Topological insulators and semimetals in classical magnetic systems

Pursuing topological phases in natural and artificial materials is one of the central topics in modern physical science and engineering. In classical magnetic systems, spin waves (or magnons) and magnetic solitons (such as domain wall, vortex, skyrmion, etc) represent two important excitations. Recently, the topological insulator and semimetal states in magnonand soliton-based crystals (or metamaterials) have attracted growing attention owing to their interesting dynamics and promising applications for designing robust spintronic devices. Here, we give an overview of current progress of topological phases in structured classical magnetism. We first provide a brief introduction to spin wave, and discuss its topological properties including magnon Hall effects, topological magnon insulators, and Dirac (Weyl) magnon semimetals. Appealing proposal of topological magnonic devices is also highlighted. We then review the collective-coordinate approach for describing the dynamics of magnetic soliton lattice. Pedagogical topological models such as the Su-Schrieffer-Heeger model and the Haldane model and their manifestation in magnetic soliton crystals are elaborated. Then we focus on the topological properties of magnetic solitons, by theoretically analyzing the first-order topological insulating phases in low dimensional systems and higher-order topological states in breathing crystals. Finally, we discuss the experimental realization and detection of the edge states in both the magnonic and solitonic crystals. We remark the challenges and future prospects before concluding this article.


Introduction
Since the discovery of the quantum Hall effect [1][2][3][4] in two-dimensional electron gas system, the topological phases of matter began to attract people's attention for their exotic physical properties. The most peculiar character of topological phase, or more precisely, the topological insulators (TIs), is that they can support chiral edge/surface states which are absent in conventional insulators. The topological edge/surface states are the modes that are confined at the boundary/surface of the system and generally have a certain chirality (clockwise or counterclockwise). These properties are topologically protected and enable them being immune from moderate disorder and/or defects, which has defined a resistance that depends only on fundamental physical constants due to the robust in-gap edge states, making possible an accurate and standardized definition of the ohm. Topological insulating phases were originally observed in electronic system [5][6][7][8][9], while the concept of TIs has been extended to a broad fields of photonics [10][11][12][13][14][15], acoustics [16][17][18][19][20][21][22], mechanics [22][23][24][25][26][27][28], electric circuits [29][30][31][32][33][34][35][36], and very recently in spintronics [37][38][39][40][41][42][43]. In the past years, the research about TIs in natural and artificial materials has become one of the most active areas in physical science and engineering because of the fundamental interest and the promising application in topological devices [5,6,13,16,17,23,[44][45][46]. The first-order TI with in-gap modes localized at the corners for = 1, along the boundaries for = 2, and on surfaces for = 3. (b) The second-order TI with in-gap modes confined at the four corners for = 2 (corner state), and along the hinges of the system for = 3 (hinge state). (c) The third-order TI with in-gap modes at corners for = 3 (corner state).
Topological semimetals (TSs) [108] are another exotic phase of matter with unusual gapless band structure. Typical TSs include Weyl semimetals [109][110][111] and Dirac semimetals [112][113][114][115]. Weyl semimetals are characterized by the twofold degenerate points (called Weyl points or nodes) resulting from the linear crossings of two bands. The Weyl points (or nodes) are described by the momentum-space monopoles of Berry curvature and they must come in pairs with opposite chirality due to the no-go theorem. The band inversion happens between two paired Weyl nodes, leading to the generation of topologically protected Fermi-arc-like surface states. It is noted that Weyl semimetal states emerge only when at least one of the symmetries (time-reversal symmetry and inversion symmetry) is broken. In contrast, the Dirac semimetals are characterized by Dirac points with fourfold degenerate band touchings. The Dirac semimetals respect both the time-reversal and inversion symmetries, while their stability requires additional crystalline symmetries, for example, the rotation symmetry [114,115]. Interestingly, it is found very recently that the Weyl semimetals can support higher-order topological edge states (hinge states) [116,117], which is referred to as higher-order Weyl semimetals. At present, the topological semimetals have been studied extensively in various systems because of their exotic properties and potential applications [118][119][120][121][122][123][124].
Another important excitation in magnetic system is the magnetic soliton. Magnetic solitons [198] are shapepreserving and self-localized structures, with typical examples including magnetic vortex [199,200], bubble [201][202][203], skyrmion [204][205][206], and domain wall [207][208][209]. These magnetic solitons have the characteristics of small size, easy manipulation, and high stability, and they are long-term topics in condensed matter physics for their interesting dynamics and promising applications [210][211][212][213][214][215][216][217]. Similar to other (quasi-)particles, the collective dynamics of magnetic solitons exhibits the behavior of waves [218][219][220][221][222][223][224][225][226][227][228][229][230][231][232]. By mapping the massless Thiele's equation into the Haldane model [233], Kim and Tserkovnyak [234] predicted that the two-dimensional honeycomb lattice of magnetic vortices (or bubbles) can support chiral edge states, which has been confirmed by full micromagnetic simulations [235]. Li et al. [236] and Go et al. [237] studied the Su-Schrieffer-Heeger (SSH) [238] states in one-dimensional magnetic soliton lattice. Li et al. predicted theoretically the second-order topological phases (corner states) in two-dimensional breathing kagome [96], honeycomb [97], and square [98] lattice of magnetic vortices and showed that the emerging corner states are very robust against disorder and defects because of the generalized chiral symmetry. The collective motion of magnetic solitons in two dimensions is described by the generalized Thiele's equation, which results in a wavelike equation in the artificial crystal, and this equation differs from the wave equations of its electronic, photonic, and acoustic counterparts in the following respects: (i) The nonvanishing topological charge induces a gyration term that is analogous to an effective magnetic field acting on a quasiparticle, thus breaking time-reversal symmetry. (ii) The inertial effect is taken into account by a mass term. A third-order non-Newtonian gyration term is included to capture the high-frequency behavior of the magneticsolitons and to allow one to determine the interaction parameters with high accuracy. (iii) The soliton-soliton coupling is strongly anisotropic. (iv) The conventional chiral symmetry in bipartite lattices is replaced by a more general chiral symmetry.
In this review, we give a detailed introduction to topological insulators and semimetals in magnonic and solitonic systems. The exposition is organized as follows: Section 2.1 describes the spin wave and magnon Hall effect; The summary of the studies about the topological magnon insulatrs is given in Section 2.2; The topological magnon semimetals (including Dirac and Weyl magnons) are discussed in Section 2.3; In Section 2.4, the higher-order topological magnons are introduced; The concepts of topological magnonic devices are discussed in Section 2.5; The topological structures and properties for different magnetic solitons are presented in Section 3.1; Section 3.2 gives a brief review about the collective dynamics of magnetic solitons; Two pedagogical topological models (the SSH and Haldane models) are introduced in Section 3.3; Sections 3.4 and 3.5 focus on the topological phases of magnetic soliton crystals. In Section 4, we present future prospects and challenges about the topology in magnetism.

Topological magnons
In magnets, due to the short-range exchange and long-range dipolar interactions, the local oscillation of magnetic moments spreads all over the magnet in the form of SWs as shown in Figs. 2(a) and 2(b). Because of their different wavelengths, SWs can be divided into three types: (i) The exchange SW, with very short wavelength, where the shortrange exchange interaction dominates. (ii) The dipolar (or magnetostatic) SW, with very long wavelength, where the long-range dipolar interaction dominates. (iii) The dipolar-exchange SW, where neither exchange nor dipolar interactions can be ignored. Figure 2(c) plots the dispersion relation of SW in magnetic ellipsoids, with representing the angle between the magnetic moment and the wave vector. Due to the limitation of experimental technology, most of the researches are about magnetostatic SW. The magnetostatic SW can be further divided into the following categories: (i) The forward volume magnetostatic spin wave (FVMSW), for equilibrium magnetization perpendicular to the film surface; For in-plane magnetized films, the SW have two types with (ii) the backward volume magnetostatic spin wave (BVMSW) and (iii) magnetostatic surface spin wave (MSSW), which allows SW propagation parallel or perpendicular to the magnetic moment, respectively. The dispersion relation for different magnetostatic SWs are shown in Fig. 2(d).
The dispersion relation of SW can be significantly modified by various approaches. One of the most effective methods is by using magnonic crystals (MCs) [239,240]. MCs are the artificial magnetic material structures with periodic variation of magnetic or geometric parameters. Similar to the photonic or phononic crystals, the band structure of SW propagating in such structures consists of a series of allowed and forbidden frequency bands [241][242][243]. Kim et al. [242] studied the one-dimensional MCs with a single nanostrip with periodic width variation, as shown in Fig. 3(a). They found that there is no band gap existing when the SW propagates into the nanostrip with uniform width. However, when the width varies periodically, a series of band gaps emerge; see Figs. 3(b) and 3(c). Furthermore, the number, position and width of the band gaps can be tuned by modifying the device geometry. Similarly, two-dimensional MCs with periodic magnetic parameters variation can also support allowed and forbidden frequency bands [244], as shown in Figs. 3(d) and 3(e). Interestingly, Ma et al. [245] reported a new type MC which consists of skyrmions. They identified band daps when the SW propagates into such skyrmion lattice, as shown in Fig. 3(f). In addition, the other type of magnetic soliton (such as domain wall) MCs also have been proposed [246,247]. Besides MCs, the spin polarized current can be used to modify the band structure of SW, too. Seo et al. [248] reported that the dispersion relation of SW have a positive or negative shift depending on the direction of current; see Fig. 4(a). Further, Zhou et al. [249] found that when suitable spin-polarized electrical currents are applied, the ferromagnetic system can support left-handed polarized SWs. Moreover, they confirmed that the right-handed and left-handed polarized SWs can coexist when the current density is larger than a critical value, with the dispersion relation under the different current density being plotted in Fig. 4(b). What's more, Moon et al. [250] showed that the interfacial DM interaction can tune the SW dispersion relation and leads to a nonreciprocal SW propagation, as showns in Figs. 4(c) and 4(d). Comprehensive summary on magonics can be found in early review articles [132,133,140,141]. We point out that, in fermionic system (such as electron), it is easy to identify the topological phase by linear transport measurements [5,6]. While magnetic systems are bosonic, they have a simple condensate or vacuum ground state when they are in topological phases [251]. As a result, it is difficult to characterize their topological nature. However, if magnetic systems are in the excited state, the situation becomes different, because magnons can carry signatures of the topological band structure. For example, by measuring the thermal Hall conductivity, one can judge if the magnetic system is in the topological insulating phase. In what follows, we will introduce the topological properties of magnons.

Magnon Hall effect
The study of the topological properties of SWs begins with the observation of MHE. It is generally known that the Hall effect occurs when the Lorentz force acts on a charge current in the presence of a perpendicular magnetic field [252]. However, magnons are neutral quasi-particles, the realization of MHE does not resort to the Lorentz force, which is similar to the anomalous Hall effect in the metallic ferromagnets [253].
In 2010, Katsura et al. [254] predicted theoretically that the intrinsic thermal Hall effect of magnons can be realized in magnets with a particular lattice structure such as kagome, which offers a promising proposal to detect the MHE via a thermal transport measurement. Later, in the same year, the MHE was observed experimentally by Onose et al. [142] in the insulating ferromagnet Lu 2 V 2 O 7 of pyrochlore lattice structures. The pyrochlore structure can be viewed as the stacking of alternating kagome and triangular lattices, as shown in Fig. 5(a). When a temperature gradient is applied longitudinally, a transverse heat current was observed. Figure 5(c) presents the measurements of transverse thermal Hall conductivity. From a broad point of view, electrons, phonons and magnons can all generate heat current. However, on the one hand, from Fig. 5(c), one can see that the thermal Hall conductivity steeply increases and saturates in the low-magnetic field region, which can not be explained by either normal Hall effect (the conductivity is proportional to the magnetic field strength) or the anomalous Hall effect due to the spontaneous magnetization. On the other hand, the emergence of the decrease of the thermal Hall conductivity in the high-field region cannot be explained in terms of the phonon mechanism [255][256][257]. Therefore, it is concluded that the transverse heat current can only be explained by the MHE. Figure 5(b) plots the schematic diagram of MHE. In the model, the MHE comes from the nonzero DM interaction (induced by the spin-orbit coupling) that breaks the inversion symmetry.
After the discovery of MHE, people try to understand the origin of the transverse thermal magnon current. Matsumoto et al. [42,143] demonstrated that a magnon wave packet [see Figs. 6(a) and 6(b)] subjected to a temperature gradient acquires an anomalous velocity perpendicular to the gradient, which is associated to the magnon edge currents. The relation between transverse thermall Hall conductivity and Berry curvature n ( ) = −i⟨ ⟩ is as follows: From Eq. (1), one can clearly see that comes from the Berry curvature in momentum space. When the energy bands are close to each other (near the band crossing), the value of reaches the maximum. The MHE can be understood as follows: when the system is in equilibrium [see Fig. 6(c)], the edge magnon currents exist due to the confining potential, and they circulate along the boundary. The amount of currents are equal at two edges, leading to a vanishing thermal current through the magnet. However, when the temperature gradient is applied [see Fig. 6(d)], the magnons will flow from the high temperature region to the low temperature region, which breaks the balance of the heat current from the two opposite edges, leading to a finite thermal Hall current. Furthermore, Zhang et al. [37] showed that these edge magnon currents are actually SW chiral edge states resulting from the nontrivial topology of magnon bands. It is also demonstrated that the one-way chiral edge transport is topologically immune from defects and disorders. In a word, the robust MHE originates from the nontrivial band structure of magnons.
Since the discovery of MHE in pyrochlore ferromagnetic insulator [142], the same effect has also been observed in other magnetic materials. Hirschberger et al. [146] report the observation of a large thermal Hall conductivity in the kagome magnet Cu(1,3-bdc), with the main results shown in Figs. 7(a) and 7(b). Surprisingly, the observed undergoes a remarkable sign reversal by changing the temperature or magnetic field, which is explained by the sign change of the Chern flux between magnon bands. Besides, Hirschberger et al. [148] also report the MHE in a frustrated pryocholore quantum magnet Tb 2 Ti 2 O 7 . The corresponding measurements of are plotted in Figs. 7(c) and 7(d). One can see that from 140 to 50 K, ∕ is −linear. Below 45 K, it develops a pronounced curvature at large , reaching its largest value near 12 K, which is the obvious signal for the thermal magnon current generating the transversal Hall conductivity. Furthermore, Tanabe et al. [145] observe the magnon Hall-like effect for sample-edge scattering in unsaturated YIG. Figures 7(e)-7(g) show the measurements of the temperature distribution with coplanar  However, when the CPW is placed under the center of the YIG, no thermal gradient is observed, as shown in Fig. 7(g). These results strongly indicate that the observed thermal gradient in Figs. 7(e) and 7(f) are attributed to the magnons at sample edges.
The MHE discussed above is based on collinear ferromagnet. While Hoogdalem et al. [175] demonstrated theoretically that noncollinear magnetic texture (skyrmion for instance) can generate a fictitious magnetic field, which can also lead to magnon thermal Hall effect. This Hall effect is solely due to the nonzero topological charge of magnetic texture, which is therefore called topological magnon Hall effect (TMHE). Subsequently, Mochizuki et al. [176] indirectly confirmed experimentally the existence of TMHE. By using Lorentz transmission electron microscope (TEM), they observed micrometre-sized crystals of skyrmions in thin films of Cu 2 OSeO 3 and MnSi exhibiting a unidirectional rotation motion, as shown in Fig. 8(a). This rotational motion can be explained below: At first, the thermal gradient was generated under the electron-beam irradiation in the Lorentz TEM experiment, then the magnon current induced by the temperature gradient is deflected by the emergent magnetic field of skyrmion lattice (TMHE), which in turn gives rise to the rotation of skyrmions through the spin-transfer torque (STT). In addition, if the sign of the temperature gradient is reversed, the direction of skyrmion rotation will reverse, too, see Fig. 8(c). The skyrmion-induced versions of the MHE are also studied by means of atomistic spin dynamics [178], in which, based on spin spiral and skyrmion lattice system, the authors predict a magnon Hall angle as large as 60%.
The discovery of MHE opens the door to the study of the topological properties of magnons. The topological magnons have great potential application prospect for designing robust and flexible spintronic devices. Over the past decade, a lot of literatures have been devoted to the topological properties of magnons, including the topological magnon insulators and semimetals, which will be reviewed in next sections.

Topological magnon insulators
Generally speaking, there are two typical systems that can support topological magnons, one is the collinear ferromagnet, while the other is noncollinear magnetic texture. Below, we will give a brief introduction about the topological magnons in these systems by several specific examples.

Collinear ferromagnet
and the topological invariant Chern number is given by Chern number is a physical quantity of particular importance for the topologically nontrivial edge modes by determining both their propagation direction and their number. There is a "bulk-boundary correspondence": the bulk property (Chern number) dictates surface/edge properties (edge magnons). The sum of Chern numbers up to the th band = ∑ ≤ is the "winding number" of the edge states in band gap . | | is the number of topologically nontrivial edge states in the th band gap and sgn( ) determines their propagation direction.
The ferromagnetic kagome lattice allows four topologically different phases by tuning parameters ∕ and ∕ [149]. and represent the Heisenberg exchange constant between nearest and next-nearest sites, respectively, while the DM parameter ( ) accounts only for the nearest-neighbor interaction. Figure 9(a) shows the semi-infinite kagome lattice, and calculated topological phase diagram is presented in Fig. 9(b), where the sign of the transverse thermal conductivity of the MHE is also indicated.   9(e), and 9(f)], similar analysis can be done as well. It is worth mentioning that the phases (-1,2,1) and (-3,4,-1) can support edge modes for both propagation directions, which leads to the change of sign in when the temperature varies. At low temperatures, edge states in the first band gap are more occupied than edge states in the second band gap. Thus the heat transport is dominated by the former edge modes. However, with the increasing of temperature, the edge states in the second band gap become increasingly populated. When the temperature is high enough, the heat current is mainly mediated by these magnons, therefore, the sign of reverses. As mentioned above, the nontrivial topology of magnon in the kagome lattice is brought about by the strong spin-orbit coupling which manifests the DM interaction. Meanwhile, Shindou et al. [170] demonstrated theoretically that the magnetic dipolar interaction can also endow spin wave volume modes with nonzeros Chern number, and the propagation direction of edge states is tunable by external magnetic fields. Figure 11(a) plots the model of periodic array of ferromagnetic islands decorated square-lattice. The energy of the system only includes the magnetostatic energy and the Zeeman energy. By using simple tight-binding descriptions, one can obtain the magnon Hamiltonian and the band structures under different field strengths, with the results shown in Fig. 10. Here the direction of the To confirm the existence of the proposed chiral spin wave edge mode, the authors performed micromagnetic simulation for the square-lattice model, as shown in Fig. 11(b). The spins are coupled via magnetic dipole-dipole interaction and no short-range exchange interaction is considered. The magnetization becomes fully polarized along direction under = 1.02 . The frequency power spectra for pulse at center and at edge are shown in Fig. 11(c). One can clearly identify the edge state and bulk state. Spatial distributon of spin-wave excitation for different modes are presented in Figs  mode is unidirectional, which can be clarified by the dispersion relation. Figure 11(h) plots the dispersion relation when the pulse field is at the center, and no edge mode is observed. However, when the pulse field locates at the edge, by taking the Fourier transformation only over the upper (or lower) side of the sample, as shown in Fig. 11

Noncollinear magnetic texture
As introduced in Section 2.1, the magnetic texture can induce the topological magnon Hall effect, which indicates that there are topologically protected magnon edge states in these systems. In 2016, Roldán-Molina et al. [177] reported the topological SWs in the atomic-scale magnetic skyrmion crystal, with the schematic diagram shown in Fig. 12(b). The Hamiltonian of the system contains a uniaxial anisotropy term, a nearest-neighbor ferromagnetic exchange coupling, the DM interaction, and the Zeeman energy. By solving the eigen equations numerically, one can obtain the SW band structures for a one-dimensional skyrmion crystal strip, as shown in Fig. 12(a). It can be clearly seen that there are several bands allowing spin wave edge states. Figure 12(c) plots the magnon occupation for the edge modes as marked in Fig. 12(a), from which the localization properties can be clearly identified. Similar results are obtained by Díaz et al. [180]. Figure 12(d) plots the magnon band structure for one-dimensional skyrmion crystal strip, where the red and blue lines represent the bands for spin wave edge states. The magnetic unit cell of the ferromagnetic skyrmion crystal and probability density of magnonic edge states are shown in Figs. 12(e) and 12(f), respectively, from which the topological magnon edge states can be observed.
Furthermore, Díaz et al. [259] show that the topological magnon also exists in antiferromagnetic skyrmion crystals. Figure 13(a) plots the bulk band structure of antiferromagnetic skyrmion crystals along the high symmetry points of the Brillouin zone (BZ). The bulk magnon gap can be clearly identified, as marked by green rectangle. If one considers a strip of infinite length along the axis with edges located at the top and bottom of the lattice, the bands for spin wave edge states will emerge, as shown in Fig. 13(b). Magnonic edge states are plotted in Fig. 13(c).
For magnetic system, the simplest two-band model that exhibits Dirac points is the Heisenberg ferromagnet or antiferromagnet on the honeycomb lattice [181]. The Hamiltonian can be expressed as: where the summation runs over nearest neighbors, ( ) and ( ) are the spins for two different sublattices, and is the exchange constant. Assuming uniform ferromagnetic interaction, i.e., = > 0, by applying the Holstein-Primakoff transformation [267], the effective quadratic magnon model can be written as In reciprocal space, letting = ∑ ⋅ ∕ √ and = ∑ ⋅ ∕ √ , one can obtain where the structure factor ( ) = − √ ∑ exp( ⋅ ⃗ ) ( = 1, 2, 3) is given in terms of the nearest-neighbor vector ⃗ [see Fig. 14(a)], and is wave vector. The eigenenergies can be derived where Ω 2 ( ) = Δ 2 + 4| ( )| 2 with Δ = − = −3 ( − ).  For Dirac materials, if the inversion symmetry is broken, a gap will open at the Dirac points, leading to a TI. Topological magnon insulator can be achieved by using similar method. For the honeycomb ferromagnets, if the Hamiltonian only contains nearest neighbors (NN) exchange interaction, the magnon band structure is gapless, even if a next-nearest neighbour (NNN) interaction is considered, which only shifts the positions of the Dirac points. Owerre [161] demonstrated theoretically that, if a next-nearest neighbour DM interaction is introduced, the time reversal symmetry of the system is broken, and a gap opens at the Dirac points. The band structure for semi-infinite system is shown in Fig. 15(a), from which one can clearly see the edge spin wave dispersion. Figure 15(b) shows the illustration of magnon edge states. From the materials point of view, chromium trihalides CrX 3 (X=F, Cl, Br and I) is a practical example of ferromagnets consisting of van der Waals-bonded stacks of honeycomb layers, which display two spin wave modes with energy dispersion similar to that for the electrons in graphene. Pershoguba et al. [183] studied theoretically the Dirac magnons in CrX 3 (X = F, Cl, Br and I) and discussed the stability of Dirac cones affected by particle statistics and interactions. They showed that honeycomb ferromagnets can display dispersive surface and edge states. Subsequently, the gap at the Dirac points in CrI 3 was observed experimentally by Chen et al. [158]. Figure 16(g) plots the crystal and magnetic structures of CrI 3 . By using inelastic neutron scattering, one can obtain the magnon dispersion relation. The results reveal a large gap at the Dirac points. The acoustic and optical spin wave bands are separated from each other by approximately 4 meV, which most likely arises from the next nearest-neighbor DM interaction that breaks the inversion symmetry of the lattice. These band gaps may lead to a nontrivial topological magnon insulator with magnon edge states. The observation of a large spin-wave gap indicates that the spin-orbit coupling plays an important role in the physics of topological spin excitations in honeycomb ferromagnet CrI 3 .
Moreover, in CoTiO 3 with ilmenite structure, by using inelastic neutron scattering experiment, Yuan et al. [184] observed Dirac magnons in this 3D quantum XY magnet. In addition, an obvious gap of order about 1 meV in the magnon dispersion is also identified. Such a gap arises from the bond-anisotropic exchange coupling, due to quantum order by disorder, which pins the order parameter to the crystal exes. The magnon spectra calculated theoretically shows that edge states connecting the bulk Dirac points can appear with zigzag edge, while vanish for armchair edge.
In addition to DM interaction which can open a gap at Dirac points, Wang et al. [173] demonstrated theoretically that the pseudodipolar exchange interaction which arises from the superexchange and atomistic spin-orbit interaction [268,269] can also open the gap at Dirac points and induce nontrivial topological magnon states. The 2D ferromagnetic spins on a honeycomb lattice [see Fig. 17(a)] is described by a classical Hamiltonian, where ⟨ , ⟩ denotes the NN sites. The first term is NN exchange interaction with exchange constant . The second term is NN dipole-dipole-like pseudodipolar exchange interaction which arises from the superexchange and atomistic spin-orbit interaction [268,269]:  ( , , ) = ( ⋅ )( ⋅ ), with the unit vector connecting sites and , and being the interaction strength. The third term is the anisotropy energy with easy axis along direction, anisotropy constant = and for sublattices A and B. The last term is the Zeeman energy from a magnetic field along direction. By neglecting damping, the LLG equation for spin becomes where = = . By solving Eq. (8), one obtains the spin-wave spectrum for an infinite system, the results are shown in Fig. 17 (b). The band gap at and ′ points is Δ = Ω − √ Ω 2 − 9 2 ∕4. When = 0, the band gap closes and Dirac cones emerge. Here, the pseudodipolar NN exchange interaction is the critical factor for band-gap opening. For a long strip with zigzag edges along direction [ Fig. 17(a)], the density plot of the spectral function on the top edge is shown in Fig. 17(c). The negative slope of the dispersion curve indicates that the propagation direction of spin-wave edge state is counterclockwise, i.e., to the left. Similarly, the states on the bottom edge propagate unidirectionally to the right. Figure 17(d) shows spatial distribution of the edge spin-wave eigenstate.

Weyl magnons
In topological magnonics, another important class of topologically nontrivial system is magnonic Weyl semimetal [185][186][187][188][189][190]. Similar to the electronic Weyl semimetals [109,270], the magnon bands in a magnonic Weyl semimetal are nontrivially crossing in pairs at special points (called Weyl nodes) in momentum space. The Weyl nodes are monopoles of Berry curvature and are characterized by the integer topological charge or chirality. Based on no-go theorem, the net topological charges in the entire Brillouin zone must be zero, the Weyl nodes thus must appear in pairs with opposite topological charges of ±1 [271,272]. The magnons around Weyl nodes can be described by effective Weyl Hamiltonians and they are thus called Weyl magnons. In Weyl semimetals, the topologically protected chiral surface states between each pair of Weyl nodes exist on the system surfaces [109,110,120]. The equal energy contour of these surface states form arcs, with the arc number equaling the number of paired Weyl nodes.
Recently, Mook et al. [185] and Su et al. [186] show that the pyrochlore ferromagnets [see Fig. 5(a)] with DM interaction are intrinsic magnonic Weyl semimetals. The effective spin Hamiltonian of the system include NN exchange interaction, NN DM interaction, and Zeeman interaction. By using the Holstein-Primakoff transformation and the Bloch theorem, one can obtain the band structures of the pyrochlore ferromagnet. Figure 18(a) shows the first bulk BZ and the first (001) surface BZ of the pyrochlore lattice. The red and blue dots schematically represent the pair of Weyl nodes with opposite topological charges. Similar to the electronic Weyl semimetal, the important hallmark of magnonic Weyl semimetal is the magnon arcs on system surfaces. The density plot of magnon spectral function on the top surface along high symmetry path Γ Γ is shown in Fig. 18(b), where one Weyl node can be identified. One can clearly see the topologically protected surface states which marked by red color with high density on the top surface. While if we consider path Γ , two Weyl nodes will appear and the pair of Weyl nodes are connected by surface states, as shown in Fig. 18(c). For fixed energies of , , and around the Weyl nodes [see Fig. 18(b)], the corresponding density plot of magnon spectral function on the top surface in the first BZ are shown in Figs. 18(d)-18(f), respectively. The magnon arcs due to topologically protected surface states are clearly displayed on the top surface. Besides, the authors [186] showed that magnonic chiral anomaly can be realized by applying inhomogeneous electric and magnetic fields that are perpendicular to each other. The electric field is used to generate magnonic Landau level according to the Aharonov-Casher effect [273], while the magnetic field is used to drive magnon flow, as shown in Fig. 18(g). The field drives magnons to move from one Weyl node to the other through the zeroth magnonic Landau level and results in the imbalance of chirality which is the signature of magnonic chiral anomaly.
Furthermore, Su et al. [187] show theoretically that the stacked honeycomb ferromagnets can also support Weyl magnons. The spin Hamiltonian of the system include NN intralayer ferromagnetic exchange interaction, anisotropy energy, NN interlayer exchange interaction, DM interaction, and Zeeman interaction. By analyzing the magnon Hamiltonian, one can obtain the phase diagram and identify five distinct phases: topological nontrivial phase which can support topologically protected in-gap surface states, trivial phase, and three different magnonic Weyl semimetal phases.  19(e) and 19(f), respectively. The surface states with high density (red color) on the front surface between Weyl nodes can be clearly seen. Near the energy of Weyl nodes, these surface states form magnon arcs on sample surfaces.
In spite of the theoretical progress, the experimental evidence of Weyl magnons is rather rare. The main reason is that Weyl points often locate far away from the center of Brillouin zone and their frequency are thus very high (hundreds of gigahertz). It is difficult to generate such high-frequency magnons by the mature microwave technology. Very recently, by using inelastic neutron scattering technology, Zhang et al. [274] observed the magnonic Weyl states  in multiferroic ferrimagnet Cu 2 OSeO 3 . They show that, in the absence of DM interaction, two pairs of degenerate Weyl nodes with the topological charge +2 and −2 are located at the Brillouin zone center and boundary. When considering the NN DM interaction, these Weyl nodes are shifted away from the high-symmetry points into a position that sensitively depends on the direction and magnitude of the DM interaction vector. Figure 20 shows the comparison of the experimental and calculated magnon spectra.

Higher-order topological magnons
So far, in magnetic system, most of the studies on topological phases are the first-order, while there are only few references discussing the higher-order topological phases. In 2019, Li et al. [96] first proposed the higher-order topological insulator in magnetic system, the discussion of which will be present in Section 3.5. Later, Sil et al. [95] reported the second-order topological magnonic phases in the ferromagnetic breathing kagome lattice. In this subsection, we focus on this model. As introduced in Section 2.2, the first-order topological magnon can be observed in kagome lattice. Sil et al. [95] found that the ferromagnetic breathing kagome lattice can support second-order topological phases under suitable conditions. Here the Hamiltonian contains Heisenberg exchange interaction, Zeeman term, and DM interaction. Figure  21(a) shows the lattice structure, with the unit cell comprising of three sites A, B, and C, where the Heisenberg exchange coupling strength is 1 for the red lines and 2 for the blue lines, the DM interaction strength 1 ( 2 ) points towards (− ) direction between upward (downward) triangles, and the external magnetic field is applied to force the magnetic moments magnetized along direction. In the absence of DM interaction, it is found that the kagome ferromagnet ( 1 = 2 ) is topologically trivial [38], while the breathing configuration ( 1 ≠ 2 ) can support the second-order topological phase. The energy spectrum of finite lattice is shown in Fig. 21(c), from which one can see that when 0 < 1 < 0.5, there exist degenerate states, with 2 being fixed to 1. The energy of the same system is plotted in Fig. 21(d) with respect to energy levels for 1 = 0.3, which clearly shows the existence of three degenerate states. Furthermore, Fig. 21(e) shows the distribution of probability density for these degenerate states, one can identify that these states are corner states localized at each corner. Intersetingly, the breathing kagome lattice with non-zero DM interaction ( 1 ≠ 2 and 1 ≠ 2 ) exhibits a rich topological phase diagram which includes distinct first-and second-order topological magnon insulating phases as well as coexistence of them. The phase diagram is presented in Fig. 21(b), with 2 and 1 being fixed to 1 and 0.1, respectively. On the one hand, by calculating the topological invariant Chern number, four different phases are identified and separated by different colors; see Fig. 21(b). On the other hand, the solid black line separates topologically nontrivial (lower portion) and trivial (upper portion) phases by considering the topological invariant bulk polarization, which was defined in Section 3.5.1. Therefore, one can obtain the first-order topological phase in red and magenta portions above the solid black line, while the green and magenta portions beneath the solid black line host both first-and second-order topological nontrivial phases, and the blue portion only supports the second-order topological phase.

Topological magnonic device
In magnon spintronics, one key topic is how to control spin wave propagation in a designed way. However, conventional spin waves are very sensitive to the device geometry, internal and external perturbations, which makes spin wave devices inflexible and fragile. Besides, it is difficult for conventional spin wave to realize unidirectional propagation. To design tunable and stable spin wave devices, the topologically robust spin waves are indispensable. Wang et al. [173] have predicted the topological chiral spin wave edge state on a ferromagnetic 2D honeycomb lattice, as shown in Fig. 17. Based on these results, they further proposed [174] the concept of topological magnonic device, including spin-wave diode, spin-wave beam splitters, and spin-wave interferometers. As demonstrated in Ref. [173], the topologically protected chiral edge spin waves exist in the band gap and propagate in a certain direction with respect to magnetization direction, i.e., counterclockwise (clockwise) for magnetization along the + (-) direction, as shown in the left panel of Fig. 22(a). Owing to the unidirectional property of topological magnons, a segment of a sample edge can be used as a spin-wave diode. The right panel of Fig. 22(b) shows a snapshot of spin waves when the excitation field is applied at position 2 ○. When a spin wave beam is excited and propagates to position 1 ○, an on state is presented. Conversely, if the excitation field is applied at position 1 ○, no spin wave can be detected at position 2 ○, which means an off state, as shown in the left panel of Fig. 22(b). Since the topological magnons propagate in opposite directions in different domains, as shown in the left panel of Fig. 22(a), their propagation towards the domain wall can neither penetrate into it nor be reflected by it. It must move along the domain wall. When the spin-wave beam reaches the other edge, it will split into two beams propagating in opposite directions, as shown in the right panel of Fig. 22(a). Thus, a domain wall is essential for a 1:2 beam splitters. Figure 22(c) illustrates an example of a 1:4 spin wave beam splitters with three domain walls that separate the = +1 domains (the pink areas) from the = −1 domains (the cyan areas). The figure shows a snapshot of the spin-wave pattern when a excitation field of frequency = 12 is continuously applied at the site marked by the inward arrow in the bottom edge. It is clearly shown that a spin-wave beam splits into four beams eventually.
Moreover, spin-wave interferometer is an important element in magnonics. By utilizing topological magnons, one can design a robust, reconfigurable spin-wave interferometer. Figure 22(d) is a proposal of a Mach-Zehnder-type spinwave interferometer with two different domains separated by domain walls. A topological magnon generated at the site marked by the inward arrow enters the first domain wall of lengh . Then the spin wave beam splits evenly to beam I ○ and II ○. After traveling a certain distance, the two beams recombine and enter the second domain wall of lengh , then spin wave can go to either 3 ○ and 4 ○. Their intensities should depend on the interference of the two beams inside the second domain wall. Remarkablely, by placing the second domain wall at different positions or by changing the length of the second domain wall, the relative phase of the two interfered spin waves can be tuned.
Interestingly, Shindou et al. [197] showed that the magnonic crystal (MC) can induce topological chiral magnonic edge mode as well. The magnonic crystal is composed of YIG and Fe, and the periodic array of holes is introduced into YIG, where Fe is filled inside every hole, as shown in Fig. 23. By using the chiral magnonic edge mode, the spin wave splitter and interferometer also can be realized. In Fig. 23(a), the MC in phase II is connected with the other MC in phase III, where the phase II and phase III are different topological nontrivial phases with Chern number  2 = 2 and  3 = 1, respectively, and is the geometric parameter. The two chiral edge modes propagating along the boundary of the theMC in phase II are spatially divided into two, where one mode goes along the boundary of the MC in the phase III, while the other goes along the boundary between these two MCs. This configuration realizes a spin-wave current splitter. The Fabry-Perot interferometer is made up of a couple of chiral spin wave edge modes encompassing a single topological MC [see Fig. 23(b)]. A unidirectional spin-wave is induced in a chiral mode ["input" in Fig. 23(b)]. Then the spin wave is divided into two chiral edge modes at a point contact (PC1). Two chiral propagations merge into a single chiral propagation at the other point contact (PC2). Depending on a phase difference between these two, the superposed wave exhibits either a destructive or a constructive interference, which is detected as an electric signal from the other antenna ("output"). Here the application of magnetic fields (PS1 and PS2) can change the velocities of the two chiral edge modes locally.
In this section, we have reviewed the topological properties of magnons, including topological magnon insulators and semimetals. The topological magnons based spintronic devices have the obvious advantages over the conventional magnonic devices. Firstly, the topological magnons are confined at the boundary of the system, while conventional magnons spread all over the system. From the energy point of view, the topological magnons based devices have lower energy consumption. By using the topological magnons, one can miniaturize the device to a greater extent. Secondly, the conventional magnons are inflexible and fragile, while topological magnons are very robust against defects and disorder, which enables topological magnons to propagate further.
From the point of view of practical application, there is a demand for low frequency mode, while the frequency of topological magnons often ranges from a few dozen to a few hundred gigahertz. Fortunately, in magnetic system, there exists another important excitation-magnetic soliton-and its collective oscillation frequency is much lower than magnons. In the next section, we will discuss the topological phases of magnetic soliton in artifical lattices.

Topological solitonic insulators
The magnetic soliton represents an important nonlinear excitation in magnetic system, which can exhibit the behavior of waves and topological phases eventually, due to the soliton-soliton interaction. Remarkably, the spintronic devices based on magnetic solitons have a lot of advantages over their electronic counterpart. For example, the nanooscillators based on magnetic vortices or skyrmions are very robust and flexible [214][215][216]; By using the skyrmion as the carrier of information, the data storage density can be greatly improved, and the current density required for encoding information can be significantly reduced [211,213,[275][276][277][278]; It is very convenient to realize various logic operations by using skyrmion [279][280][281][282][283]; There are many ways to manipulate magnetic solitons [284][285][286][287][288][289], which makes the spintronic devices reconfigurable and tunable. In this section, we focus on the collective dynamics and the topological insulator state in artificial magnetic-soliton lattice.

Structures and properties of magnetic solitons
Topology is a study of geometry or space that can keep some properties invariant under a continuous variation of the order parameter. The continuous variation means that the variations do not need to be the same for every position in physical space, but change continuously as a function of position. Magnetic soliton is a manifestation of topology in condensed matter physics. Generally, the magnetic solitons in two dimensions can be characterized by their topological charges which counts how many times the local normalized magnetization wraps the unit sphere. The typical magnetic solitons include the magnetic bubble, vortex, skyrmion, and domain wall, with the micromagnetic structures shown in Fig. 24. The topological charges for magnetic bubble and skyrmion are ±1, while it turns to ±1∕2 for vortex configurations. Topological charge is an invariant indicating that the trivial structure (for example, ferromagnetic state) can not continuously deform into a topological spin texture because of the topological protection. Magnetic solitons with the same topological charge are homotopic. The low-energy dynamics of the magnetic vortex (or skyrmion) can be described by the massless Thiele's equation [234,290] whithin the rigid approximation: where is the position vector of the vortex core,  = −4 / is the gyroscopic constant with the topological charge, is the thickness of ferromagnetic layer, is the saturation magnetization, and is the gyromagnetic ratio. is the viscous coefficient with being the Gilbert damping constant. The conservative force F = − ∕ where  is the potential energy of the system. For a single vortex, the potential energy have the parabolic type:  =  0 +  2 ∕2, where  0 is the energy of system when vortex core locates at the center of the nanodisk and  is the spring constant. By neglecting the damping term, we can derive the gyration frequency of an isolated vortex with However, it is well known that magnetic vortices and skyrmions in particular manifest an inertia in their gyration motion [201,293]. The mass effect thus should be taken into account for describing the vortex (or skyrmion) oscillation. Therefore, the Thiele's equation can be generalized as: with  being the inertial mass of magnetic soliton. Similarly, we can calculate the gyration frequency of magnetic solitons with: The positive and negative values of the in Eq. (12) indicate that there are two kinds of gyration modes with clockwise and counterclockwise direction, respectively. It is noted that the Thiele's equation containing higher-order terms can be derived from Landau-Lifshitz-Gilbert (LLG) equation (see Section 3.4.2 for details).

Collective dynamics of magnetic soliton crystals
As discussed above, the oscillation of magnetic soliton lattice [including one-dimensional (1D), two-dimensional (2D), and three-dimensional (3D) structures] have the properties of waves. For 1D case, Fig. 25(a) shows the SEM image of a sample with an array of five Py disks, which have identical dimensions and initial configuration (vortex states).
Here, the polarization and chirality of vortices array are = [+1, −1, +1, −1, +1] and = [−1, −1, −1, −1, +1], respectively. A current pulse of 1.8 ns duration is applied into the electrode stripline to trigger an excitation of vortex gyration in the first disk. The gyration motion of the first vortex can propagate to other vortices because of the dipolar interaction between disks. Then, the fast Fourier transformation (FFT) of the core position for all votices are calculated and the spectra show that the system have five discrete wave modes. Figure 25(b) shows the trajectories of the vortices cores in the individual disks for different frequencies (we choose three modes as examples). One can clearly see that the collective gyration of vortices is similar to a standing wave.
Furthermore, if we consider the 1D lattice containing more magnetic vortices, the nature of waves will be more feasible. Figures 25(c) and 25(d) show the 1D chains comprising 25 disks of pure NiMnSb and alternating NiMnSb and Py, respectively. All disks are in vortex states and have the same dimension. The spectra of collective vortexgyration excitations for pure NiMnSb and alternating NiMnSb and Py are shown in Figs. 25(e) and 25(f), respectively, which look like the dispersion relations of waves (spin wave for instance). Besides, the influence of different [ , ] orderings on the band structure are also given. Similarly, the 1D lattice containing many skyrmions can also present the band structure. Figure 26(a) plots the schematic diagram of 1D skyrmion array in nanostrip comprising 25 skyrmions, and the dependence of the dispersion relations on is shown in Fig. 26(b), respectively, where is the distance between nearest neighbor skyrmions. It can be seen that as increases, the band width Δ and the angular frequency BZ at wavevector = BZ decrease. The decrease of the total energy density with increasing would result in the decrease of Δ and BZ . The 1D skyrmions lattice can be composed of nanodisks array, as shown in Fig. 26  Considering 2D magnetic soliton systems, without loss of generality, we choose the 2D vortex lattice [224] as an example. Figures 27(a) and 27(b) show the schematic diagram and scanning electron micrograph of the vortex crystal, respectively. Figure 27(c) depicts absorption spectra depending on the phase difference between the exciting magnetic fields for homogeneous polarization patterns. The red lines represent the maxima of the absorption determined by Lorentzian fits. It can be seen that by increasing the phase difference between the exciting magnetic fields the resonance frequencies increase. Figure 27(d) plots the dispersion relation combining the measured absorption (markers) with the calculations using the extended Thiele's model (lines) and calculations for an infinite crystal without damping (dashed lines) obtained from Ref. [218] with different polarization patterns. The lower axis is the wave number while the upper axis shows the phase difference Δ . These results indicate that the band structure of 2D vortices lattice can be reprogrammed by the polarization pattern. By using ferromagnetic resonance spectroscopy and scanning transmission X-ray microscopy, the collective dynamics of 3D vortex crystals have been studied by Hänze et al. [226]. They find that the spectra of the vortex arrangements are directly linked to the chirality and polarity of the vortices.

Two pedagogical models
To have a better understanding about the topological properties of soliton systems, one can map the Hamiltonian into well-known topological models. Pedagogical topological models include the Su-Schrieffer-Heeger (SSH) model and Haldane model. In this section, we give a brief introduction about these two models to facilitate our readers interpreting the following results.

The Su-Schrieffer-Heeger model
The SSH model is a simple tight-binding model with spontaneous dimerization proposed by Su, Schrieffer, and Heeger to describe the one-dimensional polyacetylene [238]. Over the past decades, the SSH model has been generalized to various different systems [5,[294][295][296] and attracted growing interest for demonstrating the fundamental topological physics. Figure 28 (a) plots the illustration of the SSH model, where A and B represent two sublattices, and 1 and 2 are the alternating intracellular and intercellular hopping parameters, respectively. For simplicity, here we only consider the positive values of hopping parameters. The Hamiltonian reads with , (or , ) the annihilation operators localized on site (or ) of the -th cell. By using a Fourier transformation, the Hamiltonian can be written in the form of where is the wave vector, and is the lattice constant, as shown in Fig. 28(a). The spectrum of the SSH model thus can be obtained, = ± √ 2 1 + 2 2 + 2 1 2 cos . For 1 = 2 , the band structure of the system is gapless, while a gap opens at = ∕ when 1 ≠ 2 , leading to an insulating phase, as shown in Fig. 28(d). The topological invariant Zak phase can be used to judge if these insulating phases are topological: where Ψ( ) is the Bloch wave function of the energy band. Figure 28(e) plots the dependence of the Zak phase ℤ on the value of 1 ∕ 2 . It can be clearly seen that ℤ is quantized to when 1 ∕ 2 < 1 and to 0 otherwise, which indicates that the system allows two topologically distinct phases for 1 ∕ 2 < 1 and 1 ∕ 2 > 1.
Bulk-boundary correspondence indicates the existence of robust edge states when the system is in topological phase. Figure 28(f) plots the spectrum of a finite system which contains 100 lattices when 1 ∕ 2 = 0.5. One can clearly see that the system can support degenerate "zero mode" marked by red dots. Further, it is found that its wave function spatial distribution is highly localized in both end of the system, as shown in Fig. 28(h). Noticeably, these edge states are topologically protected and are immune from moderate disorder and defects. Moreover, the bulk modes (marked by black dots) are identified, with the wave function spreading all over the system, as shown in Fig. 28(i). The spectrum of the trivial system is also plotted in Fig. 28(g), where 1 ∕ 2 = 1.5. In this case, one can only observe the bulk modes. Remarkably, these results can be intuitively understood as follow: (i) For the case of 2 → 0, the system is in trivial phase [see Fig. 28(e)], one can clearly see that there are no uncoupled lattice [see Fig. 28(b)]. The system thus can only support bulk states; (ii) When 1 → 0, the system is in topological phase, and we can identify isolated lattice emerging at both ends of the system, which corresponds to the edge state [see Fig. 28(c)].

The Haldane model
Another important topological model is the Haldane model [233], which can realize the quantum Hall effect in the honeycomb lattice (graphene). Graphene is a two-dimensional form of carbon, with the conduction band and valence band touching each other at high-symmetry points (Dirac points) in the BZ [261]. Near those points, the system has a linear dispersion. The tight-binding model with the NN coupling reads: After the Fourier transformation, the tight-binding Hamiltonian takes the following form Near the high-symmetry points ± = (± 4 3 √ 3 , 0), we can expand the wave vector as = + , with | | ≪ | |.
Then the effective Hamiltonian in terms of becomes: In terms of the basis { ( ), ( )}, the Hamiltonian reads: ℎ( ) = − 3 1 2 (± − ), where and are Pauli matrixes. Then we can clearly see that the band structure of graphene is gapless at high-symmetry points ± , and has a linear dispersion near these points. The degeneracy at the Dirac points is protected by inversion (ℙ) and time-reversal ( ) symmetry. By breaking these symmetries the degeneracy can be lifted and gap will open at Dirac points, leading to a topologically non-trivial phase. Conventionally, the realization of quantum Hall state requires a strong magnetic field [1,2]. In 1988, Haldane [233] proposed that the symmetry of the graphene can be broken with a magnetic field that is zero on average in the unit cell, which brings a periodic local magnetic-flux density ( ) in thê direction normal to the 2D plane. The Haldane model was formulated by introducing the NNN hopping: The displacement of the nearest − and − hopping can be written as: 1 Fig. 29(a), and Φ 0 = |ℎ∕ | is the flux quantum. The NN hopping parameter 1 is unaffected by the magntic flux, while the NNN hopping 2 suffers from a shift 2 → 2 exp( ).
In the momentum space, by using the basis { ( ), ( )}, the Hamiltonian can be expressed as: (20) where the masses ( ) with opposite signs at high-symmetry points ± are introduced because of the ℙ symmetry.
Since the symmetry is broken by applying local magnetic-flux, a gap emerges at the high-symmetry points. To achieve the topological non-trivial phase, we require 2 < (3 √ 3 2 sin ) 2 [one can refer to Eqs. (46)-(49) in Section 3.4.2 for calculation details]. Figure 29(b) shows the phase diagram of the spinless electron model, where we have assumed | 2 ∕ 1 | < 1∕3, to guarantee that the two bands are separated by a finite gap.

First-order topological phases
The collective dynamics of magnetic solitons has received significant recent attention, as introduced in Section 3.2, while the possible topological phase is rarely discussed. In 2017, a pioneering work about the topological phase in magnetic soliton lattice was made by Kim et al. [234]. By solving the massless Thiele's equation and mapping it into the Haldane model, it is found that the solitons (vortex and bubble) arranged as a honeycomb lattice can support a chiral edge mode with the propagation direction being associated with the topological charge of the constituent solitons. Soon after that, Li et al. [235] generalized the approach by including both a second-order inertial term and a third-order non-Newtonian gyroscopic term to interpret the emerging multiband nature of chiral edge states observed in honeycomb lattice of magnetic skyrmions. Interestingly, the realization of SSH states in one-dimensional magnetic soliton lattice was also reported recently [236,237]. In this section, we aim to review the theory of the first-order topological insulating phase emerging in low dimensional magnetic soliton systems.

One-dimensional lattice
To discuss the topological insulating phases in one-dimensional magnetic soliton lattice, without loss of generality, we choose the DW as the representative example. Figure 30(a) plots the illustration of various one-dimensional magnetic soliton lattice. The Landau-Lifshitz-Gilbert (LLG) equation can be used to describe the magnetization dynamics [297,298]: where = ∕ is the unit magnetization vector with the saturated magnetization , is the gyromagnetic ratio, and is the Gilbert damping constant. The effective field ef f comprises the external field, the exchange field, the magnetic anisotropic field, and the dipolar field. st is the spin-transfer or spin-orbit torque. For the case of spintransfer torque, st = (̂ ⋅ ∇) − × (̂ ⋅ ∇) with = ∕2| | and̂ being the flow direction of the spin-polarized current. Here is the charge current density, is the spin polarization, is the -factor, is the Bohr magneton, and is the (negative) electron charge.
The collective-coordinate or { , } method provides a simple, yet accurate description of the motion of complex DWs [see Fig. 30(b)] [299,300]: where the collective coordinates and are the position and tilt angle of the -th DW, respectively, pin, includes the pinning field from both the notch and the DW-DW interaction, and are the demagnetizing factors along the -and -axis of the nanostrip, respectively, and Δ = √ 2 ∕ 2 + 0 2 ( − ) + ( − )sin 2 represents the DW width with the exchange stiffness, the magnetocrystalline anisotropy constant, and 0 being the vacuum permeability. Here we discuss the collective genuine oscillation of DW lattice near the pinning notch, the spin torque in Eq. (21) thus can be dropped tentatively.
where and are the width and thickness of the nanostrip, respectively, and is the total energy of the system: Here  is the spring constant determined by the shape of the notch and ( ) is the coupling constant depending on the distance between DWs. Generally, the DW-DW interaction can be divided into three parts: the monopole-monopole (∝ 1∕ ), the exchange (∝ 1∕ 2 ), and the dipole-dipole (∝ 1∕ 3 ) [301]. The explicit form of ( ) can be obtained from micromagnetic simulations in a self-consistent manner. Considering a small and neglecting the dissipation terms, we can arrive at the linear form of Eq. (22): where the  = 2 0 ∕ 2 ( − )Δ is the effective mass of a single DW with Δ = √ 2 ∕ 2 + 0 2 ( − ) and ⟨ ⟩ is the set of the nearest neighbors of . Here, ( ) =  1 ( 2 ) when and share an intracellular (intercellular) connection with  1,2 = ( 1,2 ) ( 1 and 2 are the alternating intersite lengths). We thus have mapped the governing equation to a Su-Schrieffer-Heeger problem [238]. The analytical formula of ( ) can be obtained by the micromagnetic simulation of a DW-DW pair separated by an arbitrary distance. Symbols in Fig. 30(c) are numerical results and the solid curve is theoretical formula ( ) = 1 ∕ + 2 ∕ 2 + 3 ∕ 3 , with 1 = −9.2635 × 10 −12 J m −1 , 2 = 2.294 × 10 −18 J, and 3 = −1.0111 × 10 −25 J m. Figure 30(d) plots the -dependence of the out-of-phase and in-phase DW-oscillation frequencies, that is, 1 and 2 respectively, in the simple two-DW system. It shows that 1 increases while 2 decreases for an increasing . One naturally expects that 1 = 2 = 0 when → ∞, with 0 = √ ∕ corresponding to the oscillation frequency of an isolated DW. By measuring 0 in experiments, one can determine the pinning-potential stiffness . This approach, however, suffers from an issue that the dynamics of a single DW can be easily modified by structure defects and material randomness, and it thus cannot precisely determine the genuine profile of the pinning potential. Below, a topological method is introduced to overcome this issue. Considering a one-dimensional DW lattice, as plotted in Fig. 30(a), where the dashed red rectangle represents the unit cell and the basis vector is a = ̂ with = 1 + 2 . The band structure of the collective DW oscillations can be computed by a plane wave expansion = exp ( + ) , where = , for different sublattices, is an integer, and is the wave vector. The Hamiltonian then can be expressed in momentum space as: Solving (25) gives the dispersion relation: where +(−) represents the optical (acoustic) branch. The bulk band structures for different geometric parameters are plotted in Fig. 30(e), where 2 is fixed to 140 nm and magnetic parameters of Ni [301] are adopted. For 1 = 2 , the two bands merge together [black curve in Fig. 30(e)], while a gap opens at = ∕ when 1 ≠ 2 [red and blue curves in Fig. 30(e)], leading to an insulating phase. Moreover, Figure 30(f) shows the dependence of the Zak phase [238] ℤ on the ratio 1 ∕ 2 . It is observed that ℤ is quantized to 0 when 1 ∕ 2 < 1 and to otherwise, indicating two topologically distinct phases in the two regions.
To verify the bulk-boundary correspondence, we consider the finite system containing an odd number (e.g., 39) of DWs. Numerical results of spectrum are shown in Fig. 31(a), where the in-gap state (red line) emerges for all ratios 1 ∕ 2 ≠ 1. We first consider the case 1 ∕ 2 = 8∕7 (> 1). Figure 31(b) plots the eigenfrequencies of the system, showing that there is one in-gap mode marked by red dot. Further, it is found that its spatial distribution is highly localized at the left end of the racetrack [see Fig. 31(c)], in contrast to its bulk counterpart shown in Fig. 31(d). We adopt the Ansätze for the localized mode as = exp( 0 ) with | | < 1. The edge state then can be solved by the equations: ( 1 +  2 ) ( ) = 0, for = 2, 3, ..., with the boundary condition  1 (1) = 0 (A-site DW is in the outmost left boundary). Because  1,2 ≠ 0, we obtain ( ) = 0 ∀ and = − 1 ∕ 2 . The wave function of A-site DWs therefore follows an exponentially decaying formula | | = | 0 |( 1 ∕ 2 ) for = 1, 2, 3, .... Analytical result agrees excellently with numerical calculations, as plotted in the inset of Fig. 31(c). Furthermore, we expect that the edge state becomes localized in the right end instead if 1 ∕ 2 < 1 and the localized modes emerge in both ends as the magnetic racetrack contains an even number of DWs. The topological robustness of the edge states can be verified by analysing the spectrum of the system including disorder and defects, with results presented in Figs. 31(e) and 31(f), respectively. Here the disorder is introduced by assuming that the coupling parameters  1 and  2 have a random variation, i.e.,  1 →  1 (1 + ),  2 →  2 (1 + ), with the disorder strength and a uniformly distributed random number between −1 and 1. As to the defects, we assume  1 and  2 suffering from a shift ( 1 → 10 1 ,  2 → 0.1 2 ) on the second and fourth DWs. From Figs. 31(e) and 31(f), we observe that the edge state is very robust against these disorder and defects, while the bulk states are sensitive to them.
The micromagnetic simulations are utilized to confirm the theoretical predictions above. A system containing 39 interacting DWs in Ni nanostrip of length 7000 nm is considered, as shown in Fig. 31(g). To obtain the spectra of DW oscillations, a sinc-function magnetic field is applied along the -axis. To find the frequency range of the edge and bulk states, we analyze the temporal Fourier spectra of the DW racetrack at two different positions (DW 1 and DW 20, for example). Figure 31(h) shows the results, with peaks of the red and black curves denoting the positions of edge and bulk bands, respectively. We then apply a sinusoidal magnetic field h( ) = ℎ 0 sin(2 )̂ over the whole system to excite the edge and bulk modes by choosing two frequencies = 1.863 and 2.145 GHz, respectively, as marked in Fig. 31(h). The spatial distribution of DW-oscillation amplitude for these two modes are plotted in Figs. 31(i) and 31(j), respectively, from which one can clearly identify the localized and extended nature of the edge and bulk states, respectively. Full micromagnetic simulations are well consistent with the analytical results.
By including the STT term in Eq. (22), we obtain the generalized Landau-Lifshitz-Gilbert equation: By linearizing Eq. (28) and neglecting the dissipation terms, we have: The solution of (29) can be written as: From Eq. (30), we find that the STT does not modify the DW-oscillation frequency but causes a shift to its equilibrium The non-adiabaticity parameter can therefore be accurately quantified by experimentally measuring the slope of − curve, i.e., = ( − ) ∕ 2 0 . The rubust topological edge of DW lattice can be utilized as the DW frequency standard, which can be used to accurately measure the pinning profile and to finally resolve the controversy about the parameter. Noticeably, these general results are applicable to other types of soliton (e.g., magnetic vortex, skyrmion, etc). Recently, Go et al. [237] studied a metamaterial composed of the magnetic soliton disks structured in a onedimensional bipartite chain, as shown in Fig. 32

Two-dimensional lattice
To analysis the collective dynamics of magnetic vortices on honeycomb lattice, Kim et al. [234] begin with the massless Thiele's equation [Eq. (10)]. Different from the single isolated vortex, the potential energy  should include the contributions both from the confinement of a single disk and the interaction between disks: [218,234,302]. Here,  ∥ and  ⟂ are the longitudinal and transverse coupling constants due to the anisotropic nature of dipole-dipole interactions, respectively. Impose = ( , ) and defining = + , Eq. (10) can be simplied as follow (here the topological charge is chosen to −1∕2): where = ∕||, = ( ∥ −  ⟂ )∕2||, = ( ∥ +  ⟂ )∕2||, is the angle of the direction̂ from the -axis, = ( 0 − 0 )∕| 0 − 0 |, and ⟨ ⟩ is the set of nearest neighbors of . Here we have neglected the dissipation. We then expand the complex variable as: For vortex gyrations with = −1∕2, one can justify | | ≫ | |. By substituting Eq. (33) into Eq. (32), one can obtain: wherē = − is the relative angle from the bond → to the bond → with between and , ⟨⟨ ⟩⟩ is the set of the second-nearest neighbors of . Equation (34) is similar to the Haldane model for electrons in a honeycomb lattice [233,303], where the last term in the right-hand side represents the next-to-nearest hopping that breaks the time reversal symmetry, leading to the existence of a chiral edge state. Figures 33(a) and 33(c) plot the schematic illustrations of vortices in a honeycomb lattice with zigzag edges for = 1∕2 and = −1∕2, respectively. The corresponding band structures for gyration modes are shown in Figs. 33(b) and 33(d). One can clearly identify the chiral edge states. Further, it can be seen that the chirality of the edge modes reverses when the topological charge of vortices switches the sign. In addition, if the vortices are replaced by magnetic bubbles [see Fig. 33(e)], the chiral edge states still exist, as shown in Fig. 33(f). However, if the last term in the right-hand side of Eq. (34) vanishes, e.g. we set̄ = ± ∕2 [see Fig. 33(g)], the time reversal symmetry of the system maintains and the dispersion relation is gapless, leading the disappearance of chiral edge states, see Fig. 33(h).
It is well known that the magnetic bubbles and skyrmions manifest an inertia in their gyration motion [201,288,293,304]. Therefore, if we want to develop an accurate theory about the coupled magnetic soliton (including vortex, bubble, and skyrmion) oscillations, a second-order inertial term and higher-order corrections should be taken into account. Next, we derive the generalized form of the Thiele's equation from the original LLG equation. In terms of the tensor notation, the LLG equation can be written as We can write an alternative form of the LLG equation with t the total effective magnetic field and t = 1 + 2 + 3 . Here 1 = − 1 ̇ , 2 = − ̇ , and 3 = eff are the gyroscopic equivalent field, the dissipative equivalent field, and the effective field, respectively.
Besides, we can define the local force density = − , with = 1, 2, 3. For all , we require the balance of forces 1 + 2 + 3 = 0. Next, we assume that the steady-state magnetization depends on not only the position of the guiding center but also its velocity and acceleration, and we thus have = [ − ( ),̇ ( ),̈ ( )], where = ( , ) is the position of the magnetic soliton guiding center. We havė Then the different forces can be calculated. First of all, 1 = 1 ̇ + 2 ̈ + 3 ⃛ , with 1 = − , 2 = − ̇ , and 3 = − ̈ . Secondly, the dissipation term 2 can be ignored due to the small damping. Finally, since the spins propagate in a steady manner, only externally applied fields contribute to the reversible energy force.
For the gyroscopic term, we can define vector = − 1 2 , with = 1, 3 such that 1 ̇ = 1 ̇ and 3 ⃛ = 3 ⃛ . Then a new vector can be defined It is obvious that = (0, 0, ) when a two-dimensional system is considered. Then we have Likewise, there exists a third-order gyroscopic term of the magnetic soliton as where We can also define a mass tensor Here we assume that  =  = − and  =  = 0, where Eventually, the LLG equation is simplified to the generalized Thiele's form: where = − 0 is the displacement of the magnetic soliton center from its equilibrium position 0 ,  is gyroscopic parameter,  is the effective mass of the magnetic soliton [201,288,293,304],  3 is the third-order non-Newtonian gyroscopic coefficient [305][306][307], and = ∫ ex is the external force. It should be noted that the above derivation is purely phenomenological and detailed microscopic mechanisms are still needed to clarify the origin of the solition mass and its non-Newtonian behavior.
From the aspect of micromagnetic simulations, we consider a two-dimensional honeycomb lattice with 984 identical magnetic nanodisks to demonstrate the chiral edge states. Figure 34(a) shows the sketch. Each nanodisk contains a Bloch-type skyrmion made of MnSi [308] which supports the bulk Dzyakoshinskii-Moriya interaction [309,310]. Here, the distance between nearest neighbor disks is equal to the disk diameter, indicating that skyrmions can strongly interact with each other mediated by exchange coupling. Figure 34(b) plots the band structure of the collective skyrmion oscillations when the exciting field (sinc-function magnetic field) locates in the lattice center [marked by green cross in Fig. 34(a)]. It can be clearly seen that there are no bulk states in the gaps (the shaded areas). Interestingly, when the exciting field is located at the edge of the lattice [marked by yellow cross in Fig. 34(a)], the band structures are obviously different. Figures 34(c) and 34(d) show the dispersion relations of the system by performing the FFT over the upper ( 2 ∕2 < < 2 ) and the lower (0 < < 2 ∕2) parts of the lattice, respectively. One can easily find four edge states appear in the gaps, labeled as ES1-ES4. Furthermore, by analysing the group velocity ∕ of these edge states, the chirality can be identified: ES1 and ES2 counterclockwise propagate, while ES4 behaves oppositely, ES3 shows a bidirectional propagation and it is thus non-chiral.
By plotting the propagation of gyration motion of skyrmions in real space for different modes, one can further confirm the chirality of these edge states. The excitation of edge modes can be realized by applying a sinusoidal field h( ) = ℎ 0 sin(2 )̂ on one nanodisk at the top edge, indicated by the blue arrows in Figs. 35(a)-35(d). Here, four representative frequencies are chosen to visualize the propagation for different edge states. One can clearly observe the unidirectional propagation of these modes with either a counterclockwise manner [ES1 and ES2 shown in Figs.  35(a) and 35(b), respectively] or a clockwise one [ES4 shown in Fig. 35(d)]. In contrast, the propagation of ES3 is bidirectional, as shown in Fig. 35(c). This non-chiral mode can be simply explained in terms of the Tamm-Shockley mechanism [311,312] which predicts that the periodicity breaking of the crystal potential at the boundary can lead to the formation of a conducting surface/edge state. Furthermore, the propagation of the edge states is shown to be immune from the defects, while the Tamm-Shockley mode is not.

Higher-order topological phases
In the previous sections, we have discussed the topological insulating phases in magnetic soliton system. All these phases, however, are first order by nature. In this section, we move on to the higher-order topological phase in magnetic soliton crystals, by presenting thorough calculations details in the breathing kagome [96], honeycomb [97], and square [98] lattices of magnetic vortex.

Kagome lattice
We first consider a breathing kagome lattice of nanodisks with vortex states, as shown in Fig. 36(a) (the vortex topological charge = 1∕2), with 1 and 2 indicating the alternate distance between vortices. We start with the generalized Thiele's equation derived in Section 3.4.2 to describe the collective dynamics of vortex lattice. Because there are two different bond distances ( 1 and 2 ) in this model, the equation should be modified as: The coupling strengths ∥ and ⟂ strongly depend on the parameter ( = ′ ∕ with ′ the real distance between two vortices and being the radius of nanodisk) [313][314][315]. The analytical expression of ∥ ( ) and ⟂ ( ) are very important for calculating the spectra and the phase diagram. The eigenfrequencies of coupled two-vortex system can be expressed as = 0 √ (1 ± ∥ ∕ )(1 ∓ 1 2 ⟂ ∕ ) [313], where 1 (or 2 ) is either +1 or −1 depending on the vortex polarity. Therefore, once the frequencies of coupled modes for different combinations of vortex polarities ( 1 2 = 1 or 1 2 = −1) are determined from micromagnetic simulations, we can derive ∥ and ⟂ according to the dispersion relation. Moreover, it is found that coupling strengths ∥ and ⟂ are the functions of −3 , −5 , −7 and −9 . Therefore, under the help of micromagnetic simulations for two vortices system with different combinations of vortex polarities, one can obtain the best fit of the numerical data [96]: 6.48907∕ 7 + 13.6422∕ 9 ), as shown in Fig. 36(b), where the symbols and curves represent the simulation results and analytical formulas, respectively. In the calculations, the material parameters of Permalloy (Py: Ni 80 Fe 20 ) [316,317] were used. Therefore, we have = −3.0725 × 10 −13 J s rad −1 m −2 . Besides, the spring constant , mass , and non-Newtonian gyration 3 can be obtained through the following relations [306,318]: 0 = ∕ , 3̄ 2 = , 3 (Δ + 0 ) = , 2̄ = 1 + 2 , and Δ = | 2 − 1 |, where 0 is the frequency of the gyroscopic mode, 1 and 2 are the frequencies of the other two higher-order modes with opposite gyration handedness [306]. By analyzing the dynamics of a single vortex confined in the nanodisk [96], we have: = 1.8128 × 10 −3 J m −2 , = 9.1224 × 10 −25 kg, and 3 = −4.5571 × 10 −35 J s 3 rad −3 m −2 . With these parameters, by solving Eq. (51) numerically, one can obtain the eigenfrequencies of the breathing kagome lattice for different values 2 ∕ 1 , as shown in Fig. 36(c), where 1 is fixed to 2.2 . By analyzing the spatial distribution of the eigenfunction for different modes, one can see that the second-order topological edge states (corner state) can exist only if 2 ∕ 1 > 1.2, this conclusion holds for different values of 1 . Furthermore, the complete phase diagram can be obtained by systematically changing 1 and 2 , with results plotted in Fig. 36(d). It can be seen that the boundary separating topologically non-trivial and metallic phases lies in 2 ∕ 1 = 1.2, while topologically trivial and metallic phases are separated by 1 ∕ 2 = 1.2. When 2 ∕ 1 > 1.2, the system is topologically non-trivial and can support second-order topological corner states.
Topological corner states have the property of being immune from the bulk disorder. Figure 36(e) plots the eigenfrequencies of the triangle-shape breathing kagome lattice of vortices under different strengths of disorder, with the geometric parameters 1 = 2.08 and 2 = 3.60 ( 2 ∕ 1 = 1.73 > 1.2). The disorder is introduced by assuming the resonant frequency undergoes a random shift, i.e., 0 → 0 + 0 , where indicates the strength of the disorder and is a uniformly distributed random number between −1 to 1. It can be seen from Fig. 36(e) that with the increasing of the disorder strength, the spectra for both edge and bulk states are significantly modified, while the corner states are quite robust.
The same geometric parameters as Fig. 36(e) are chosen to visualize the different modes (including corner , edge, and bulk states). The eigenfrequencies and eigenmodes of the system are plotted in Figs. 37(a) and 37(b)-(e). It is found that there are three degenerate modes with the frequency equal to 927.6 MHz, represented by red balls. These modes  [96] are indeed second-order topological states (corner states) with oscillations being highly localized at the three corners; see Fig. 37(d). The edge states are also identified, denoted by blue balls in Fig. 37(a). The spatial distribution of edge oscillations are confined on three edges, as shown in Fig. 37(c). However, these edge modes are Tamm-Shockley type [311,312], not chiral, which was confirmed by micromagnetic simulations [96]. Bulk modes are plotted in Figs. 37(b) and 37(e), where corners do not participate in the oscillations.
The other type of breathing kagome lattice of vortices (parallelogram-shape) also supports the corner states, with the sketch plotted in Fig. 37(f). Here, the same parameters as those in the triangle-shape lattice are adopted. Figure  37(g) shows the eigenfrequencies of system. Interestingly, it can be seen that there is only one corner state, represented by the red ball. Edge and bulk states are also observed, denoted by blue and black balls, respectively. The spatial distribution of vortices oscillation for different modes are shown in Figs. 37(h)-37(k). From Fig. 37(j), one can clearly see that the oscillations for corner state are confined to one acute angle and the vortex at the position of two obtuse angles hardly oscillates. The spatial distribution of vortex gyration for edge and bulk states are plotted in Figs. 37(i), 37(h), and 37(k), respectively. Further, the robustness of the corner states are also confirmed [96].
The higher-order topological properties can be interpreted in terms of the bulk topological index, i.e., the polarization [99,100]: where is the area of the first Brillouin zone, = − ⟨ | | ⟩ is Berry connection with = , , and is the wave function for the lowest band. It is shown that ( , ) = (0.499, 0.288) for 1 = 2.08 and 2 = 3.60 and ( , ) = (0.032, 0.047) for 1 = 3 and 2 = 2.1 . The former corresponds to the topological insulating phase while the latter is for the trivial phase. Theoretically, for breathing kagome lattice, the polarization ( , ) is identical to  [96] the Wannier center, which is restricted to two positions for insulating phases. If Wannier center coincides with (0, 0), the system is in trivial insulating phase and no topological edge state exists. Higher-order topological corner states emerge when the Wannier center lies at (1/2, 1/2 √ 3) [51,70]. Micromagnetic simulations can be used to verify the theoretical predictions of corner states. The triangle-shape and parallelogram-shape breathing kagome lattice of vortices are considered, as shown in Fig. 36(a) and Fig. 37(f), with the same geometric parameters as those in Fig. 37(a) and Fig. 37(g), respectively. Figure 38(a) shows the temporal Fourier spectra of the vortex oscillations at different positions. One can immediately see that, near the frequency of 940 MHz, the spectrum for the corner has a very strong peak, which does not happen for the edge and bulk. It can be inferred that this is the corner-state band with oscillations localized only at three corners. Similarly, one can identify the frequency range that allows the bulk and edge states, as shown by shaded area with different colors in Fig. 38(a).  Figure 38(i) shows only one corner state at only one (bottom-right) acute angle. Spatial distribution of vortices gyration for bulk and edge states are shown in Figs. 38(g) and 38(h), respectively. Interestingly, the hybridization between bulk mode and corner mode occurs as well in parallelogram-shaped lattice, see Fig. 38(j).
In recent years, nano-oscillators in magnetic systems have attracted great attention for potential applications. However, the working frequency of these oscillators is very sensitive to external disturbances. If the HOTI phase (corner state) is used, vortex-based nano-oscillators should have extraordinary stability against defects and disorder and should therefore have broader prospects for application as topological microwave sources.
In condensed matter physics, besides kagome lattice, the topological properties in honeycomb lattice are also studied extensively. The rich topological phases (including first-and second-order) are confirmed in breathing honeycomb lattice of vortices, which will be introduced in the next section.

Honeycomb lattice
It is well known that the perfect graphene lattice has a gapless band structure with Dirac cones in momentum space [261]. When spatially periodic magnetic flux [233] or spin-orbit coupling [319] are introduced, a gap will open at the Dirac point, leading to a FOTI. Interestingly, it have been shown that the gap opening and closing can be realized by tuning the intercellular and intracellular bond distances in photonic [57] and elastic [83] honeycomb lattices, in which the HOTI appears. In this section, we show that the higher-order topological insulating phase do exist in a breathing honeycomb lattice of vortices.  ) + 1 ∑ The band structure of system can be calculated by diagonalizing the Hamiltonian, where the elements can be expressed explicitly as.
The topological invariant Chern number is usually adopted to judge whether the system is in the FOTI phase [173,320]. However, to determine whether the system allows the HOTI phase, another different topological invariant should be considered. In addition to the bulk polarization, it has been shown that ℤ Berry phase [84,[102][103][104][105][106][107] is a powerful tool to characterize the HOTI.
In the presence of six-fold rotational ( 6 ) symmetry, the ℤ 6 Berry phase is defined as follow: where A(k) is the Berry connection: Here, Ψ(k) = [ 1 (k), 2 (k), 3 (k)] is the 6 × 3 matrix composed of the eigenvectors of Eq. (54) for the lowest three bands. 1 is an integral path in momentum space ′ → Γ → ′ ; see the green line segment in Fig. 39(b). In addition, the six high-symmetry points , , ′ , ′ , ′′ , and ′′ are equivalent, because of the 6 symmetry. Therefore, there are other five equivalent integral paths ( 2 ∶ ′ → Γ → ′′ , 3 ∶ ′′ → Γ → ′′ , 4 ∶ ′′ → Γ → , 5 ∶ → Γ → , and 6 ∶ → Γ → ′ ) leading to the identical . It is also straightforward to see that the integral along the path 1 + 2 + 3 + 4 + 5 + 6 vanishes. Thus, the ℤ 6 Berry phase must be quantized as = 2 6 ( = 0, 1, 2, 3, 4, 5). By simultaneously quantifying the Chern number  and the ℤ 6 Berry phase , the topological phases and their transition can be determined accurately. , the highest three bands and the lowest three bands merged separately, leaving a next-nearest hoppinginduced gap centered at 927 MHz. In this case, the FOTI phase was anticipated [234,235]. However, the six bands is the width of the nanoribbon. Source: The figures are taken from Ref. [97] are separated from each other when considering the parameters 1 ≠ 2 [see Figs. 39(c) and 39(e)], indicating that the system is in the insulating state. These insulating phases and the phase transition point can be further distinguished by calculating Chern number and ℤ 6 Berry phase. Figure 40(a) shows the dependence of the Chern number () and the ℤ 6 Berry phase ( ) on the parameter 2 ∕ 1 . Here the material parameters of Py (Ni 80 Fe 20 ) [316,317] are used and 1 is fixed to 2.5 . In addition, the eigenfrequencies for a parallelogram-shaped [see Fig. 40(b)] structure are also shown in Fig. 40(c). One can see that the system is in the trivial phase when 2 ∕ 1 < 0.9 and 1.08 < 2 ∕ 1 < 1.49, in the FOTI phase when 0.9 < 2 ∕ 1 < 1.08, and in the HOTI phase when 2 ∕ 1 > 1.49. The complete phase diagram of system can be obtained by systematically changing 1 and 2 , with the results plotted in Fig. 40(d). The boundary for the phase transition between trivial and FOTI phases depends only weakly on the choice of the absolute values of 1 and 2 but is (almost) solely determined by their ratio, as indicated by dashed black lines ( 1 ∶ 2 ∕ 1 = 0.94 and 2 ∶ 2 ∕ 1 = 1.05) in the figure. While the boundary for the phase transition between trivial and HOTI phases is a linear function 3 ∶ 2 = 2.24 1 − 1.88. From Eq. (53), we can see that the topological charge of the vortex has no influence on higher-order topology for the reason that the sign of topological charge just determines the direction (clockwise or anti-clockwise) of gyration. However, it indeed can affect the chiral edge state (first-order topology). Namely the chirality of edge state will be reversed if the topological charge changes.
The existence of symmetry-protected states on boundaries is the hallmark of a topological insulating phase. Figures . For 1 = 2 = 3.6 , the lattice considered is identical to a magnetic texture version of graphene. In contrast to the gapless band structure for perfect graphene nanoribbons, the imaginary second-nearest hopping term opens a gap at the Dirac point and supports a topologically protected first-order chiral edge state [234,235]. For 1 = 2.08 and 2 = 3.6 , one can clearly see two distinct edge bands, in addition to bulk ones, as shown in Fig.  40(h). These localized modes are actually not topological because they maintain the bidirectional propagation nature, which is justified by the fact that the wave group-velocity ∕ can be either positive or negative at different points. However, the higher-order topological corner states will emerge around these edge bands when the system is decreased to be finite in both dimensions.
A parallelogram-shaped vortex lattice is considered to visualize the second-order corner states, where 1 = 2.08 and 2 = 3.6 . From the spectrum [see Fig. 41(a)], one can clearly see that there exist a few degenerate modes in  [97] when the moderate defects and disorder are introduced into the system, corner state 3 at the obtuse-angled corner is well confined around 927 MHz, which means that this corner state is suitably immune from external frustrations. This feature is due to the topological protection from the generalized chiral symmetry [97]. However, the frequencies of other corner modes have obvious shifts, revealing that these crystalline-symmetry-induced modes are sensitive to disorder. The origin of the edge state is attributed to the so-called Tamm-Shockley mechanism [311,312].
To verify theoretical predictions, one can implement full micromagnetic simulations. Here, the parallelogramshaped breathing honeycomb lattice of magnetic vortices with an armchair edge is considered, as shown in Fig. 40(b). Figure 41(g) shows the temporal Fourier spectra of the vortex oscillations at different positions. It can be seen that around the frequency of 944 MHz (948 MHz), the spectra for acute-angled corner (obtuse-angled corner) have an obvious peak, which does not happen for the spectra for edge and bulk bands. Therefore, these two peaks denote two different corner states that are located at acute-angled or obtuse-angled corners. Similarly, the frequency range for bulk and edge states also can be identified. Further, to visualize the spatial distribution of the vortex oscillations for different modes, four representative frequencies are chosen: 872 MHz for the bulk state, 934 MHz for the edge state, 944 MHz for the acute-angled corner state, and 948 MHz for the obtuse-angled corner state, respectively. We then stimulate their dynamics by applying a sinusoidal field to the whole system. The 10 ns gyration paths of all vortices are plotted in Figs. 41(h)-41(k) when the excitation field drives a steady-state vortex dynamics. The spatial distribution of vortices motion for the bulk and edge states are shown in Fig. 41(h) and Fig. 41(i), respectively. We observe type I corner state with vortex oscillation localized at the acute-angled corner in Fig. 41(j). Interestingly, one can note a strong hybridization between the type II and type III corner states, as shown in Fig. 41(k), which is because their frequencies are very close to each other and their wavefunctions have a large overlap [see Figs. 41(d) and 41(f)].
Corner states are topologically protected and are deeply related to the symmetry of Hamiltonian (54). Below, we prove that the emergence of topological zero modes is protected by the generalized chiral symmetry. First of all, because ( 2 1 + 2 2 2 )∕2 ≪ 0 , the diagonal element of  can be regarded as a constant, i.e., 0 = 0 , which is the "zero-energy" of the original Hamiltonian. 1,2,3,4,5,6 are the next-nearest hopping terms. At first glance, the system does not possess any chiral symmetry to protect the "zero-energy" modes because the breathing honeycomb lattice is not a bipartite lattice. Here, we generalize the chiral symmetry for a unit cell containing six sites by defining where the chiral operator Γ 6 is a diagonal matrix, and  1 =  − 0 I. Here, to prove the system has generalized chiral symmetry, we divide the system into six subgroups with the components of matrix Hamiltonian being nonzero only between different subgroups, such a property is essential for chiral symmetry and indicates no interaction within sublattices. Upon combining the last equation with the previous five in Eqs. (58), we have Γ −1 6  6 Γ 6 =  1 , implying that [ 1 , Γ 6 6 ] = 0; thus, Γ 6 6 = I, which is completely analogous to the SSH model [238]. Hamiltonians  1,2,3,4,5,6 each have the same set of eigenvalues 1,2,3,4,5,6 . The eigenvalues of Γ 6 are 1, exp(2 ∕6), exp(4 ∕6), exp( ), exp(8 ∕6), and exp(10 ∕6). Therefore, we can write in the same bases as that for expressing Hamiltonian (54). By taking the trace of the sixth line from Eqs. (58), we can obtain ∑ 6 =1 Tr( ) = 6Tr( 1 ) = 0, which indicates that the sum of the six eigenvalues vanishes Given an eigenstate that has support in only sublattice , it will satisfy  1 = and Γ 6 = exp[2 ( − 1)∕6] with = 1, 2, 3, 4, 5, 6. From these formulas and Eqs. (58), we obtain = 6 = 0, indicating = 0 for any mode that has support in only one sublattice, i.e., zero-energy corner state.
The corner states are protected by the generalized chiral symmetry. To prove this point experimentally, one can observe whether the frequencies of corner states are robust when the generalized chiral symmetry is broken by introducing the NNN hopping terms in the specific lattice. However, it is rather difficult to introduce NNN hopping by a designed manner in magnetic and condensed matter systems. Very recently, it is shown that the emerging topolectrical circuits can solve this problem for the reason that the coupling between any two lattice can be easily realized by adding extra circuit elements (such as capacitor and inductor). Based on the breathing kagome topolectrical circuit, Yang et al. [92] observed the symmetry-protected zero modes (corner states). They proved that the frequency of corner states suffers from a obvious shift when the NNN hopping is introduced by connecting the capacitor within the sites in the corner. The illustration and experiment setup are shown in Fig. 42 (a). Figure 42 (b) plots the experimental . Source: The figures are taken from Ref. [92] measurements of impedance (between corner and bulk) with and without . Furthermore, it is also comfirmed that the frequency of corner states does not change when the NNN hopping is located in the edge or bulk. These experimental results therefore substantiate the conclusion that corner states are indeed protected by the generalized chiral symmetry.

Square lattice
In previous sections, we have discussed the HOTI phase in the breathing kagome and honeycomb lattice of vortices. On the one hand, the square lattice is also widely studied in different systems [56,80,87]. On the other hand, it is well known that topological states of many Hamiltonians, which support topological states in honeycomb lattice or other lattices, disappear immediately when the lattice deform into a square lattice. Thus the study of topological states in square lattice is important and may be non-trivial in this aspect. Moreover, previous studies are focused on corner states only with a single frequency. Since the generalized Thiele's equation contains higher-order terms, the topologically stable multimode corner phases may exist in magnetic soliton lattice. In this subsection, we show that the multimode HOTI phase indeed emerges in a breathing square lattice of vortices. Besides, it is demonstrated that the HOTI phase based on square lattice is convenient for the application of display.
The breathing square lattice of magnetic nanodisks with vortex states is shown in Fig. 43(a). Similarly, the generalized Thiele's equation is adopted to describe the collective dynamics of the vortex lattice and the eigenvalue equation of the system can be obtained: For an infinite lattice, the dashed black rectangle indicates the unit cell, as shown in Fig. 43(a). a 1 = ̂ and a 2 = ̂ are two basis vectors, with = 1 + 2 . The matrix form of the Hamiltonian in momentum space can be obtained by considering a plane wave expansion of = exp( ) exp ( ⋅ a 1 + k ⋅ a 2 ) , where k is the wave vector, and are two integers:  = ⎛ ⎜ ⎜ ⎜ ⎝ 0 2 + 1 exp(− k ⋅ a 1 ) 2 + 1 exp( k ⋅ a 2 ) 1 2 + 1 exp( k ⋅ a 1 ) 0 2 2 + 1 exp( k ⋅ a 2 ) 2 + 1 exp(− k ⋅ a 2 ) * 2 0 2 + 1 exp(− k ⋅ a 1 ) * 1 2 + 1 exp(− k ⋅ a 2 ) 2 + 1 exp( k ⋅ a 1 ) The bulk band structures with various geometric parameters ( 1 and 2 ) are shown in Figs. 43(c)-43(e). For Fig. 43(d)], all bands merge together, leading to a gapless band. However, when 1 ≠ 2 , two gaps open and they locate between 1st and 2nd bands, 3rd and 4th bands, respectively. Interestingly, the 2nd and 3rd bands are always merged no matter what values 1 and 2 take. The topological invariants Chern number and ℤ 4 Berry phase can be used to further distinguish whether these insulating phases are topologically protected. Figure 43(f) plots the dependence of the Chern number  and the ℤ 4 Berry phase on the ratio 2 ∕ 1 with 1 fixed to 3 . One can clearly see that the ℤ 4 Berry phase is quantized to 0 when 2 ∕ 1 < 1 and to otherwise, showing that 2 ∕ 1 = 1 is the phase transition point separating the trivial and topological phases. Furthermore, the Chern number vanishes for all ratios 2 ∕ 1 , indicating that the system has no first-order TI phase. A simple way to understand the zero Chern number is that all elements of the lattice Hamiltonian are real numbers apart from the phase factor exp(± ⋅ a 1,2 ), which naturally leads to a vanishing Chern number. Therefore, we conclude that the system is in the HOTI phase when 2 ∕ 1 > 1, and in the trivial phase when 2 ∕ 1 < 1. This conclusion holds independent of the 1 value, since the 2D breathing square lattice can be viewed as two copies of SSH chains along the horizontal and vertical directions, respectively.
The dynamics of finite vortex lattice can be used to directly confirm the existence of corner states. The eigenfre-  Fig. 44(a). The bulk, edge, and corner states are marked by black, blue, and red arrows in Fig. 44(a), respectively. The intuitive understanding why these corner states only appear in the special parameter region ( 2 ∕ 1 > 1) is as follow: on the one hand, the configuration shown in Fig. 44(b) is in the HOTI phase. In such a case, one can clearly identify four isolated vortices at corners. Thus the localized corner states will appear; on the other hand, in the limit 1 → ∞ [see Micromagnetic simulation results are plotted in Fig. 45 for a comparison. The spectra of the vortex oscillations at different positions are shown in Fig. 45(a). One can clearly see that near the eigenfrequencies of a single vortex gyration (0.939 GHz and 11.941 GHz), the spectrum for the corner has two very strong peaks, which do not exist for edge and bulk bands. Similarly, the frequency range supporting the bulk and edge states also can be identified. Interestingly, for the 14.189 GHz peak, although the spectrum in the corner has a strong peak, the oscillation amplitude at the edge is sizable as well, which indicates a strong coupling between edge and corner oscillations. The emerging HOTI in vortex lattice can be used to design topological devices. Figure 46 show a display device based on vortex lattice. The desired display "H" in the HOTI phase is surrounded by another vortex lattice in the trivial phase, as shown in Fig. 46(a). The display points are marked by arabic numbers 1 − 7. The collective dynamics of the  [98] whole system is stimulated by applying a sinusoidal magnetic field with the frequency = 0.939 GHz. Figure. 46(b) plots the spatial distribution of the oscillation amplitudes, from which one can clearly see that only the vortices at the desired display points have sizable oscillations, while the other vortices do not participate in the display. We point out that other display shapes can be realized by a similar method, too.

Conclusion and outlook
We have reviewed the recent progress on topological insulator and semimetal phases in magnon and soliton based crystals. These studies not only deepen our understanding on topological physics and its manifestation in classical magnetism, but also hold promise for future robust spintronic devices.
The topological insulating phases of magnons appear in gapped system and can support topologically protected spin wave modes confined in the boundaries or corners. Dirac and Weyl magnons emerge in gapless band structures and magnon arc states are topologically protected in the surface. The generation and detection of topological edge spin wave can be achieved with the same techniques used in conventional magnonic devices. For example, topological edge spin wave can be excited by the antenna microwave magnetic field and can be detected by the Brillouin light-scattering spectroscopy [132,141]. Remarkably, Bonetti et al. [321] report that the real-space spin wave movie can be created by using a high-sensitivity time-resolved magnetic X-ray microscopy, which can be conveniently adopted to detect the topological edge spin wave. For application, the spin-wave edge states can be used to design various topological magnonic devices, including spin-wave diode, spin-wave beam splitters, and spin-wave interferometers, etc. The predicted second-order topological insulating phase based on magnetic soliton lattice can facilitate the design of different spintronic devices: (i) For the application aspect, it is still a challenging issue to realize stable display function in natural and artificial materials. The main obstacle lies in the difficulty for precisely controlling both the position and the frequency of the local oscillations. Topological insulators provide a new route for that purpose. The chiral topological edge states can perfectly localize the energy at the boundary of the system. If the boundary of the system is set to a specific shape, the display function can be realized under the external excitation of chiral edge modes. However, the proposal suffers from some disadvantages. On the one hand, to form a clear picture, a very large system needs to be conceived. On the other hand, the target picture must be continuous and it is difficult to display discrete pictures. The emerging HOTI can well solve these problems. Zhang et al. [77] designed the programmable imaging device based on the acoustic second-order topological insulators, as shown in Fig. 47. The imaging device consists of two subwavelength digital elements of "0" and "1", which correspond to trivial and secondorder topological nontrivial state, respectively. Figure 47(a) illustrates a heart-like acoustic profile. When the corner states [labeled by red dots in Fig. 47(a)] are excited at the specific frequency, the simulated and experimental data verify that most energy is confined at the corners, which realized stable topological acoustic imaging device, as shown in Figs. 47(b) and 47(c). Similarly, the acoustic imaging of characters can be realized, see Figs. 47(d)-47(g). For magnetic system, the imaging device based on HOTI in magnetic vortices lattice is discussed as well [see Fig. 46]. Remarkably, these imaging device are topologically protected and can immune from external disturbances, which makes magnetic HOTIs have potential applications for designing topological spintronic imaging elements. (ii) Moreover, in recent years, nano-oscillators in magnetic systems have attracted great attention for potential applications [322][323][324][325]. In particular, oscillators based on magnetic-soliton structures [214][215][216][326][327][328] can be used as good microwave sources owing to their outstanding characteristics of small size, easy manipulation, and high tunability. However, the working frequency of these oscillators is very sensitive to external disturbances. If the HOTI phase is used, vortex-based nano oscillators should have extraordinary stability against defects and disorder and should therefore have broader prospects for application as topological microwave sources. (iii) The multiband nature of the corner modes (with a spectrum ranging from less than 1 GHz to dozens of gigahertz) is very useful for designing broadband topological devices. From an experimental point of view, we note that both fabricating the metamaterials of magnetic solitons and detecting the highly spatially localized corner modes are already within the reach of current technology. On the one hand, by using electron-beam lithography [224,229,329] or X-ray illumination [330], the artificial magnetic soliton with different lattices can be created. On the other hand, by tracking the nanometer-scale vortex orbits using the ultrafast Lorentz microscopy technique in a time-resolved manner [331], one can directly observe the second-order topological corner states of magnetic soliton lattice. Figure 48 shows the schematic representation of time-resolved Lorentz microscopy and the related experimental results for tracking vortex core.
Moreover, the detection of soliton lattice edge states can be realized by using the interaction between edge and bulk waves which is similar to the nonlinear three-magnon process [332]. Figure 49(a) shows the schematic picture of nonlinear three-magnon processes in the DM interaction nanostrip, where a propagating spin wave ( , ) interact with the localized spin wave ( , ) bounded in the nanostrip. In general, two kinds of three-magnon processes (confluence and splitting) can occur, as illustrated in Fig. 49 (a). By comparing and (or 2 ), one can obtain the information of bounded state spin wave. Similarly, if we send both a bulk mode ( 2 , 2 ) and an edge mode ( 1 , 1 ) simultaneously in the soliton lattice, by probing the reflected bulk mode ( , ) as shown in Fig. 49(b), we can identify the information of edge state based on the energy-momentum conservation law = 2 + 1 and ( 2 + 1 − ) ⋅̂ = 0, and = 2 − 1 and ( 2 − 1 − ) ⋅̂ = 0 for three-wave confluences and for three-wave splittings, respectively.
Finally, we comment that the study on topological phase and phase transitions in magnon and soliton metamaterials Figure 49: (a) Schematic picture of nonlinear three-magnon processes in the DM interaction nanostrip. In the dashed red square, it shows the three-magnon confluence of and into . In the dashed blue square,it plots the stimulated three-magnon splitting of into two modes = and 2 , assisted by a localized magnon . (b) Illustration for detecting the edge state of skyrmion lattice. Source: The figures are taken from Ref. [332].
is quite active but is still in the initial stage, and many open questions are yet to be answered and more new phenomena is to be disvovered: (1) In the original calculations of magnetic soliton HOTIs, identical nanodisks are assumed. However, if the translational symmetry is broken, for instance by introducing Kekulé distortions into the disk sizes, one may realize topological devices supporting robust Majorana-like zero modes localized in the device's geometric center [333].
(3) The twisted bilayer graphene structure has attracted a lot of attention over the past few years for the exotic physical properties [334,335]. We envision that the topological property of twisted bilayer of honeycomb lattice based on magnetic solitons (or spins) is also an appealing research topic.
(4) The practical applications of topological insulating phases (especially for HOTI) in magnetic system are still lacking, since the experimental detection of these topological phases and the device design are challenging.
(5) The LLG equation describing the dynamics of magnetic moment is intrinsically nonlinear, however, the analytical theory about the topological magnons is based on the linear approximation. When the oscillation amplitude of magnetic moment or soliton is large enough, the nonlinear effect should be considered. The influence of nonlinearity on the topology phase and phase transition is an interesting issue for future study. (6) In order to observe the Weyl points in magnonic system, the excitation sources with very high frequency (hundreds of gigahertz) are demanded, which, is not compatible with the mature microwave antenna technology. Magnetic soliton as another important excitation in magnetic system, holds much lower frequency (by one order of magnitude) than magnons. Therefore, realization Weyl semimetals in magnetic soliton crystals is an appealing research topic, too.

Declaration of competing interest
The authors declare no competing financial interests that could have appeared to influence the work reported in this paper.