## Abstract

Electronic states in magnets are characterized by the quantum mechanical Berry phase defined in both the real and momentum spaces. This Berry phase constitutes the gauge fields, i.e. the emergent electromagnetic fields in solids, and affects the motion of the electrons. In momentum space, the band crossings act as the magnetic monopoles, i.e. the sources or sinks of the gauge flux. In real space, the spin textures with non-coplanar spin configurations produce the gauge field by the solid angle leading to the spin chirality. Skyrmion is the representative structure supporting this gauge field. A typical phenomenon reflecting this gauge field is the anomalous Hall effect, i.e. the Hall effect produced by the spontaneous magnetization combined with the relativistic spin–orbit interaction. We discuss a few examples recently studied related to these issues with some new results on skyrmion formation.

## 1. Introduction

Non-collinear spin configurations are the arena of rich physical phenomena ranging from the microscopic to macroscopic length scales. The most fundamental interaction between the spins is the exchange interaction due to the combination of the electron–electron interaction and the fermi statistics of the electrons, which is either ferromagnetic or antiferromagnetic. Therefore, the magnetic order is usually the collinear parallel or antiparallel configuration. The sources of the non-collinear spin configurations are (i) frustrated exchange interactions, (ii) spin–orbit interactions (SOIs), and (iii) long-range magnetic dipolar interactions. These interactions in common introduce the frustration in a broad sense. In the case of (i), the Fourier transformation of the exchange interaction *J*_{ij} between the ionic sites *i* and *j*
1.1
has the minimum at a certain incommensurate vector **q** different from **q**=**0** or **q**=(*π*,*π*,*π*) (antiferromagnetic vector) [1]. In the case of fixed length of the magnetic moment, i.e. |**S**_{i}|=*S*, it is shown that the ground state is given by the helical spin structure as
1.2
where **e**_{1} and **e**_{2} are two arbitrary orthogonal unit vectors independent of the wavevector **Q**.

An example of the case (ii) is the Dzyaloshinskii–Moriya (DM) interaction in non-centrosymmetric systems [2,3]. This interaction can be described by the following Hamiltonian in the continuum approximation:
1.3
where **M**(**r**) is the magnetization, **B** the magnetic field, *J* the ferromagnetic interaction and *α* the DM interaction constant. The lattice constant is taken to be unity. The ground state under zero magnetic field **B**=0 is the helical spin structure
1.4
where **e**_{1}×**e**_{2}=**e**_{3}=**Q**/|**Q**|, |**Q**|=|*α*|/*J* and the sign ± is determined by the sign of the DM interaction *α*. Another example of the case (ii) is due to the single spin anisotropy. If the easy-axis is pointing to different directions for different spin sites, the non-collinear spin structure is expected.

For the last case (iii), Garel & Doniach [4] discussed the formation of the incommensurate modulation of the magnetization in a thin film of an Ising ferromagnet with the long-range magnetic dipolar interaction. In the case of uniform magnetization, the magnetic charge is accumulated at the surface of the sample, which costs the magnetic energy. Therefore, the non-uniform domain structures of the magnetization are stabilized in the ground state. In the case of thin film, the volume-to-surface ratio becomes large, and this issue is of vital importance.

Now let us consider the relevance of these non-collinear spin configurations to quantum mechanical waves. Namely, even though the spin order can be treated classically, the motion of electrons coupled to it should be treated quantum mechanically. Suppose the conduction electron hops between the two sites *i* and *j* with the constraint that its spin direction is enforced to be along that of the local spin **S** at each site. In this case, the effective matrix element of the transfer is given by
1.5
where |*χ*_{i}〉 is the two-component spin wave function and *θ*_{ij} is the angle between the two local spins at *i* and *j*. The phase factor e^{iaij} is analogous to the Peierls phase in the presence of the external magnetic field, but it is associated with the spin structure here. Let **S**_{i}, **S**_{j} and **S**_{k} be the spins at sites *i*, *j* and *k*, respectively. When a conduction electron moves around the three sites *i*→*j*→*k*→*i*, the product of its phase factors is given by e^{iΩ/2}, where *Ω* is the solid angle subtended by the three spins on the unit sphere. This corresponds to the magnetic flux, and is called the (scalar) spin chirality, which is proportional to **S**_{i}⋅**S**_{i}×**S**_{k} for slowly varying spins. Therefore, the non-coplanar spin configurations will lead to the Aharonov–Bohm effect and Hall effect of topological origin [5]. We will discuss some of the recent advances in this respect.

## 2. Magnetic monopoles in momentum space

As discussed §1, the scalar spin chirality acts as an effective magnetic field and leads to the Hall response [6,7]. First, we consider the case where the magnetic unit cell is smaller than the mean free path, and the Bloch wave functions are well defined in the first Brillouin zone (1st BZ). The total gauge flux penetrating the unit cell is zero or an integer multiple of 2*π* owing to the periodicity because the contour integral of **a**(*r*) along its boundary vanishes (mod 2*π*). Then, the question is ‘Is there any consequence of the spin chirality on the nature of the Bloch wave function?’. Ohgushi *et al.* [7] considered this problem for a model on the two-dimensional Kagome lattice with the non-coplanar spin configuration. Because the Kagome lattice contains the three atoms in the unit cell, there are three bands in the limit of strong Hund’s coupling. With the parallel spin configuration, the band structure is simply that of the tight-binding model on the real transfer integral. Lowest energy flat band is realized that touches at the *Γ*-point with the second band, which crosses at the corners of the 1st BZ with the third band (figure 1*a*). These crossings are described as the Dirac fermions similar to those in graphene [8]. Now, we tilt the spins on the triangles slightly to point towards the centre of each triangle, which results in the solid angle subtended by the three spins. According to equation (1.5), the flux *ϕ* (−2*ϕ*) acting on the electrons moving along the triangle (hexagon) is produced. This will create the energy gaps between the bands as shown in figure 1*b*, and also the Berry phase [9] of the Bloch wave functions. The vector potential for Berry connection is defined as
2.1
for the crystal momentum **k** and band index *n*, and the corresponding curvature analogous to the magnetic field as
2.2
The Chern number (*Ch*_{n}) defined below characterizes the global property of the Berry curvature:
2.3
which becomes non-zero by the spin tilting. Because the Hall conductivity *σ*_{xy} is given by
2.4
it is quantized when the chemical potential is within the energy gap. This is a realization of the ‘zero field quantum Hall effect’ first proposed by Haldane for a model on the honeycomb lattice [10]. Roughly speaking, this phenomenon occurs when more than two different types of loop coexist in the unit cell, and each band feels the fluxes with different weight.

Let us examine more carefully the mechanism of how the finite Chern number is produced by the flux *ϕ*. When we focus on one of the band crossings at *ϕ*=0 at the corners of 1st BZ, the electronic states near this crossing can be described by the 2×2 Dirac Hamiltonian as
2.5
with the appropriate scaling and the mass *m* is proportional to . It is well known that the Hall conductivity of the Hamiltonian equation (2.5) when the chemical potential is within the gap is
2.6
in the context of ‘parity anomaly’ [11], and also formulated based on a thermodynamic identity by Strĕda [12]. One can consider the three-dimensional space of **k**=(*k*_{x},*k*_{y},*m*) and calculate the Berry curvature distribution in this **k**-space to obtain [9]
2.7
which is exactly that of a magnetic (anti)monopole sitting at the origin. The Hall conductivity is the integral of the flux over the plane *k*_{z}=*m*= fixed, and has the discontinuous change of e^{2}/*h* when the plane crosses the (anti)monopole. Therefore, the lesson here is that the divergence of the momentum space magnetic field **b**_{n}(**k**) can be non-zero in sharp contrast to the usual electromagnetic field, and the band crossings correspond to the magnetic (anti)monopoles.

Then, the next question is whether any real materials show this scenario of Hall effect or not. We will discuss below that the anomalous Hall effect (AHE) in Nd_{2}Mo_{2}O_{7} (hereafter we denote it as NMO) with pyrochlore structure fits well to this story [13]. Pyrochlore lattice can be regarded as the three-dimensional generalization of the Kagome lattice, i.e. it contains the Kagome lattices and triangular lattices normal to the [111] direction. In the material NMO, there are two interpenetrating sublattices composed of the tetrahedrons of Nd and Mo atoms shifted along the *c*-axis as shown in figure 2*a* [13]. The conduction electrons are in the d-orbitals of Mo atoms, whereas the Nd atoms have localized spins with strong easy-axis anisotropy in the direction connecting the centre of the tetrahedron and Nd atom. This results in the non-coplanar spin configurations of Nd spins, which are coupled to the conduction electron spins antiferromagnetically through *J*_{df}. This transmits the scalar spin chirality of Nd spins to conduction electrons (figure 2*b*). In figure 2*c* is shown the temperature dependence of the Hall resistivity *ρ*_{H}. *ρ*_{H} shows a steep increase as the temperature is lowered corresponding to the increase of the ordering of Nd spins and associated increase in the exchange field and spin chirality acting on the conduction electrons. The absolute value of the low temperature *ρ*_{H} is consistent with the tight-binding calculation, taking into account the tilting angle of the Mo spins estimated from the neutron scattering experiment [13]. Here, we used the generalized expression of the Hall conductivity for the metallic case:
2.8
Note that this corresponds to the intrinsic contribution of the AHE [5]. In addition, the experimental observation of the sign change in *σ*_{xy} under the external magnetic field along [111] direction [14] supports this scenario, i.e. the sign of the spin chirality changes due to the [111] magnetic field.

The earlier-mentioned scenario for the magnetic monopoles in momentum space is not limited to the case of non-coplanar spins, but is more general. Even in the usual collinear ferromagnet, the effect of the SOI can be regarded as the complex phase associated with the transfer of the electrons. This can be seen when we write the SOI as
2.9
where the first line gives the interpretation that the SOI can be regarded as the effective momentum-dependent Zeeman magnetic field, whereas the second line says that the spin-dependent vector potential **A**_{SO} is acting on the electrons. In the magnetically ordered state, **s** can be regarded as the c-number, and hence the electron is subject to a constant effective magnetic field. It should be noted that again the total magnetic flux in the magnetic unit cell is zero owing to the periodicity, and the multi-band effect is essential for the non-zero Hall effect.

Concerning this issue, a simple two-dimensional three-band model made from *t*_{2g} orbitals with uniform magnetization and SOI was studied [15]. In this model, the non-zero Chern number for each band is realized similar to the case discussed earlier. In the absence of the SOI, the two spin components are decoupled and shifted to each other by the exchange energy splitting, and there occur several band crossings. The Chern numbers are zero in this case because each spin band does not feel any effect of the time-reversal symmetry breaking. The SOI lifts the degeneracies and generates the mass term *m* as discussed earlier, and produces the finite Chern numbers. This means that the SOI gives essentially a non-perturbative and topological effect.

This story has been confirmed both in the first-principles band calculation and in the experiment in a metallic ferromagnet SrRuO_{3} [16]. Figure 3 shows the magnetic and transport properties of SrRuO_{3}. Figure 3*a*–*d* shows the temperature dependence of the spontaneous magnetization, resistivity, transverse resistivity *ρ*_{xy} and the Hall conductivity *σ*_{xy}, respectively. It is noted here that the Hall resistivity and conductivity are dominated by the anomalous contribution. Just below *T*_{c}, *σ*_{xy} is negative and turns into positive and again has a maximum. This characteristic behaviour strongly suggests that the perturbative expansion in *λM* (*λ*: SOI, *M*: magnetization) is not allowed, and the AHE is a fingerprint of the Berry phase of the Bloch wave function. We plot in figure 3*d* the transverse conductivity *σ*_{xy} as a function of the spontaneous magnetization *M*_{s} compared with the first-principles band structure calculation. The non-monotonic dependence comes from the relative positions of the band crossings and the Fermi energy, which vary as *M*_{s} changes. Similar conclusion is obtained also for the AHE in Fe [17], where the sharp spiky structure of the Berry phase distribution in momentum space corresponding to the band (anti)crossings has been found.

If these band (anti)crossings play a crucial role, the low-energy scale typical of that of the SOI enters into the problem of the AHE. Therefore, the dynamical AHE, i.e. *σ*_{xy}(*ω*) in ferromagnetic materials, should provide important clues. Owing to the rapid progress of terahertz spectroscopy [18,19], the low-energy structures of *σ*_{xy}(*ω*) are being revealed. Figure 4 shows an example of the low frequency data for *σ*_{xy}(*ω*) in the ferromagnetic phase of SrRuO_{3}. It is seen clearly that non-trivial structure exists at around 4 meV, which is well fitted by a model assuming the band (anti)crossing near the Fermi energy [19].

## 3. Berry phase in the spin textures

The analysis in **k**-space mentioned earlier is justified when the magnetic unit cell is small compared with the mean free path ℓ. In this case, the Berry phase in real space is translated to that in **k**-space. In the other limit, i.e. when ℓ is shorter than the size of the spin texture *ξ*, the Berry phase acts as the **r**-space magnetic field directly. This situation is realized in the spatially slowly varying spin textures, and the recently discovered skyrmion crystals in non-centrosymmetric magnets with B20 structure give an ideal example of this case [20,21]. This system is described by the Hamiltonian equation (1.3) with the magnetic field energy −**B**⋅**M** added.

The so-called A-phase in the *T*–*B* phase diagram (*T*: temperature, *B*: magnetic field) of MnSi has been a mystery. Recent neutron-scattering experiments [20] identified this A-phase as a triangular crystal of skyrmions. A theoretical analysis concluded that while the conical state with the spins tilted and *q*-vector being along the external magnetic field is the stable state in most of the phase diagram, the thermal fluctuation stabilizes this skyrmion crystal in a narrow region near the critical temperature *T*_{c}. Compared with this three-dimensional case, on the other hand, when one reduces the thickness of the sample smaller than the wavelength of the spiral, the conical state becomes energetically higher than the skyrmion crystal when the external magnetic field is perpendicular to the film. Actually, a Monte Carlo simulation of a two-dimensional magnet with DM interaction concluded that the skyrmion crystal state is stable in a much wider region of the (*T*,*B*)-plane including the zero-temperature case [22]. This study triggered a recent real space observation of the skyrmion crystal using Lorentz tunnelling electron microscopy (TEM) in a thin film of (Fe,Co)Si [21]. Figure 5 shows the phase diagram and the Lorentz TEM images of the skyrmions and skyrmion crystals. Figure 5*a*–*c*,*g* indicates the experimental results, while figure 5*d*–*f*,*h* shows the results obtained by Monte Carlo simulations. The agreement between the two is excellent.

Skyrmion structure is relevant to the Berry phase and fictitious magnetic field discussed in §1. When the skyrmion is regarded as the mapping from the two-dimensional space to the unit sphere, i.e. the direction of **M**, it wraps the sphere once, which means that the integral of the magnetic flux over one skyrmion is 2*π*. Therefore, the effective magnetic field induced by the skyrmion is estimated as ≅4000 T when the size of the skyrmion *ξ*=1 nm, and is ∝*ξ*^{−2}. This fictitious magnetic field **b** acts on the conduction electrons in MnSi, (Fe,Co)Si and MnGe and gives rise to the Hall effect due to the Lorentz force, with **b** replacing the external magnetic field **B**. This scenario has been confirmed by recent experiments [23].

An important direction related to the skyrmions is their dynamics. Especially, a recent experiment [24] revealed that a current density as low as *j*_{c}≅10^{2} A cm^{−2} can drive the motion of the skyrmion crystal in MnSi. Motivated by this experiment, we have recently developed a theoretical framework on the coupled dynamics of the skyrmions and conduction electrons. The spin textures characterized by **n**=**M**/|**M**| satisfy the Landau–Lifshitz–Gilbert equation [25]
3.1
where **j** is the conduction electron current and *α* is the Gilbert damping coefficient. On the other hand, the conduction electrons are subject to the effective electromagnetic field as
3.2
Especially, because the skyrmion carries the effective magnetic flux, its motion will induce d**b**/d*t* and hence **e** due to the electromagnetic induction. This **e** field leads to the additional Hall effect to the topological AHE due to static **b** as discussed above. Many other effects are predicted from equations (3.1) and (3.2) such as a new mechanism for the damping of the skyrmion motion, and the motion of the skyrmion transverse to the current (skyrmion Hall effect) [25].

## 4. Spin textures due to magnetic dipolar interaction

As mentioned in §1, the magnetic dipolar interaction in ferromagnetic thin films combined with the magnetic anisotropy would produce the non-collinear spin configurations. Garel & Doniach [4] showed that the phase diagram in the *T*–*B* plane consists of the three phases, i.e. the striped phase, triangular bubble array and the ferromagnetic states. Their model treats only the *z*-component of the magnetization, and based on the GL expansion of the free energy. Taking into account also the *x*- and *y*-components of the magnetization, the bubble turns out to be a skyrmion. In this section, we report some recent new developments on the spin textures in a ferromagnet film with dipolar interaction, which shows richer structures than the helimagnet with DM interaction discussed above [26]. Figure 6*a* is the experimental phase diagram in the *T*–*B* plane and figure 6*b*–*f* show the Lorentz TEM images of magnetic structures in Sc-doped hexaferrite thin film. The magnetic field *B* is perpendicular the film, i.e. along *z*-direction. There are a variety of spin structures appearing, i.e. (i) the longitudinal conical spin structure (C) with the wavevector **q** along the *z*-axis, where the magnetization is mostly in-plane except the small *z*-component induced by *B*, (ii) the stripe structure (S), where the wavevector **q** is within the plane and the magnetization is rotating within the spin plane perpendicular to **q**, and (iii) the magnetic bubble structure (B) essentially the same as the skyrmions discussed above. The star symbols in figure 6*a* correspond to the values of (*T*,*B*) for the TEM images in figure 6*b*–*f*, respectively. Figure 6*b* is for the conical state, and the magnetization is mostly uniform in the *xy*-plane, figure 6*c* for the mixture of the conical and bubble structures, and figure 6*f* for the periodic array of the bubbles. Note that the black and while colours of the bubble correspond to the chirality of the in-plane component of the magnetization, which is the unique feature of the skyrmions owing to the dipolar interaction compared with that of the DM interaction. Namely, the chirality degrees of freedom are determined by the sign of the DM interaction, while they survive in the case of the dipolar interaction. As the magnetic field *B* is reduced from figure 6*f*, the phase of the coexisting stripe structures and bubbles appears (figure 6*e*), and turns to that of pure stripe at zero magnetic field (figure 6*d*).

Figure 7 shows the direction of the in-plane magnetization obtained by the magnetic transport-of-intensity method for the two sets of (*T*,*B*) values, i.e. in Figure 7*a* *T*=270 K, *B*=0 and in figure 7*b* *T*=270 K, *B*=80 mT. Inhomogeneous stripes with topological defects such as stripe branch (around area A in figure 7*a*) and stripes with end points (points B1–B4 in figure 7*a*) have been observed. Red and right green lines show the in-plane magnetic components in Bloch walls with opposite spin rotation direction (chirality), and dark lines represent the out-of-plane magnetic domains. In such a 30 nm thick sample, the width of magnetic domain wall is comparable to domain size. It is clearly seen that the chirality of walls changes its sign around topological defects. As a perpendicular magnetic field was applied to the sample, magnetic bubbles including skyrmions with chirality (+1 or −1) and pinched stripes with the Bloch lines without skyrmion number were generated (denoted by s1–s4). However, no long-range correlation of spin chirality exists even though the positional order of skyrmions in triangular lattice emerges.

Therefore, the thin film ferromagnets with long-range dipolar interaction offer rich possibilities of spin textures with non-trivial topology. Various topological phenomena related to these spin textures together with their manipulation by current and/or optical excitations are intriguing problems left for future studies [27].

## 5. Conclusions

We have discussed the topological aspects of the spin textures from the viewpoint of Berry phase in momentum and real spaces. As a consequence of this Berry phase, we considered the Hall responses of topological origin. When the magnetic unit cell is much smaller than the mean free path, the momentum is well defined and the Berry curvature in the 1st BZ is characterized by the band crossings acting as ‘magnetic monopoles’. In the other limit, i.e. when the mean free path is shorter than the length scale of the spin textures, the real space Berry curvature acts as the fictitious magnetic field, and its dynamics can be described as the emergent electromagnetism. As for the spin textures, various origins can be considered, i.e. (i) frustrated exchange interactions, (ii) SOIs such as DM interaction, and (iii) long-range magnetic dipolar interactions. The nature of the spin textures depends sensitively on their origin, and as an example we showed a recent Lorentz microscope observation of the stripe, skyrmions and Bloch lines in (Fe,Co)Si and Sc-doped hexaferrite corresponding to (ii) and (iii), respectively. Theoretical design of the spin textures and associated novel electronic structures will be a fruitful field in the future from the viewpoints of both basic science and applications in the coming decades.

## Acknowledgements

The authors thank H. Katsura, M. Mochizuki, S. Murakami, N. Furukawa, A. V. Balatsky, M. Onoda, S. Onoda, H. J. Han, C. Jia, K. Nomura, M. Mostovoy and S. C. Zhang for collaborations, and T. Arima, N. Kida, M. Kawasaki, D. I. Khomskii and A. Aharony for useful discussions. This work was supported by priority area grants, grants-in-aid under grant nos. 19048015, 19048008 and 21244053, and NAREGI Nanoscience Project from the Ministry of Education, Culture, Sports, Science and Technology, Japan, Strategic International Cooperative Programme (joint research type) from Japan Science and Technology Agency, and by Funding Programme for World-Leading Innovative R and D on Science and Technology (FIRST Program)D.

## Footnotes

One contribution of 8 to a Theo Murphy Meeting Issue ‘Emergent magnetic monopoles in frustrated magnetic systems’.

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