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Z. Li et al.
photonic [30–35], acoustic [36–40], and electric-circuit [41–45] systems. Different
from first-order topological insulators (FOTIs), a HOTI has (n − 2) or even (n − 3)dimensional (second-order or third-order) topological boundary states, which goes
beyond the conventional bulk-boundary correspondence and is characterized by several new topological invariants, such as the nested Wilson loop [46], Green’s function
zeros [47], the quantized bulk polarization (Wannier center) [25, 36, 37], and the
Z N Berry phase (quantized to 2π/N ) [48–52]. The HOTIs thus have broadened our
understanding on topological insulating phases of matter.
There are two important excitations in magnetic systems. One is the spin wave (or
magnon) [53, 54], i.e., the collective motion of magnetic moments. The topological
phase of magnons in magnetic materials is of great current interest in magnetism
because of its fundamental significance as well as in spintronics because of its practical utility for robust information processing [20, 22, 55–60]. The other one is the
collective oscillation of magnetic solitons (such as the magnetic vortex [61, 62],
bubble [63–65], skyrmion [66, 67], and domain wall [68–70]), which are long-term
topic in condensed matter physics for their interesting dynamics and promising application. These magnetic solitons generally have the characteristics of small size, easy
manipulation and high stability. The spintronic devices based on magnetic solitons
thus have advantages over other electronic devices. For example, the racetrack memory made of magnetic domain walls can greatly improve the data storage density and
the reading speed [71, 72]; the critical current density required for encoding information can be significantly reduced by using the skyrmion as the carrier of information
[73, 74]; the magnetic oscillators based on vortices or skyrmions are very robust and
flexible [75–77]. On the other hand, it has been shown that the collective gyration
motion of magnetic solitons can exhibits the behavior of waves [78–82]. Furthermore,
the first-order topological chiral edge states based on two-dimensional honeycomb
lattices of magnetic solitons (vortices and bubbles) have been predicted by Kim and
Tserkovnyak [83]. In a word, the metamaterials based on topological spin texture are
attracting more and more attention for both fundamental interest and the potential
applications in spintronics and quantum computing [84–89].
In this chapter, we report the realization of TIs in magnetic soliton lattices. The
exposition is organized as follows: Sect. 15.2 introduces the topological structures,
properties, and applications of magnetic solitons; the first-order topological edge
states of skyrmion lattice are discussed in Sects. 15.3, 15.4 and 15.5 focus on the
second-order TIs in the breathing kagome and honeycomb lattices of magnetic vortices, respectively. We summarise the results in Sect. 15.6.
15.2 Topological Structures, Properties, and Applications
of Magnetic Solitons
Topology is a study of geometry or space which can keep some properties invariant
under a continuous transformation. Topological magnetic soliton is an application
of topology in condensed matter physics. More precisely, magnetic solitons are the
spin textures characterized by a topological charge [83],
Z. Li et al.
photonic [30–35], acoustic [36–40], and electric-circuit [41–45] systems. Different
from first-order topological insulators (FOTIs), a HOTI has (n − 2) or even (n − 3)dimensional (second-order or third-order) topological boundary states, which goes
beyond the conventional bulk-boundary correspondence and is characterized by several new topological invariants, such as the nested Wilson loop [46], Green’s function
zeros [47], the quantized bulk polarization (Wannier center) [25, 36, 37], and the
Z N Berry phase (quantized to 2π/N ) [48–52]. The HOTIs thus have broadened our
understanding on topological insulating phases of matter.
There are two important excitations in magnetic systems. One is the spin wave (or
magnon) [53, 54], i.e., the collective motion of magnetic moments. The topological
phase of magnons in magnetic materials is of great current interest in magnetism
because of its fundamental significance as well as in spintronics because of its practical utility for robust information processing [20, 22, 55–60]. The other one is the
collective oscillation of magnetic solitons (such as the magnetic vortex [61, 62],
bubble [63–65], skyrmion [66, 67], and domain wall [68–70]), which are long-term
topic in condensed matter physics for their interesting dynamics and promising application. These magnetic solitons generally have the characteristics of small size, easy
manipulation and high stability. The spintronic devices based on magnetic solitons
thus have advantages over other electronic devices. For example, the racetrack memory made of magnetic domain walls can greatly improve the data storage density and
the reading speed [71, 72]; the critical current density required for encoding information can be significantly reduced by using the skyrmion as the carrier of information
[73, 74]; the magnetic oscillators based on vortices or skyrmions are very robust and
flexible [75–77]. On the other hand, it has been shown that the collective gyration
motion of magnetic solitons can exhibits the behavior of waves [78–82]. Furthermore,
the first-order topological chiral edge states based on two-dimensional honeycomb
lattices of magnetic solitons (vortices and bubbles) have been predicted by Kim and
Tserkovnyak [83]. In a word, the metamaterials based on topological spin texture are
attracting more and more attention for both fundamental interest and the potential
applications in spintronics and quantum computing [84–89].
In this chapter, we report the realization of TIs in magnetic soliton lattices. The
exposition is organized as follows: Sect. 15.2 introduces the topological structures,
properties, and applications of magnetic solitons; the first-order topological edge
states of skyrmion lattice are discussed in Sects. 15.3, 15.4 and 15.5 focus on the
second-order TIs in the breathing kagome and honeycomb lattices of magnetic vortices, respectively. We summarise the results in Sect. 15.6.
15.2 Topological Structures, Properties, and Applications
of Magnetic Solitons
Topology is a study of geometry or space which can keep some properties invariant
under a continuous transformation. Topological magnetic soliton is an application
of topology in condensed matter physics. More precisely, magnetic solitons are the
spin textures characterized by a topological charge [83],