromaget with NNN DMI [142, 143],
and so on. We call them topological magnonic systems. Here, we provide the basic
ingredients and a theoretical framework for topological magnonic materials.
As a representative, we consider ferromagnetic spins on a honeycomb lattice as
shown in Fig. 14.14a. a 1(2,3) are three neighbouring site-vectors of length a. The
blue and red arrows represent the magnetic moments μ i = μm i of magnitude μ and
direction m i at site i of sublattices A and B, respectively. The system is described
by a Hamiltonian,