Topological Phenomena in Spin Systems: Textures and Waves

Abstract

This chapter reviews the implications of topology in the static and dynamics of magnetic systems. Our focus is twofold. In the first part, we describe how the application of topology allows an understanding of the structure and dynamics of magnetic textures that separate different magnetic domains in magnetic materials. Topological textures are rationalized in terms of elementary topological defects that determine complex magnetic orders and magnetization dynamics processes in the underlying magnetic systems. The second part studies topological phases and topological phenomena associated with the band theory of linear magnetic excitations. Topological spin waves are usually accompanied by exotic phenomena in magnetic materials such as the emergence of chiral edge states and the magnon Hall effect.

1. Introduction

Topology in condensed matter physics categorizes the robustness of classical or quantum states under smooth deformations. Dissimilar physical states may manifest such robustness in terms of a common set of preserved quantities. The preserved quantities, or invariants, affect the static properties of such systems and their dynamics and allow the classification of their physical states. Crucially, the study of topological invariants in such systems allows the definition of robust rules that facilitate the understanding and modeling of the associated physical phenomena.

The application of topology to condensed matter physics began with the study of lattice defects like dislocations and disclinations. In magnetism, the topological methodology allowed the systematic study of complex magnetic structures that take the form of localized magnetic defects in real space. In a medium with an underlying magnetic order, a defect is a point or a collection of points where the order parameter, usually a function of the magnetization vector, is not well defined. The defect is topological when it cannot be removed, whatever the continuous modifications exerted on the order parameter distribution over space. Depending on the system’s geometry, the external fields, and the interactions between its constituents, topological spin textures may arise aimed to separate distinct magnetic phases in a sample. Examples of such textures are domain walls (DWs). In other cases, the spin textures can be part of the magnetic order and define the magnetic landscape of the whole system. An example could be a lattice of periodically arranged skyrmions, or a lattice that relaxes into chiral spin textures periodically arranged. In either case, the study of self-localized topological spin textures and the rules associated with their creation, merging, and motion, strongly rooted in their winding numbers, has been extremely useful in characterizing the ground state and dynamics of magnetic systems.

Research related to the transport of topological textures in real space has set the basis for spintronics and racetracks memory devices. The first part of this chapter is devoted to reviewing the main results regarding topological textures in real space. Though they may arise in ferromagnets, antiferromagnets, or ferrimagnetic systems, here, we focus on topological spin textures on ferromagnetic samples, which constitute a mature field of research.

More recently, topology has caused renewed interest because of the study of topological phase transitions and topological phases of matter [1]. These phases usually manifest in reciprocal space, where the nontrivial wrapping of the Brillouin zone around the Hamiltonian space leads to exotic topological states. In this context, topological magnetic excitations in the form of spin waves or magnons (the quanta of spin waves) have recently become a very active and broad field. Transport with spin waves can be accomplished without electronic degrees of freedom, reducing the Joule heating, and making the spin a promising route to engineer the transmission of information. When spin waves propagate unidirectionally along the surface of a system, they are topologically protected by the bandgap in the reciprocal space. These chiral spin waves very often imply robustness because they can only be destroyed through external forces that consume a finite amount of energy. Topological bands are usually associated with exotic transport phenomena like the thermal Hall effect. The study of topological spin excitations has led to a new field called topological materials, including topological insulators and topological semimetals.

We begin by briefly summarizing the main interactions among spins in magnetic systems. The hierarchy of such interactions together with the geometrical properties of the underlying magnetic system gives rise to anisotropies that constrain the spin vector along specific directions.

1.1 Magnetic interactions

The different coupling mechanisms between spins can give rise to various ground states of spin textures and dynamics. In the following, we consider a semiclassical model of spins represented by vectors localized at lattice sites. In the so-called micromagnetic limit, the system of localized spins is transformed into a smoothly varying spin texture, described by a set of continuous vector fields. Next, we describe the most relevant magnetic interactions involved in the stabilization of topological defects and the realization of topological bands in magnetic materials. These include the exchange, Dzyaloshinskii-Moriya (DM), magnetostatic (dipolar), and Zeeman interactions.

1.1.1 Exchange interaction

It originates from the concomitance of the Pauli exclusion principle and the Coulomb interactions. In the classical Heisenberg model, where the exchange interaction is the only term, the Hamiltonian is generally written as

$$H_E=\sum _{\mathit{\alpha \beta }=x,y,z}\sum _{⟨\mathit{r}{\mathit{r}}^{\prime }⟩}{J}_{\mathit{r}{\mathit{r}}^{\prime }}^{\mathit{\alpha \beta }}{S}_{\mathit{r}}^{\alpha }{S}_{{\mathit{r}}^{\prime }}^{\beta },$$E1

with $J_{\mathit{r}{\mathit{r}}^{\prime }}$ the exchange coupling constant among spins $S_r$ and $S_{r^{\prime }}$ localized at positions $\mathit{r}$ and ${\mathit{r}}^{\prime }$. Here, $⟨\mathit{r}{\mathit{r}}^{\prime }⟩$ denotes nearest-neighbor (NN) spins, revealing the interaction’s short-ranged nature. For materials that are isotropic, the spatial and spin dependence of the exchange tensor can be suppressed, and the exchange coupling can be treated as a homogeneous coupling constant between spins at neighboring sites. The sign of $J$ determines, among other ingredients, the emergence of ferromagnetism ($J<0$) or antiferromagnetism ($J>0$).

1.1.2 Dzyaloshinskii-Moriya interaction

The Dzyaloshinskii-Moriya (DM) interaction is an asymmetric exchange coupling that favors a noncollinear orientation of spins. It was proposed to be the source of a nonzero spontaneous magnetic moment in certain magnetic states of the magnetic material Hermatite, by Dzyaloshinskii in $1958$ [2]. The microscopic origin of the effect was later found by Moriya [3] who showed how spin-orbit coupling gives rise to an antisymmetric interaction mechanism in systems with low magnetic order. The DM spin Hamiltonian reads

$$H_{\mathit{DM}}=\sum _{⟨\mathit{r}{\mathit{r}}^{\prime }⟩}{\mathit{D}}_{\mathit{r}{\mathit{r}}^{\prime }}\cdot \left({\mathit{S}}_{\mathit{r}}\times {\mathit{S}}_{{\mathit{r}}^{\prime }}\right),$$E2

with ${\mathit{D}}_{\mathit{r}{\mathit{r}}^{\prime }}$ the DM vector coupling. The direction and magnitude of ${\mathit{D}}_{\mathit{r}{\mathit{r}}^{\prime }}$ are determined by the symmetry class the crystalline lattice belongs to. The DM interaction favors a canting among neighbor spins that competes with the direct exchange interaction. Minimizing the magnetic energy (containing DM) leads to twisted magnetic structures, such as skyrmions.

1.1.3 Magnetostatic interaction

The magnetostatic energy is also called dipolar interaction energy. In a crystal, each magnetic moment creates a dipolar field, and each moment is exposed to the magnetic field created by all other dipoles. Denoting by ${\mathit{H}}_{\mathit{r}}$ the magnetic field created by all spins other than spin $r$ in the position of ${\mathit{S}}_{\mathit{r}}$, the magnetostaic energy can be written as follows:

$$E_M=\frac{-{\mu }_0}{2}\sum _r{\mathit{S}}_{\mathit{r}}\cdot {\mathit{H}}_{\mathit{r}},$$E3

where ${\mu }_0$ is the magnetic permeability. In a continuum model, the magnetostatic energy corresponds to the long-ranged interaction energy, equal to the work made against the magnetic field generated by a continuous magnetic moments distribution, to bring an elementary magnetic moment from infinity to its actual position. The magnetostatic field at a given location within the body depends on the contributions from the whole magnetization vector field. The magnetostatic energy is a nonlocal interaction and can be taken into account by introducing the appropriate magnetostatic field according to Maxwell equations for magnetized media. In a discrete model, the dipolar interactions between spins are computed by considering spins as point dipoles located at the sites of a lattice [4].

1.1.4 Zeeman interaction

It is the interaction energy of the magnetic moment of a system with the external field ${\mathit{H}}_{\mathit{ext}}$. It is defined as

$$E_z=-{\mu }_0\sum _{\mathit{r}}{\mathit{S}}_r\cdot {\mathit{H}}_{\mathit{ext}}.$$E4

2. Static of magnetic textures

2.1 Topological defects

Ordered phases of matter, such as magnetism or superconductivity in solid state systems, are characterized by a parameter space $V$. This is the space where the order parameter lives in and shapes the spatial and temporal features of the phase. In magnetism, $V={\mathbb{S}}^{n-1}$, $n$ is the number of magnetization components, and a single spin in three dimensions (3D) is represented by a vector $\mathbf{S}$. For $n=3$, the parameter space is the 2-sphere (two dimensional) ${\mathbb{S}}^2$. A $m$-sphere is the space of all elements, $x$, in ($m+1$)D satisfying $\mid x\mid =1$ [5].

A topological defect represents regions where the order parameter is not well defined or regions where it describes more complex structures in space. These objects can be explored by mathematical tools from homotopy theory [6, 7]. For every parameter space $V$, the $r\mathrm{th}$ homotopy group ${\pi }_r\left(V\right)$ exists, and each group element is a homotopy class. Topologically different spaces have different classes of equivalent objects. Members from different topological classes cannot be deformed into another [5].

One of the simplest homotopy groups is ${\pi }_1\left({\mathbb{S}}^1\right)$, the group of mappings from the 1-sphere (a circle) to the 1-sphere. ${\pi }_1\left({\mathbb{S}}^1\right)$ is called the fundamental group and is isomorphic to $\mathrm{\mathbb{Z}}$, which is the fundamental group of the circle. This means that we assign the number of times it goes around the circle to any loop [8]. Physically, it indicates all the possible ways to wrap a circle with a loop ($r=1$). A simple class is the trivial one that corresponds to the possibility of shrinking the loop to a point. The existence of nontrivial classes in a particular homotopy group indicates the presence of topological defects in the physical systems that realize it. Vortices, magnetic textures in two or three dimensions where the order parameter is null at the core, are good examples of the realization of ${\pi }_1\left({\mathbb{S}}^1\right)$.

In higher dimensions, the space of parameters features $V={\mathbb{S}}^m$, and the homotopy group becomes ${\pi }_n\left({\mathbb{S}}^m\right)$. The most notorious example in magnetism occurs in two dimensions and corresponds to skyrmion textures [9, 10]. Here, the parameter space is a $2$-sphere, and two angles determine the coordinate space. Together they configure the second homotopy group ${\pi }_2\left({\mathbb{S}}^2\right)$ which is, again, isomorphic to $\mathrm{\mathbb{Z}}$. In other words, the order parameter $\mathbf{S}$ takes values from the coordinate space, parameterized by the spherical angles $\theta$ and $\varphi$, that is, ${\mathbb{S}}^2$, and sends them to parameter space ${\mathbb{S}}^2$. The winding number is the number of times the order parameter wraps ${\mathbb{S}}^2$. Extensions of it include higher-dimensional skyrmions [11], monopoles [12], and Hopfions [13].

Generally, for Ising, XY, and Heisenberg magnets, the nontrivial homotopy groups are the following. Ising: ${\pi }_0\left({\mathbb{S}}^0\right)={\mathrm{\mathbb{Z}}}_2\equiv 0,1$. In the case of Ising spins, topological defective surface exists in 3D, topological line defects in two dimensions, and topological point defects in 1D. For XY spins ${\pi }_1\left({\mathbb{S}}^1\right)=\mathrm{\mathbb{Z}}\equiv 0,+1,-1,+2,-2,\dots$ there exist topological linear defects in 3D, and topological point defects in 2D, examples of topological point defects in two dimensions are the Kosterlitz-Thouless vortices and antivortices [14]. For Heisenberg spins, ${\pi }_2\left({\mathbb{S}}^2\right)=\mathrm{\mathbb{Z}}$. This includes topological defects in the form of points in 3D, as the Bloch point [15]. The Bloch point and the vertical Bloch line are elementary topological defects in topological textures like domain walls. In the Bloch point, every magnetization orientation appears once exactly. The vertical Bloch line is a topological soliton and separates two different magnetization directions inside domain walls. The singular vortex, a topological point defect for XY spins, consists of a cut through the length of a Bloch point [15].

Magnetic solitons, nonlinear excitations in magnetic systems, are shape-preserving and self-localized magnetic structures. Typical examples include the magnetic vortex, bubble, skyrmion, and domain wall [12]. They have a small size and high stability and can be reconstructed from elementary topological defects. The topological constraints from the associated homotopy group directly influence the properties of materials by catalyzing or inhibiting the switching between different magnetic-ordered states by means of topological defects present in magnetic solitons [16]. Consider the homotopy group ${\pi }_2\left({\mathbb{S}}^2\right)$. For Heisenberg spins, it considers a topological point defect in a three-dimensional medium and a topological soliton for a medium in 2D. The transformation from one topological soliton into another occurs through the exchange of topological defects with topological indices equal to the difference between the indices of the two solitons.

In the following, we will focus on the origin, stabilization, and dynamics of domain walls and two-dimensional skyrmions.

2.2 Domain walls

Domain walls (DWs) are transition regions between different domains. A domain wall arises when the boundary conditions are not uniform. This is the case of finite samples, where boundary conditions are imposed by the energetically preferred magnetization orientations at the edges [17], like in disks and nanostrips.

Whether or not a DW can be continuously transformed one into the other depends on the nature of the first homotopy group of the order parameter space ${\pi }_1\left(V\right)$: if it is trivial, then all walls are topologically equivalent, but if it is not, then topologically different walls are allowed to exist. The only nontrivial case occurs for the XY spins, ${\pi }_1\left({\mathbb{S}}^1\right)$.

2.2.1 Domain walls in wires

For XY spins, there is an infinite number of topologically different DWs. This is not the case for Heisenberg spins where the magnetization vector is allowed to tilt out of the plane and each DW path can continuously deform into another within the unit sphere surface of the magnetization vector $\mathit{M}$. An example of DW made out of XY spins in a wire is the $180-$DW, which separates two adjacent oppositely oriented domains. In the case of domains with collinear spins, they can be of the head-to-head or tail-to-tail types [18]. For domains with parallel moments, there are the up-down, down-up Neel DW and the up/down, down/up Bloch DW [19]. Using a polar angle to an axis orthogonal to the wire, a topological index or winding number (or topological charge) for these DW can be defined

$$W=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }\frac{\mathit{d\theta }}{\mathit{dz}}\mathit{dz}=\frac{\theta \left(\infty \right)-\theta \left(-\infty \right)}{2\pi }.$$E5

W describes how many times $\overrightarrow{m}$ wraps the unit circle when the spatial coordinate sweeps the magnetic wire. For head-to-head (tail to tail) walls, $W=1/2$ ($W=-1/2$). In the presence of an external field along the wire, a DW with $W=1/2$ ($W=-1/2$) moves along (against) the field to lower the Zeeman energy. When present at the edges of magnetic films, these domain walls, combined with other topological defects, may give rise to composite domain walls as the transverse and vortex domain walls. For Heisenberg spins in a wire, all DWs with the same boundaries have the same winding number.

$W$ of a single domain is zero, and thus, it is defined as topologically trivial. The 1D model is a good approximation for small-diameter (much less than exchange length and the DW width) nanowires or large bulky magnets whose magnetization varies only along one direction.

2.3 Elementary defects in flat magnets

In nanomagnets, elementary defects consist of vortices with integer winding numbers and edge defects with half-integer winding numbers. In this framework, domain walls are composite objects made out of two or more elementary defects [20].

For two-dimensional XY finite magnets, the bulk winding number along a path $\mathcal{C}$ in ${\mathrm{ℝ}}^2$ is $W=\frac{1}{2\pi }{\int }_{\mathcal{C}}\nabla \theta \cdot \mathit{dr}$. If $W=0$, there is no singularity inside $\mathcal{C}$. If $W=\pm 1$, there is a singularity inside $\mathcal{C}$ called vortex (antivortex). At the edge of the strip, $\mathcal{C}$ can be a segment of such edge, and the winding number is modified to $W=\frac{1}{2\pi }{\int }_{\mathcal{C}}\nabla \left(\theta -{\theta }_c\right)\cdot \mathit{dr}$, where ${\theta }_c$ is the angle of the sample edge. $W$ takes fractional values of $\pm 1/2$ in edge defects. The sum of all winding numbers over the bulk and edge of the sample is conserved. The total winding number of bulk and edges is a topological invariant [21].

In a ferromagnet without intrinsic anisotropy, the magnetic energy consists of the exchange contribution and the magnetostatic energy. The magnetic field and magnetization vector are related through Maxwell equations. Analytical treatment aimed to find magnetic configurations of the lowest energy is possible in a thin film limit where magnetization lies in the plane of the film. In this case, the magnetic energy becomes a local functional of the magnetization composed of two terms: the exchange term, which corresponds to the XY model with the ground states obeying the Laplace equation ${\nabla }^2\theta =0$ in bulk, and the term that expresses the magnetostatic energy due to magnetic charges at the film edge and sets the boundary conditions [22]. In the limit when the width $w$ of the sample (with thickness $t$ and exchange length $\lambda$) is larger than the effective magnetic length $\mathrm{\Lambda }=4{\mathit{\pi \lambda }}^2/tlog\left(w/t\right)$, magnetization at the edge is forced to be parallel to the boundary, in the thin film geometry $\left(t\ll w\right)$ [20].

Solutions in this limit have been constructed by analogy with electrostatic in two dimensions [20]. Angle gradients are associated with components of the electric field, and topological defects with winding numbers $+1$ and $-1$ become positive and negative point charges with unit strength. In the presence of an edge, there are two classes of solutions: (1) single vortices, which are repelled from the boundary, and (2) half vortices, also called edge defects, with a singularity at the edge and winding number $\pm 1/2$. A superposition of two edge defects with winding number $-1/2$ and one vortex with charge $+1$ has zero total topological charge [20]. The conservation of a topological charge constrains the creation and annihilation of DW.

2.4 Composite domain walls in soft magnetic strips and rings

Domain walls in magnetic strips and rings are composite objects made of two or more elementary defects: vortices with integer winding numbers ($W=\pm 1$) and edge defects with fractional winding numbers ($W=\pm 1/2$) [22]. This framework provides a basic understanding of the complex switching processes observed in ferromagnetic nanoparticles [21]. In ferromagnets, the competition between exchange and magnetic dipolar energies creates nonuniform magnetization patterns in the ground state: whereas the exchange energy favors a state with uniform magnetization, magnetic dipolar interactions align the magnetization vector with the surface. In a large magnet, a compromise is reached by forming uniformly magnetized domains separated by domain walls.

2.4.1 Transverse DW

In a strip of with $w$ and thicness $t$, a domain wall interpolates between the $\theta =0$ and $\theta =\pi$ ground states and can be constructed out of two edge defects with opposite winding numbers. Its solution is $tan\left(\theta \left(x,y\right)\right)=\pm \frac{\mathit{cos}\left(\mathit{\pi y}/w\right)}{sinh\left(\pi \left(x-X\right)/w\right)}$ [22]. From the electrostatic analogy, the two defects experience an attractive Coulomb force (not strong enough to overcome edge confinement). This force holds the composite domain wall together [21] (see Figure 1). In wider and thicker strips ($\mathit{wt}>{\lambda }^2$), the magnetostatic energy breaks the symmetry between the defects with positive and negative winding numbers. In particular, the $+1/2$ edge defect has higher magnetostatic energy than their $-1/2$ counterparts [23].

Figure 1.

Vortex DW consisting of two edge defects and one vortex defect. Transverse DW consisting of two edge defects with the same winding number. Narrow nanoring after the applied field $\overrightarrow{B}$ has created two domain walls made out of two edge defects each.

2.4.2 Vortex DW

Vortex walls are stabilized when a strip’s width and thickness substantially exceed the exchange length [24]. In this limit, the magnetostatic energy is the dominant contribution to the energy of a domain wall and the primary force determining the shape of topological defects. In the geometry of a thin film, the magnetostatic energy is minimized if the density of magnetic charges vanishes in bulk, and if on the surface the magnetization follows the contour. Domain walls in this limit are made of two $-1/2$ edge defects and a $+1$ vortex between them. Although different in shape, the $-1/2$ edge defects in the exchange and magnetostatic limits have identical topological properties (see Figure 1).

Transverse and vortex walls have the same total winding number equal to zero. Thus, the transverse wall and the vortex wall are topologically equivalent.

Conservation of topological charge has important implications for magnetization dynamics in nanomagnets. When a magnetic field is more significant than a specific value, but well below the Walker breakdown [25], a series of DW structure transitions occur to preserve the total winding number of the DW. For instance, edge defects with $-1/2$ winding number can only give birth to an antivortex of winding number $-1$ after changing its own winding number to $+1/2$ [26]. In rings, the ground state contains no topological defects and has zero magnetic dipole moment. The ring can be magnetized by applying an in-plane magnetic field. Switching off the field leaves the ring in a metastable state with remnant magnetization containing two composite domain walls [26]. By applying the magnetic field in the opposite direction, the walls can be set in motion on a collision course and may annihilate, leaving the magnet in the ground state. However, direct annihilation of two domain walls is impossible since both have edge defects with winding number $-1/2$ at the inner edge of the ring. Therefore, in thin and narrow rings where vortices are forbidden energetically, the two domain walls do not annihilate but instead form a $360-$DW (see Figure 1). In thicker and wider rings, annihilation does occur, and the ring returns to a ground state. In this case, since the edge defects cannot migrate into the bulk, they alter their signs by exchanging a vortex. The $+1/2$ defect emits a $+1$ vortex into the bulk and converts into a $-1/2$ defect [27].

2.5 Defects in disks with XY and Heisenberg spins

In soft magnetic films in two dimensions, the magnetostatic energy dominates the anisotropy terms, so the magnetization is confined to the plane of the film. Perfect confinement means that spins are $\mathit{XY}$. As confinement is not perfect in actual samples (with finite magnetization and nonzero exchange), topological defects as the vortex with topological charge $W=+1$ and the antivortex with $W=-1$ are regularized by having the core magnetization in the third dimension, over a size proportional to the micromagnetic exchange length $\mathrm{\Lambda }=\sqrt{2A/\left({\mu }_0{M}_s^2\right)}$, where A is the exchange constant and $M_s$ is the saturation magnetization. Owing to the nonzero winding number, an isolated vortex cannot be a localized object in an infinite film. However, they can exist locally in confined geometries such as nanodisks. Indeed, these structures are topologically stable if the magnetization is assumed to stay in the plane at infinity. This is the case of a disk with magnetization at the edge in the plane. Physically, as the edge constraint arises from magnetostatics, the avoidance of surface magnetic charges dictates that $\overrightarrow{m}$ be tangent to the edge. Therefore, for a disk-shape sample, the stabilized winding number is 1. For Heisenberg spins inside a disk, the core of the vortex is regular with purely out-of-plane moment $m_z\sim p=\pm 1$, with p the polarity of the vortex. Because of the topology of the edge magnetization, the vortex cannot be continuously erased into a uniform structure, and the polarity cannot continuously change from $+1$ to $-1$. Thus, the constraints at the magnetic film’s edge let other topological structures appear. The Heisenberg vortex or antivortex is also known as meron [7].

2.6 Skyrmion textures

Skyrmions, swirling spin textures appearing in chiral magnets, are magnetic structures in which the spins point in all the directions wrapping a sphere [9, 28]. These topological spin textures have ignited a growing interest in spintronics [10] due to their rich phenomenology as well as novel potential applications [29]. Their nanoscale size, topologically protected stability [28], and the very low electric current densities needed to displace them [30] are among the best qualities that make them attractive candidates for information carriers in high-density data-storage technologies. They have been explored in many magnetic materials, also named chiral magnets, for example, MnSi [30, 31, 32], Fe${}_{1-x}$Co${}_x$Si [33, 34, 35], Mn${}_{1-x}$Fe${}_x$Ge [36], FeGe [37], La${}_{0.5}$Ba${}_{0.5}$MnO${}_3$ [38], and CuOSeO${}_3$ [39], among others [12]. Spontaneous skyrmion phases have been synthesized using temperature and external magnetic fields. Their detection is typically carried out by neutron scattering [31], Lorentz transmission electron microscopy (LTEM) [37], and spin-resolved scanning tunneling microscopy [40] experiments.

The spin texture of magnetic skyrmions can originate from various mechanisms. Depending on the particular magnetic system, skyrmions can be stabilized by long-ranged dipolar, Dzyaloshinskii-Moriya, frustrated exchange, or four-spin exchange interactions [10]. In chiral magnets, spin orbit coupling and lack of inversion symmetry cause the appearance of the Dzyaloshinskii-Moriya interaction [2, 3]. This interaction is responsible for the stability of static skyrmions in two and three dimensions [28]. In magnetic systems, single skyrmions as well as the skyrmion crystal can become the lowest energy configuration by adjusting temperature and the applied magnetic field.

A (single) skyrmion configuration is a texture described by the direction of spin $\mathit{S}\left(\mathit{r}\right)$ at spatial position $\mathit{r}=\left(x,y\right)$, where the spin at the core points down, while at the perimeter point up, as shown in Figure 2. The topological skyrmion number is defined as a measure of the wrapping of $\mathit{S}\left(\mathit{r}\right)$ around a unit sphere [5]

Figure 2.

Schematic profile for single (left) and lattice (right) of magnetic skyrmions. The color indicates the relative direction of each spin respect to the vertical axis. Spins in dark red are pointing outward and those in blue are directed inward.

$$Q=-\frac{1}{4\pi }\int \mathit{m}\cdot \left(\frac{\partial \mathit{m}}{\partial x}\times \frac{\partial \mathit{m}}{\partial y}\right)d^2\mathit{r}$$E6

where $4\pi$ accounts for the surface of the sphere and $\mathit{m}$ is the local normal to the sphere, both partial derivatives of $\mathit{m}$ belong to the local tangent plane of the sphere and the modulus of their vector product is the sine of their angle. In Eq. (6), the integrand is the solid angle spanned by $\mathit{m}$ when one moves all spins to the Bloch sphere center in the spin space. In the case of cylindrical symmetry, with the spherical angles relative to the magnetization at infinity being sole functions of the radius r and the in plane angle, a useful relation is obtained between $Q$, $p$ (the polarity of the core of the structure) and $S=\frac{1}{2\pi }{\int }_0^{2\pi }\frac{d\mathrm{\Phi }}{\mathit{d\varphi }}\mathit{d\varphi }$, the winding number of the planar magnetization component: $Q=\mathit{Sp}$.

Topological protection does not mean absolute stability. The energy barrier between a skyrmion state and the single domain state is finite. Skyrmions can be generated from the edges of a sample without breaking the conservation of topology and from the bulk with the broken conservation of topology [41].

Up to now, we have discussed the statics of magnetic textures. We show next that the dynamics of magnetic textures can be affected by topology. Magnetization dynamics in continuous media is governed by the Landau-Lifshitz-Gilbert (LLG) equation supplemented by additional torque terms in the presence of spin-polarized currents. In this equation, the energy density gives rise to an effective magnetic field defined by a variational equation. This field originates from all the energetic contributions described above. Consequently, in most cases, the Landau-Lifshitz-Gilbert equation is not solvable analytically. Thiele’s equation gives an approximate analytical solution for the dynamics of magnetic textures. To illustrate these ideas, we consider the dynamics of skyrmions next.

2.7 Dynamics and fluctuations of skyrmion textures

Skyrmions can be driven by external forces derived from charge currents, thermal gradients, and magnetic fields, among others. Particularly, current-driven skyrmion dynamics displays interesting topological transport properties, such as the Skyrmion Hall effect [32]. This effect results from the so-called spin-transfer torques phenomena, which is a torque exerted by carrier spins on the magnetization [42]. Driven skyrmion dynamics can also be triggered by mechanisms such as thermal gradients [43], inhomogeneity in the fields [29, 44], or magnon currents [45].

A first approximation captures the evolution of skyrmions in terms of a reduced set of collective coordinates. This approach is general and helps to parameterize the magnetization of the texture in terms of few degrees of freedom. Through the Landau-Lifshitz-Gilbert (LLG) equation, the skyrmion dynamics is mapped to a particle-like equation of motion, also known as Thiele’s eq. [46]. Next, we outline this approach.

The starting point is the Landau-Lifschitz-Gilbert (LLG) eq. [47, 48, 49] for the magnetization direction $\mathit{M}$, which incorporates spin-transfer torques [42, 50]. This torque splits into adiabatic and nonadiabatic [42, 51, 52, 53] contributions, defined as $-{\mathit{v}}_s\cdot \nabla \mathit{M}$ and $\beta {\mathit{v}}_s\cdot \nabla \mathit{M}$, respectively. Here, ${\mathit{v}}_s=-\left({\mathit{pa}}^3/2\mathit{eM}\right)\mathit{j}$ is the spin velocity of the conduction electrons, $p$ is the spin polarization of the electric current, $e\left(>0\right)$ the elementary charge, and $\beta$ a dimensionless parameter that accounts for the nonadiabaticity of the spin transfer.

The LLG equation reads

$$\left(\frac{\partial }{\partial t}+{\mathit{v}}_s\cdot \nabla \right)\mathit{M}=\mathit{M}\times {\mathit{H}}_{\mathrm{eff}}+\alpha \mathit{M}\times \left(\frac{\partial }{\partial t}+\frac{\beta }{\alpha }{\mathit{v}}_s\cdot \nabla \right)\mathit{M},$$E7

where ${\mathit{H}}_{\mathrm{eff}}$ is the effective field and $\alpha$ is the Gilbert damping constant. To describe the dynamics of single skyrmions, a particle-like motion [54, 55, 56] is considered (this approach is also useful for the case of domain walls [24]). It consists of a parametrization of the magnetization vector $\mathit{M}$ in terms of collective coordinates. For a single skyrmion moving rigidly along the trajectory $\mathit{x}\left(t\right)$, we take as an ansatz the profile $\mathit{M}\left(\mathit{r},t\right)={\mathit{M}}_0\left(\mathit{r}-\mathit{x}\left(t\right)\right)$, where ${\mathit{M}}_0$ represents the static skyrmion texture centered at the origin. The skyrmion profile ${\mathit{M}}_0$ is obtained by minimizing the magnetic energy. Plugging in the ansatz on the LLG equation and integrating over complete space, we get for the skyrmion dynamics the Thiele’s equation

$$a_{\mathit{ij}}{\dot{x}}^j\left(t\right)+F_i\left(\mathbf{x}\left(t\right)\right)=0,$$E8

with the $2\times 2$-matrix $a_{\mathit{ij}}=-{\epsilon }_{\mathit{ikj}}{g}_k+\alpha {\mathcal{D}}_{\mathit{ij}}$. Here, indices $i,j,\text{and}k$ denote coordinates $x,y,\text{and}z$, respectively, ${\epsilon }_{\mathit{ijk}}$ is the Levi-Civita symbol, and summation over repeated indices is assumed. The first term in $a_{\mathit{ij}}$ contains the gyromagnetic vector defined by

$$g_i=-\frac{1}{2}{\epsilon }_{\mathit{ijk}}{G}_{\mathit{jk}},$$E9

where $G_{\mathit{ij}}=\int d\mathit{r}{\epsilon }_{\mathit{klm}}{M}_0^k{\partial }_i{M}_0^l{\partial }_j{M}_0^m$. This term in the equation of motion describes the Magnus force [30] exerted by flowing electrons. For a single skyrmion $g_i=g{\delta }_{\mathit{iz}}=4\mathit{\pi Q}$, where $Q$ is the skyrmion charge, that for our case is $Q=-1$. On the other hand, the second contribution represents the dissipative force whose components are $D_{\mathit{ij}}=\int d\mathit{r}{\partial }_i{M}_0^k{\partial }_j{M}_0^k$, which obeys for the single-skyrmion case $D_{\mathit{ij}}=D{\delta }_{\mathit{ij}}$ because of the symmetry of the spin configuration.

The drift velocity of the skyrmion is affected, on one side, by a force $\mathit{F}$ given by $F_i\left(\mathit{x}\right)=\left[{\epsilon }_{\mathit{ijk}}{g}_j-\beta \mathcal{D}{\delta }_{\mathit{ik}}\right]v_s^k-\partial V\left[\mathbf{x}\right]/\partial x_i,$ that is explicitly written as

$$F_x=\left(-{\mathit{gv}}_s^y-\beta {\mathit{Dv}}_s^x\right)-\partial V/\partial x,$$E10

$$F_y=\left({\mathit{gv}}_s^x-\beta {\mathit{Dv}}_s^y\right)-\partial V/\partial y,$$E11

containing both the gyrotropic and dissipative contribution due to the electron’s coupling and a force due to the potential $V\left[x\right]$ from the surrounding environment, for example, magnetic impurities, local anisotropies, or geometric defects. Potential $V\left(\mathit{x}\right)=V_H\left(\mathit{x}\right)+V_A\left(\mathit{x}\right)$ derives from an inhomogeneous magnetic field $\mathbf{H}\left(\mathbf{x}\right)$ coupled to the magnetization of the ferromagnet and a position-dependent perpendicular anisotropy $A\left(\mathbf{x}\right)$, with $V_H\left(\mathit{x}\right)=-\frac{1}{\mathrm{\hslash }}\int d\mathbf{r}{M}_k\left(\mathbf{r}-\mathbf{x}\right)H_k\left(\mathbf{r}\right)$ and $V_A\left(\mathit{x}\right)=-\frac{1}{\mathrm{\hslash }}\int d\mathit{r}{M}_z^2\left(\mathbf{r}-\mathit{x}\right)A\left(\mathit{r}\right)$, respectively. The motion of single skyrmions is, therefore, determined by the solutions of Thiele’s equation, with the forces given by Eqs. (10) and (11). Generalizations are straightforward and include random dynamics due to thermal fluctuations or disorder and noncoherent dynamics, that is, the motion with deformable shapes. The last leads to inertial terms in Thiele’s equations, with an effective mass quantifying the degree of deformation.

We conclude this section by remarking that skyrmions have better stability than vortices. However, as we have seen, their nontrivial topological number leads to a finite gyrovector in the Thiele equation and, consequently, to the existence of a transverse motion in longitudinal driving force. Thus, there exists a threshold current density above which skyrmions can annihilate at the film edge. This edge effect strongly limits the speed of skyrmion propagation, which is vital for real applications.

2.8 3D skyrmions and beyond

Generalizations of 2D skyrmions are diverse. Particular efforts have been directed to realize three-dimensional textures in magnetic materials [12]. As an example we mention the magnetic hopfion [13, 57] and the three-dimensional skyrmion [11], a class of textures classified according to homotopy groups ${\pi }_2\left({\mathbb{S}}^3\right)$ and ${\pi }_3\left({\mathbb{S}}^3\right)$. These topological textures are characterized by the Hopf charge $Q_H$, given by

$$Q_H=\int d\mathit{rA}\left(\mathit{r}\right)\cdot \mathit{B}\left(\mathit{r}\right),$$E12

where $\mathit{B}\left(\mathit{r}\right)={\epsilon }_{\mathit{ijk}}\mathit{S}\cdot \left({\partial }_j\mathit{S}\times {\partial }_k\mathit{s}\right)$ is the emergent magnetic field and $\mathit{A}\left(\mathit{r}\right)$ is the associated vector potential $\mathit{B}=\nabla \times \mathit{A}$. And the skyrmion number $Q_s$

$$Q_s=\frac{1}{24{\pi }^2}\int d\mathit{r}{\epsilon }^{\mathit{\mu \nu \lambda }}\mathrm{tr}\left({\mathcal{L}}_{\mu }{\mathcal{L}}_{\nu }{\mathcal{L}}_{\lambda }\right),$$E13

where ${\mathcal{L}}_{\mu }={\mathcal{R}}^{-1}{\partial }_{\mu }\mathcal{R}$ and $\mathcal{R}$ is a rotation matrix that locally aligns each spin along the quantized axis. These spin textures represent sophisticated structures even in their simplest form $Q_H=Q_s=1$. Hopfions and 3D skyrmions lead to interesting dynamical properties such as the current-induced dynamics [13], field-driven resonance modes [58], and thermally activated drag of spin current [11]. 3D structures provide a versatile platform to explore the role of emergent phenomena. However, questions regarding the stability, nucleation, and low-energy dynamics are still under scrutiny and deserve special attention.

3. Topology in the dynamics of linear magnetic excitations

In Landau’s formalism of phase transition theory [59], all phases of matter can be characterized by a local order parameter, and the phase transition between different phases corresponds to a symmetry breaking. The discovery of the integer and fractional quantum Hall effects enabled the realization of a new phase of matter beyond Landau’s paradigm. This new phase should be described by topological invariants and was dubbed the topological phase. The topological invariants remain unchanged under the adiabatic deformations of the system as long as the bulk gap is not closed. Topological invariants arise from geometrical phases, as we show next.

3.1 Geometric phases, Zak phase, and Chern number

Whenever a quantum system undergoes a cyclic evolution governed by a slow change of parameters, it acquires a phase factor: the geometric phase. Unlike dynamical phases, geometrical phases are not attributed to forces applied to quantum systems. Instead, they are associated with the connection of space [60]. The Aharonov-Bohm phase [61] is a particular case of the geometric phase. It is the relative phase acquired by two electronic wavepackets encircling a magnetic field confined to a solenoid, so that along their paths, the magnetic field is zero, but the vector potential is nonzero. The Aharonov-Bohm phase affects their interference pattern when closing the loop and is proportional to the enclosed magnetic flux. The Aharonov-Bohm phase is topological. As such, it does not depend on the shape or the geometric properties of the particle path, but only on its topological invariants, provided that the particle is moving in a field-free region.

The general form of the geometric phase, the Berry phase, was introduced by Michael Berry [62], who showed that geometric phases arise purely from the adiabatic evolution of a quantum state $\mid n\left(\mathit{R}\right)⟩$ upon the modification of the parameters $\mathit{R}$. Geometrical phases are related to the topological structure formed by the Hilbert space of a quantum system with a Hamiltonian $\mathrm{\mathscr{H}}$, and the space of its adiabatically varied parameter $\mathit{R}$, $\mathrm{ℳ}$. The evolution of the state $\mid n\left(\mathit{R}\right)⟩$ is tied to the Berry connection, a generalization of the vector potential $\mathit{A}=i⟨n\left(\mathit{R}\right){\nabla }_{\mathbf{R}}|\mathbf{n}\left(\mathit{R}\right)⟩$. $A$ gives rise to the Berry phase, $\gamma$, after integration along a closed path. Connections allow the differentiation of wavefunctions over $\mathit{R}$ by providing a unique way of dragging them from one Hilbert space in $\mathrm{ℳ}\times \mathrm{\mathscr{H}}$ to another. This process is known as parallel transport and can occur provided that a smooth path between both spaces is provided.

For electrons in crystalline systems, the Bloch theorem is applicable regardless of the form of the potential. The Bloch theorem states that the energy eigenstates of an underlying periodic Hamiltonian, known as Bloch states, are given by the wavefunction $\mid {\psi }_{n,\mathit{k}}\left(\mathit{r}\right)⟩=e^{i\mathit{k}\cdot \mathit{r}}\mid u_{n,\mathit{k}}\left(\mathit{r}\right)⟩$. Here, $n$ denotes the energy band index, $\mathit{k}$ is the electronic quasi-momentum, and $u\left(\mathit{r}\right)$ is a periodic function with the periodicity of the underlying Bravais lattice vector $\mathit{R}$, $u\left(\mathit{r}+\mathit{R}\right)=u\left(\mathit{r}\right)$. Consider an isolated band. Following $\mathit{k}\mathbf{=}\mathit{k}\left(\mathit{t}\right)$ throughout the Brillouin zone, the Bloch state of an electron traverses a closed path in the reciprocal quasi-momentum parameter space, and a geometric phase emerges. Now, space of parameter $\mathit{R}$ corresponds to the adiabatic evolution of the original Bloch state $\mid {\psi }_{n,\mathit{k}}\left(\mathit{r},t_0\right)⟩$ through a closed loop in the first Brillouin zone [63]. This results in the geometric (Berry) phase, $\gamma =\int \mathit{A}\left(k\right)\cdot \mathit{d}k$, where $\mathit{A}$ is the Berry connection for a given energy band, $-i⟨u_k|{\mathbf{\text{∇}}}_{\mathit{k}}|u_k⟩$, a gauge-dependent parameter that does not correspond to an observable. Nevertheless, the geometric phase is gauge invariant. The Berry curvature, defined as the curl of the Berry connection $\mathbf{\Omega }\left(\mathit{k}\right)={\mathbf{\text{∇}}}_{\mathit{k}}\times \mathit{A}\left(k\right)$, is a measure of the local rotation of the electronic wavepacket as it transverses the Brillouin zone and like the connection is gauge dependent. Through the Stokes theorem, one can write the geometric phase as an integral over the manifold of the Berry curvature. If such a manifold is closed (a torus, for instance), then the result is topological and quantized by a $2\pi$ multiple of the topological index named Chern number.

The Berry phase presented above applies to generic systems described by a parameter-dependent Hamiltonian. It is a topological invariant that can be used to judge whether or not the system is a first-order topological insulator.

If the Berry phase is computed over a noncontractible loop of the Brillouin zone torus, it is known as the Zak phase, $\mathcal{Z}={\int }_{L}d\mathit{k}\cdot {\mathit{A}}_n\left(k\right)$(mod$2\pi$). Zak [64] introduced the concept of geometric phase for Bloch electrons in one-dimensional periodic lattices in 1989. Zak phase is useful for identifying the topological phase in a one-dimensional system, where the integral interval goes from 0 to $2\pi /a$, with $a$ the lattice constant. When the one-dimensional system has inversion symmetry, the nonzero Zak phase indicates that the system is in a topological phase. Besides the Berry phase, the Chern number is another typical topological invariant adopted to describe first-order topological insulators. It is expressed as the integration of the Berry curvature in the Brillouin zone $C_n=\frac{1}{2\pi }{\int }_{\mathit{BZ}}{d}^{2}k{\mathrm{\Omega }}_{n,z}\left(\mathbf{k}\right)$ [63].

The Chern number is an integer. For $C_n=0$, the system is in a trivial phase. If the system supports a nontrivial topological phase, then $C_n$ is a nonzero integer [65]. For time-reversal symmetric systems, the Chern number is always zero. To characterize first-order topological insulator in time-reversal symmetric systems, the $Z_2$ invariant is often used [63]. For higher-order topological insulators, typical topological invariants include bulk polarization and ${\mathrm{\mathbb{Z}}}_N$ Berry phases.

The definition of the Chern number presented above is used in fermionic systems. Because magnons are bosons, the expression of $C_n$ must be adapted for magnetic systems. The Hamiltonian of bosons can be expressed in terms of a para-unitary matrix $T_k$ instead of a unitary matrix. A projection operator in the vector space can be defined in terms of $T_k$, $Q_n=T_k{\mathrm{\Gamma }}_n{\sigma }_3{T}_k^{†}{\sigma }_3$, where ${\mathrm{\Gamma }}_n$ is a diagonal matrix taking $+1$ for the $n$th diagonal component and zero otherwise, and a diagonal matrix ${\sigma }_3$ takes $+1$ in the particle space while $-1$ in the hole space. Then, the Berry curvature for bosons is given by $B_n=i{\epsilon }_{\mathit{\mu \nu }}\mathit{Tr}\left(Q_n\left({\partial }_{k_{\mu }}{Q}_n\right)\left({\partial }_{k_{\nu }}{Q}_n\right)\right)$, and the Chern number in magnonic systems becomes $C_n=\frac{1}{2\pi }{\int }_{\mathit{BZ}}d\mathbf{k}{B}_n\left(\mathit{k}\right)$ [63, 66].

3.2 Spin waves and magnons

The most peculiar characteristic of topological phases, like the topological insulator [67], is that they can support chiral edge (in 2D lattices) and surface (in 3D lattices) states, which are absent in conventional insulators. The topological edge and surface states are wave modes that are confined at the boundary/surface of the system and generally have a specific chirality. These properties are topologically protected, enabling them to be immune to moderate disorders and defects.

The bulk-boundary correspondence dictates that the bulk property of topological insulators [68] determines the character of edge or surface modes. A first-order $m$-dimensional topological insulator has $\left(m-1\right)$-dimensional topological edge/surface states. Higher-order $m$-dimensional $k$-order topological insulators, on the other hand, allow for $\left(m-k\right)$-dimensional topological boundary modes $\left(2\le k\le n\right)$, such as corner states and hinge states.

In the magnetic arena, there has been intensive research on topological magnetic states in the last decade. Magnons, quanta of spin waves, propagate localized spin fluctuations in solids. Research associated to the detection, and manipulation of magnons is often called magnonics. Topologically protected unidirectional surface spin waves present advantages over their trivial counterparts because they are robust against internal and external perturbations. The study of the topological properties of magnons began in 2010 with the experimental observation of the magnon Hall effect (MHE) in the insulating pyrochlore ferromagnet ${\mathrm{Lu}}_2{\mathrm{V}}_2{\mathrm{O}}_7$ [69]. The topological transport of charge-neutral excitations in insulators produces a thermal analog of the topological Hall effect by electrons. A model of uncompensated net magnon edge currents was proposed to explain the emerging MHE where the longitudinal temperature gradient induces a transverse thermal current, [66, 70]. The concept of a topological magnon insulator was then adopted to describe a large class of systems that support chiral magnon current circulating around their boundaries. The MHE was subsequently observed in garnet magnets (for example, yttrium iron garnets), kagome and pyrochlore magnets, and frustrated pyrochlore quantum magnets [66, 68].

Topological bands also arise in gapless magnetic systems. This is the case of topological semimetals, which include the Dirac and Weyl semimetals. Weyl semimetals are characterized by degenerate points, denoted Weyl points resulting from the linear crossing of two bands. Weyl points come in pairs with opposite chiralities. The band inversion happens between two paired Weyl nodes, leading to the generation of topologically protected Fermi-arc-like surface states. Weyl semimetals can also support higher-order topological edge states [71] and emerge when at least one of the two, that is, time reversal symmetry or inversion symmetry, is broken. On the other hand, Dirac points with fourfold degenerate band touchings characterize Dirac semimetals. Dirac semimetals respect both the time-reversal and inversion symmetries, while their stability requires additional crystalline symmetries, for example, the rotational.

Besides the topological magnon insulator and the Dirac and Weyl semimetals, there are other sources of topological effect in magnon systems, as is the case of the topological magnon polarons resulting from the spin-lattice coupling and the topological phases of magnon Bogoliubov-de Gennes systems [68]. Nowadays, the search for topological magnons in antiferromagnet and ferrimagnet is an active field of research [72].

The experimental and theoretical advances on topological magnons over the past decade have focused on finding the topological magnon insulator state in lattices with triangular motifs such as the kagome (or pyrochlore) and honeycomb lattices. Simultaneously, different microscopic mechanisms leading to nontrivial topology have been identified for such lattices. Examples include the Dzyaloshinskii-Moriya (DM) interaction due to the inversion symmetry broken, the magnetic dipolar interaction, the pseudodipolar exchange interaction, and internal fields from intrinsic magnetic textures.

3.3 Topological transport with magnons: The Magnon Hall effect

The MHE was observed experimentally in the insulating pyrochlore ferromagnet ${\mathrm{Lu}}_2{\mathrm{V}}_2{\mathrm{O}}_7$ [69]. In the experiment, a transverse heat current was observed when a temperature gradient was applied longitudinally. The experiment showed two peculiarities: i) the thermal Hall conductivity steeply increased and saturated in the low magnetic field regime. This behavior could not be explained by the normal Hall effect where the conductivity is proportional to the magnetic field strength. It resembles instead the anomalous Hall effect [73] due to spontaneous magnetization. ii) the thermal Hall conductivity decreased in the high field regime, which could not be attributed to phonons. It was concluded then that the transverse heat current was due to the MHE. The model included the ferromagnetic exchange interaction and a nonzero DM interaction induced by the spin-orbit coupling, which breaks the inversion symmetry. Subsequently, it was demonstrated that a magnon wave packet subjected to a temperature gradient acquires an anomalous velocity perpendicular to the gradient, which is associated with the magnon edge currents [74].

The thermal Hall conductivity originates from the Berry curvature ${\mathrm{\Omega }}_n\left(\mathit{k}\right)$ in momentum space. Therefore, the transverse thermal Hall conductivity${\kappa }^{\mathit{xy}}$ is given by

$${\kappa }^{\mathit{xy}}=-\frac{k_B^{2}T}{\mathrm{\hslash }v}\sum _{n,\mathit{k}}{c}_2\left({\rho }_n\right){\mathrm{\Omega }}_{n,z}\left(\mathit{k}\right)$$E14

where ${\rho }_n\left[{\epsilon }_n\left(\mathit{k}\right)\right]$ is the Bose distribution function, $k_B$ is the Boltzman constant, $v$ is the volume, and $T$ is the temperature. ${\mathrm{c}}_2\left({\rho }_n\right)=\left(1+\rho \right){\left(log\frac{1+\rho }{\rho }\right)}^2-{\left(log\rho \right)}^2-2{\mathrm{Li}}_2\left(-\rho \right)$, with ${\mathrm{Li}}_2\left(z\right)$ the polylogarithm function. In the MHE, when the system is in equilibrium, the edge magnon currents exist due to the confining potential, and they circulate along the boundary. The currents are equal at the two edges of a sample leading to a vanishing thermal current through the magnet. If a temperature gradient is applied, the magnons will flow from the high-temperature region to the low-temperature region, which breaks the balance of the heat current in the two opposite edges, leading to a finite thermal Hall current. Such edge magnon currents are spin waves chiral edge states resulting from the nontrivial topology of magnon bands. The one-way chiral edge transport is topologically immune to defects and disorder. Since the discovery of MHE in pyrochlore ferromagnetic insulators, the same effect has been observed in other magnetic materials. Examples include the frustrated pyrochlore quantum magnet ${\mathrm{Tb}}_2\mathrm{Ti}2{\mathrm{O}}_7$, YIG, and the kagome magnet $\mathrm{Cu}\left(1,3-\mathrm{bcd}\right)$ [63].

Noncollinear magnetic textures like skyrmions can generate a fictitious magnetic field, leading to the magnon thermal Hall effect. This Hall effect is due to the nonzero topological charge of magnetic texture and is called the topological magnon Hall effect (TMHE) [75].

Over the past decade, much effort has been devoted to studying the topological properties of magnons. Magnetic systems and models include a plethora of materials, lattices, interfaces, and heterostructures. Next, we present a few examples of topological magnon insulators and semimetals.

3.4 Magnetic topological matter: Magnon insulators, Dirac, and Weyl semimetals

3.4.1 Topological magnon insulators

Chern number is a property of the bulk of a system and determines the propagation direction and the number of topologically nontrivial edge modes. The sum of Chern numbers up to the $n$th band ${\nu }_n={\sum }_{j\le n}{C}_n$ is the winding number of the edge states in bandgap $n$. ${\nu }_n$ corresponds to the number of topologically nontrivial edge states in the $n$th bandgap and $sign\left({\nu }_n\right)$ determines their propagation direction.

The topological magnon insulator was predicted in the ferromagnetic kagome lattice with Heisenberg and Dzyaloshinskii-Moriya (DM) interactions [68]. The ferromagnetic kagome lattice allows four topologically different phases by tuning the parameters $J_{\mathit{NN}}/J_N$ and $D/J_N$, where, in this context, $J_N$ and $J_{\mathit{NN}}$ denote nearest-neighbor and next-nearest-neighbor (NNN) symmetric exchange interaction and $D$ is the nearest-neighbor antisymmetric interaction. In this case, the nontrivial topology of magnon in the kagome lattice is brought about by the strong spin-orbit coupling.

The magnetic dipolar interaction can also endow spin wave volume modes with nonzero Chern number. Recently, it has been found theoretically that dipolar zig-zag chains, one-dimensional lattices of point dipoles with a two-point basis, are building blocks of two-dimensional lattices with topological magnon bands. The system is a two-band model where the only coupling between spins is due to the long-ranged dipolar interactions. Dipolar coupling brings about topology by locking the spin’s degrees of freedom to the lattice, giving rise to an effective spin-orbit interaction Stripes, built from a finite number of chains, host topological chiral edge states. Tuning the vertical distance between zig-zag chains allows for the direct control of magnonic frequencies, the velocity of the chiral edges modes, and the transverse thermal conductivity. The Chern numbers of the two spin-wave volume bands are exchanged for a critical vertical distance due to two band touchings, which yield two monopoles and endow the Berry phase with a divergence. This triggers the exchange of the Chern numbers between the bands. Such topological phase transition causes the change of sign of both the Hall conductivity and the sense of motion of edge states in the system [76].

In artificial systems, the observation of topological insulator states has been recently realized in a tunable superconducting qubit chain, which exhibits both nontrivial topological invariants and topological edge states [63].

3.4.2 Dirac and Weyl semimetals

The Heisenberg magnet on the honeycomb lattice exhibits Dirac points. The Hamiltonian can be expressed as $H=-\sum J_{\mathit{ij}}{\mathit{S}}^A\cdot {\mathit{S}}^B$, where the summation runs over nearest neighbors, ${\mathit{S}}^A$ and ${\mathit{S}}^B$ are the spins for two different sublattices $A$ and $B$, and $J_{\mathit{ij}}$ is the exchange constant. Around the band degeneracy points $K$ and $K^{\prime }$, the dispersion relation is linear. If the system is an antiferromagnetic honeycomb lattice, the energy dispersion degenerates only at the $\mathrm{\Gamma }$ point, but the dispersion remains linear. For magnons, the time-reversal operator can be defined for the Bogoliubov Hamiltonian. As long as the pseudo spin time-reversal symmetry is preserved, the Dirac points exist in the Brillouin zone, but if the time-reversal symmetry is broken, a gap will open at the Dirac points, leading to a topological insulator. If the Hamiltonian only contains NN exchange interaction, the magnon band structure is gapless for the honeycomb ferromagnets, and if a NNN exchange interaction is considered, it shifts the positions of the Dirac points. However, if a DM interaction is introduced, the pseudo spin time-reversal symmetry of the system is broken, and a gap opens at the Dirac points. Chromium trihalides ${\mathit{CrX}}_3$ (X = F, Cl, Br, and I) are 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. The gap at the Dirac points in ${\mathit{CrI}}_3$ was observed experimentally [68] by using inelastic neutron scattering. The observation of a sizeable spin-wave gap indicates that the spin-orbit coupling plays a vital role in the physics of topological spin excitations not only in bulk honeycomb ferromagnet ${\mathit{CrI}}_3$ but also in its monolayers. By using inelastic neutron scattering method, Dirac magnons have also been observed in the three-dimensional quantum XY magnet ${\text{CoTiO}}_3$. A gap arises from the bond-anisotropic exchange coupling, due to quantum order by disorder, which pins the order parameter to the crystal axes [77].

The magnon bands in a magnonic Weyl semimetal are crossed in pairs at the points dubbed Weyl nodes. The Weyl nodes are monopoles of Berry curvature and are characterized by the integer topological charge, or chirality [68]. Because the net topological charges in the entire Brillouin zone must be zero, the Weyl nodes appear in pairs with opposite topological charges of $\pm 1$. In Weyl semimetals, the topologically protected chiral surface states between each pair of Weyl nodes exist on the system surfaces, and the equal energy contour of these surface states forms arcs, with the arc number equaling the number of paired Weyl nodes. The vital hallmark of magnonic Weyl semimetal is the magnon arcs on system surfaces. The pyrochlore (${\mathit{Lu}}_2{V}_2{O}_7$) ferromagnet with DM interaction is an intrinsic magnonic Weyl semimetal with the pair of Weyl nodes having opposite topological charges. In the model of such a system, the effective spin Hamiltonian includes NN exchange interaction, NN DM interaction, and Zeeman interaction. The magnonic chiral anomaly could be realized by applying inhomogeneous electric and magnetic fields perpendicular to each other: when a magnon moves its magnetic moment can interact with the electric field and it acquires an AC phase, while the magnetic field is used to drive the magnon flow. 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 [78, 79].

4. Conclusion

Topology is a wide and deep-rooted concept in physics. It manifests as defects in the order parameter, topological order in the ground state, or in the spectrum of excitation of ordered phases. Tools from topology and homotopy theory have proven indispensable for the prediction of a variety of new phenomena. In this chapter, we have focused on the topological properties in ordered media such as magnetic textures and low energy spin fluctuations.

We have introduced basic elements of homotopy theory to classify self-localized magnetic textures in two and three dimensions. As examples, we considered magnetic domain walls and skyrmions in two dimensions.

The complex structures of domain walls can be understood in terms of elementary topological defects, with winding numbers that determine the switching processes in nanomagnets. Skyrmions emerge as a single entity but also in ordered arrays. The dynamics of skyrmions, as a response to external forces from electric currents, thermal gradients, and magnetic fields, realizes deflected trajectories in analogy to the Magnus effect. The dynamics of skyrmions is determined by the Landau-Lifschitz-Gilbert equation, which, through reduction of degrees of freedoms, is simplified to Thiele’s equation. This framework embodies a rigid single-skyrmion dynamics, which can be generalized in multiple ways, in particular to include size deformations leading to inertial motions.

Spin waves, propagating excitations in magnetic materials, manifest too topological characteristics. This time the topology is rooted in the band spectrum of the system’s linear excitations rather than in the order parameter. The Chern number is one of the topological invariants that help out to characterize the topological character of the bands. Topological spin wave bands are usually accompanied with exotic phenomena such as the emergence of chiral edge states and the magnon Hall effect. Research on topological band theory constitutes the central tool for studying either bosonic or fermionic topological phases of matter.

Acknowledgments

P.M. thanks support from Fondecyt under Grant No. 1210083. R.T. would like to thank Sebastian Diaz for the preparation of images and fruitful discussions.

Conflict of interest

The authors declare no conflict of interest.