In this Perspective, we present some important aspects of two fundamental concepts of modern spintronics, namely, spin–orbit torque and topology. Although these two fields emerged separately in condensed matter physics, in spintronics they show a deep connection, which requires further theoretical and experimental investigation. The topological features can arise both from momentum space via the wave functions as well as from real space via complex magnetic configurations. These features manifest themselves as unique aspects of different equilibrium and non-equilibrium properties. Physical interactions of such a topological origin can open new possibilities for more efficient mechanisms for manipulating magnetic order with electrical currents, which, in turn, can lead to faster and more efficient spintronics devices.

In recent years, topology has become an integral part of modern spintronics and materials with nontrivial topology have become an indispensable part in device engineering. Although their superior performance is unambiguous, the actual role of topology in the physical observables is often not well understood and require further theoretical investigation. The nontrivial topological properties can arise from both real space (i.e., due to non-collinear magnetic configuration) and reciprocal space (i.e., due to wave function and band structure), each of which bears its distinct features. Among different entities, spin–orbit torque (SOT) is one of the most important physical observables that is significantly influenced by both real space and reciprocal space topology. While in systems with uniform magnetization the non-trivial behaviors arise from the reciprocal space via the spin-texture, in systems with real space magnetic texture (i.e., skyrmion) they can originate from a complex mixture of topological features from both real space and reciprocal space. In this Perspective, we address some of the crucial aspects of SOT and their connection to different topological features of the system from both real and reciprocal space. At first glance, each of these aspects may appear independent of each other and may be regarded as different branches of spintronics. However, a closer look would reveal that they are strongly intertwined with each other and often difficult to isolate. This Perspective would elaborate the strong interdisciplinary nature of different branches of spintronics and provide a unified picture of different physical mechanisms.

The key feature that connects these different aspects is coming from the topology of the system, which is characterized by the Berry curvature of the system (Fig. 1). Most of the cases, especially for systems with uniform magnetic profile such as ferromagnets or antiferromagnets and their heterostructures, the Berry curvature arises from the reciprocal space and manifests itself via different transport properties and response functions. Section I provides a detailed description of the theoretical frameworks used to study the transport as well as SOT features along with the recent advances for different classes of materials. These studies are often done in the clean limit, which can significantly differ from experimental observation due to the presence of impurities in realistic systems. This has been addressed in Sec. II, which provides a detailed description of SOT in topological insulators in the presence of impurities and, thus, provides a connection between ideal theoretical predictions and realistic observations. To understand the connection between different physical observables and the topology of the system, one must look at the underlying geometric structure of the quantum states which is elaborated in Sec. III. It also shows that in the presence of magnetism one can extend the dimension of the underlying phase space, which allows us to define mixed Berry curvature by exploiting the complex connection between the electronic and magnetic subspace. This also provides a better understanding of the origin of different magnetic interactions such as the Dzyaloshinskii–Moriya interaction (DMI). The presence of such asymmetric magnetic exchange interactions leads to the formation of a magnetic texture, which can demonstrate more complex behavior of SOT. This has been described in Sec. IV. In two dimensions, these complex interactions can give rise to non-trivial magnetic textures like skyrmions, which can be characterized by a real space topology and, therefore, can demonstrate more complex behavior of SOT originating from the complex interplay between real and reciprocal space topological properties. The nature of SOT in the presence of real space non-trivial magnetic texture is the main focus of Sec. V, which also elaborate the nature of topological Hall effect (THE) emerging from such real space magnetic texture utilizing the methods described in Sec. II. Thus, each section represents a unique branch of spintronics and their interdependence also shows how one can construct a complete picture of different spintronic phenomena by unifying different approaches.

## I. EFFECT OF RECIPROCAL SPACE TOPOLOGY AND SPIN TEXTURE ON SPIN ORBIT TORQUE

The study of electrical manipulation of magnetization via spin torque is strongly motivated by its potential application in magnetic storage devices, which provide most reliable and robust performance. The first generation of magnetic storage devices consists of two magnetic layers with different magnetic alignment separated by an insulator. A charge current, polarized by the first magnetic layer, is utilized to manipulate the magnetic orientation of the second layer via spin transfer torque (STT). This is one of the first example of electrical manipulation of magnetic order. This process is not very efficient since the current has to pass through a resistive junction. This hurdle was overcome with the introduction of spin–orbit torque (SOT) where a non-equilibrium (NEQ) spin density is produced by passing a charge current through a material with strong spin–orbit coupling (SOC).^{1,2} There are mainly two types of torques that play a major role in manipulating the magnetization, namely, the field like (FL) ( $ T F\u223c m ^\xd7( z ^\xd7 j ^ e)$) and damping like (DL) ( $ T D\u223c m ^\xd7[( z ^\xd7 j ^ e)\xd7 m ^]$) torque.^{3,4} There is, however, a strong debate behind the origin of these two main torques. Initially, while the Rashba–Edelstein effect was considered to be the driving mechanism behind the $ T F$,^{5,6} the $ T D$ was assumed to be originated from the Spin-Hall effect.^{7–9} Apart from that, other effects such as spin swapping^{10,11} and magnetoelectric effect^{12,13} can also play a crucial role in specific systems. The true origin of these torques can be quite complex. It is worth mentioning that the interface between a material with SOC and a magnetic material can give rise to new types of interactions,^{14} which can lead to new types of torques. For example, in a heavy-metal ferromagnet interface such as Co–Pt or Co–Cu, one can observe different interface generated spin current,^{15,16} which can arise from spin dependent scattering amplitude and can have the same signature as spin Hall current emerging from bulk. Understanding the true nature of SOT at interfaces, therefore, requires more in depth theoretical as well as experimental investigation.

Another aspect that dominates a vast extend of modern spintronics is topology. The concept of topological insulators was first investigated in the context of two-dimensional materials.^{17–19} In two dimensions, the topological characterization can be done in terms of bulk Chern number, winding number of edge states,^{20} or the $ Z 2$ invariant.^{21} In three dimensions, such characterization becomes even more complicated.^{22–24} Materials with such non-trivial topology can demonstrate exotic transport features. One should be careful with the notion of *topological effect* in this regard. A topologically non-trivial system is characterized by a non-zero topological index, which describes the features of its quantum states. Such topological index can manifest itself as a physical observable. For example, in a Chern insulator, the Chern number is nothing but the Hall conductance of the system^{25} which can be verified by using Berry curvature formula.^{26} Therefore, quantum Hall effect can be considered a topological effect, which can also be identified by their quantized values.^{27} In general, the term *topological effect* is often used to denote an effect, which has a contribution from Berry curvature or an effect observed in a topologically non-trivial system. Material with non-trivial reciprocal space topology was initially considered a source of dissipationless transport via quantized conducting channel, which has been experimentally observed first in HgTe–CdTe quantum well.^{28} Three-dimensional topological insulators were first observed in $Bi$ compounds such as $ Bi 2 Se 3$ and $ Bi 2 Te 3$ in 2009.^{29–31} Instead of quantized conducting channels, these materials demonstrate topologically protected surface state with strong spin-momentum locking and, therefore, quickly become an ideal playground for studying physical phenomena driven by Rashba like interactions.^{32} As a result, these new class of materials naturally become a suitable candidate for studying SOT,^{33} which also opens new possibilities for next generation spintronic devices.^{34}

### A. Theoretical studies

Theoretical studies regarding SOT are mainly focused on understanding the origin of different components of torques and their connection to the topological features of the system for different material as well as complex heterostructures. To identify the physical origin of the torques, one can look for different response functions such as longitudinal conductance, transverse or Hall conductance, spin response function, or the torkance. These studies can be done within semiclassical Boltzmann formalism^{9,35} or Green’s function based approach such as Kubo or Keldysh formalism. In this section, we are going to give a brief overview of the Kubo and Keldysh formalism, whereas the Boltzmann formalism will be discussed in detail in Sec. II. Often the spin response function and, therefore, the NEQ spin accumulation can be anticipated from the equilibrium spin texture in reciprocal space with an effective Dirac Hamiltonian,^{36,37} tight binding model,^{38–40} or with first-principle calculation.^{41–43} However, the spin texture does not provide a complete picture of the physical origin of the torque. For a proper understanding of $ T D$, one needs to study its topological features as well using proper NEQ formalism.

#### 1. Kubo formalism

^{39,40,44}the NEQ observables are calculated by going through an imaginary time domain via Matsubara frequencies. Within the Kubo–Bastin formula, the response function of an observable $ O ^$ is given by (Ref. 45)

^{25}which is a topological invariant of the system provided the Fermi level, lies within a gap. This expression can be further simplified by integration by parts and can be written as

^{46}

^{,}$\u27e8 O ^ \u27e9 s u r f a c e$ and $\u27e8 O ^ \u27e9 s e a$ are known as Fermi surface and Fermi sea terms.

^{47,48}For topologically trivial systems, the surface term, coming from the states at the Fermi level, is more dominating whereas for topologically non-trivial system the sea term becomes dominant (Fig. 2). The torque is often decomposed into an intraband and interband contribution as

^{49,50}

^{12}and contributes to the DL torque.

#### 2. Keldysh formalism

^{51,52}and, therefore, is often called the Landauer Keldysh formalism.

^{53,54}This method can be incorporated within the tight binding

^{55}as well as the first-principle framework.

^{56–61}Within Keldysh formalism, the NEQ expectation value of an observable $ O ^$ for a two terminal device configuration subjected to a bias voltage $ V B$ can be defined as

^{62}one can decompose the total contribution into Fermi sea and surface terms

^{60}as

^{63,64}

#### 3. Symmetry of torques

In general, SOT can be divided into two classes, namely, field like (FL) and damping like (DL) torque. Since the FL torque changes sign under reversal of magnetization direction while the DL torque retains its sign, they are often called odd and even torque, respectively.^{65,66} While the odd/FL torque originates only from the surface term, the even term contains both sea and surface contribution. The sea term is more robust against impurity^{39,40} and thermal fluctuation.^{60} Depending on the structural symmetry of the system, the resulting torques can have a more complex angular dependence (Fig. 3) manifested by a higher order anisotropy of SOT,^{40,60,67–69} which often originates from the warping of band structure.^{69–71} In addition to that, the interaction between different degrees of freedom such as valley and spin can also cause a distortion in reciprocal space spin texture,^{72} which can produce additional components of SOT. Such unconventional torques can open new channels for field free switching.^{73}

#### 4. Impurity scattering

Impurity scattering has a profound effect on SOT, which is different form the effect generated due to scattering from the interface. This stems from the spin dependent scattering amplitude, which can be caused by a spin dependent^{74} or spin independent impurity.^{38,72,75,76} The impurity can cause a significant enhancement of DL torque.^{38,74} Higher order impurity corrections can also distort the symmetry of the resulting SOT and can give rise to new components.^{60} Impurity scattering may also lead to a complete cancellation of different torques.^{75} However, this might also originate from the underlying theoretical framework. Most of these theoretical studies are done with an effective Rashba model where the effect of impurity is considered with suitable vertex correction in the ladder approximation. For the Rashba Hamiltonian, the vertex correction also leads to a vanishing Spin Hall effect,^{77} which can be an artifact of the parabolic dispersion^{78} and can be avoided with alternative models.^{79} A detailed study of the effect impurity scattering considering the higher order corrections with proper diagrammatic technique is, therefore, very crucial to understand the experimental observation of SOT. Such study can be done with the effective model^{72,75} or within *ab initio* framework.^{80,81} The situation can be quite different when considering an interface with topologically nontrivial material since the scattering is different in comparison to normal metals. However, depending on the band dispersion and impurity concentration, quantum mechanical and semi-classical calculations can produce very different results.^{82,83} Although a quantum mechanical calculation with proper diagrammatic correction or real space impurity average with Keldysh formalism can produce better result compared to semi-classical Boltzmann approach, they are computationally more expensive. At low impurity concentration away from the Dirac point, the Boltzmann approach performs quite well. This is discussed in detail in Sec. II.

#### 5. Spin orbit field

For practical purpose, the torque is often characterized with an effective magnetic field such as $ T= m\xd7 B$,^{84} which can be measured experimentally.^{85–89} The effective $ B$ field is proportional to the non-equilibrium spin accumulation $ s$,^{67} which can be calculated using suitable response theory. For example, the FL torque is proportional to NEQ $ s y$ component and the DL torque is proportional to the NEQ $ s x$ component for a current along the $x$ axis.^{39,40,90} In such case, one can calculate the response tensor for the effective magnetic field as well. The response tensor for the effective field is usually not proportional to the response tensor for the torque and require proper evaluation.^{50,91} One can further decompose the torques into components, which are odd or even under time reversal symmetry or simply under reversal of magnetization. For example, FL torque is odd in $ m ^$, while DL torque is even in $ m ^$. It is, therefore, often useful to compute the magnetization-even and magnetization-odd components of the response functions separately in theoretical calculations of the SOT,^{92} which is also beneficial for experimental observation.^{66,67}

#### 6. Berry curvature

Berry phase is a central idea modern condensed matter physics.^{13,93} Here, we focus on the abelian Berry phase only and skip the discussion on non-abelian Berry connection, which plays a crucial role in different fields, especially in quantum computation.^{94} Different physical observables such as polarization and orbital magnetism are known to originate from the Berry curvature. Note that the response function for the transverse current or the anomalous Hall coefficient is a topological invariant itself.^{25} For an insulating system, the Berry curvature effect is dominated by the Fermi sea or the interband term. For the metallic system, however, it can be more complicated. For example, the quantum anomalous Hall effect (QAHE) is known to come from the Berry curvature of the Fermi surface and can be shown to be a topological Fermi liquid property.^{95} For a system with Rashba spin orbit coupling, it shares a direct connection with DL.^{12} In addition, it also contributes to the DMI.^{96} Theoretically, it can be understood by evaluating different response functions such as spin response function or torkance,^{92} where the Berry curvature like terms usually appear in the interband or Fermi-sea like terms. A suitable decomposition scheme^{97} is, therefore, crucial for a topological characterization of the physical observables. Note that the Berry curvature is a generic term, which can connect any two physical observables provided the Hamiltonian is a periodic function of them which allows to extend the Berry curvature formalism to higher dimension as well.^{98,99} More details on the connection between the response functions and the topological invariants with Berry curvature in both reciprocal and real space are discussed in Secs. III and IV.

### B. 3D topological insulators

Topological insulators are favored as SOT device for their strong spin-charge conversion efficiency.^{100} Research in this field is pioneered by the experimental demonstration of superior SOT in TI by Fan *et al.*^{101} (at 1.9 K) and Mellnik *et al.*^{102} (at room temperature). This opens up a new possibility for utilizing topological insulators for room temperature application, which subsequently stimulated a large number of studies. Subsequent studies^{103,104} show that in a TI based device, magnetization switching can be achieved with a current density of $ 10 5 A / cm 2$, which is two orders of magnitude smaller compared to the current needed for a heavy metal based device (Table I).

Material . | θ
. | σ _{ s }
. | J _{ SW }
. |
---|---|---|---|

Pt^{105} | 0.08 | 3.4 | 2.85 × 10^{7} |

Ta^{7} | 0.15 | 0.8 | 7.8 × 10^{6} |

W^{106} | 0.40 | 1.9 | 1.6 × 10^{6} |

Be_{2}Se_{3}^{103} | 0.16 | 0.15 | 2.8 × 10^{6} |

Bi_{x}Se_{1−x}^{107} | 18.62 | 1.45 | 4.3 × 10^{5} |

Bi_{0.9}Sb_{0.1}^{108} | 52 | 130 | 1.5 × 10^{6} |

(BiSb)_{2}Te_{3}^{109} | 2.66 | 1.06 | 3 × 10^{5} |

WTe_{2}^{110} | 0.2 | 0.11 | 7.05 × 10^{5} |

Material . | θ
. | σ _{ s }
. | J _{ SW }
. |
---|---|---|---|

Pt^{105} | 0.08 | 3.4 | 2.85 × 10^{7} |

Ta^{7} | 0.15 | 0.8 | 7.8 × 10^{6} |

W^{106} | 0.40 | 1.9 | 1.6 × 10^{6} |

Be_{2}Se_{3}^{103} | 0.16 | 0.15 | 2.8 × 10^{6} |

Bi_{x}Se_{1−x}^{107} | 18.62 | 1.45 | 4.3 × 10^{5} |

Bi_{0.9}Sb_{0.1}^{108} | 52 | 130 | 1.5 × 10^{6} |

(BiSb)_{2}Te_{3}^{109} | 2.66 | 1.06 | 3 × 10^{5} |

WTe_{2}^{110} | 0.2 | 0.11 | 7.05 × 10^{5} |

This superior efficiency can be understood from the nature of the surface states. In a 3D TI, the low energy states, i.e., states close to the Dirac cone, are localized near the surface. These states have a strong spin-momentum locking, which originates from the bulk topology of the system (Fig. 4). As a result, most of the charge current is utilized in charge-spin conversion maximizing the resulting torque.^{39} Compared to that, in a heavy metal most of the current flows through the bulk while the spin-momentum locking is finite near the interface only. This is also reflected in the spin mixing conductance^{111} and spin pumping.^{112–114} The efficiency of the spin charge conversion is measured in terms of spin Hall angle,^{115} which is the ratio of spin to charge current. This definition is, however, valid for a two-dimensional electron gas and one should be careful while characterizing the surface of a three-dimensional system. One can obtain a spin-Hall angle much larger than 1,^{101,107,108,116} which might be an artifact coming from an ambiguous definition of surface current density.^{7,39,117} Conventionally, the surface current density is often defined as the bulk current divided by the thickness of the TI layer. This approach works well for heavy metals like Pt or W, where the wave functions near the Fermi level are distributed all over the thickness. For a TI, the near Fermi level states are strongly localized near the surface^{39,118} and to obtain the 2D current density one should divide the total current with the depth of penetration of the surface states rather than the width of the sample.

Another challenge in characterizing the efficiency of 3D TI is the presence of a bulk current. Initial study suggests that at low temperature the current flow is mostly two dimensional^{119} and remains independent of the bulk.^{120} At higher temperature, there can be small bulk current causing a shunting effect.^{121} However, such bulk current does not have any significant impact on the efficiency of the SOT. In 2018, DC *et al.*^{107} demonstrated superior switching efficiency of topological insulators by using sputtered $ Bi 2 Se 3$ (Table I). They explain this effect in terms of enhancement of spin-charge conversion efficiency due to the confinement effect. Theoretically, it has also been demonstrated that random scalar impurity can enhance the SOT,^{38} which may arise from an emergent Berry curvature.^{122} The nature of SOT in topologically non-trivial system in the presence of defects, therefore, requires a more in-depth study. This has been further discussed in Sec. II within semi-classical Boltzmann formalism.

### C. 2D topological insulators and semimetals

2D topological insulators have recently made a profound appearance as a source of SOT. The strong SOC leading to the generation of SOT can either come from the buckling of the planner structure, such as silicene,^{123–125} germanene,^{126} stanene,^{127} and phosphonere,^{128} or from the presence of a heavy metal such as W or Ta or the transition metal dichalcogenides.^{129} These materials show a strong charge-spin conversion efficiency which, as we discussed earlier, is not sufficient to generate strong SOT relevant for switching. The charge-spin conversion efficiency results from the Rashba-like interaction that contributes to the FL torque, whereas the switching requires DL torques that originate from interfacial as well as topological features of the system. Contrary to 3D topological insulators, in a 2D topological insulator the bulk gap corresponds to edge states which demonstrate quantized conductance. The SOT can be further enhanced in the presence of random impurity,^{38,130} which can have an intrinsic origin as well.^{122}

Another class of 2D material that has shown promising performance are the 2D semimetals such as $WT e 2$^{131,132} and $TaT e 2$.^{133} Unlike 2D topological insulators, in a 2D semimetal there is no global gap and the bands touch each other at specific points, which distinguishes from normal 2D metal. At the surface, these points give rise to gap less arcs (Fig. 5), which is considered to be a strong source of charge-spin conversion.^{35,134} In addition, the topology of the system plays a crucial role in generating SOT in this system, which can facilitate efficient switching. The SOT in these materials can be governed by a complex mixed Berry curvature,^{135,136} which is explained in detail in Sec. III. The topological feature can be further tuned with an additional electric field,^{132} which gives them additional flexibility in manipulating SOT.^{137}

### D. Antiferromagnets

Antiferromagnets are comparatively new members as a potential candidate for SOT which also demonstrate fascinating topological features.^{138,139} Theoretically, the applicability of SOT to manipulate magnetization in AFM was first proposed in 2014 by Železný *et al.*,^{140} which was subsequently demonstrated experimentally in CuMnAs^{141} and $ Mn 2 Au$.^{142} Although the components of the torques are restricted by the symmetry of the structure, one can obtain more exotic components by suitable magnetic ordering.^{143} Among different classes of AFM, the non-collinear and non-coplanar AFMs have recently caught significant attention due to their unique topological features.^{144–147} These unique topological features lead to efficient SOT and switching in different classes antiferromagnets.^{148} In addition, layered AFM such as $\alpha Fe 2 O 3$^{149} or $ CrI 3$^{150} can be used in combination of another topological insulator^{40,149} to provide an efficient source of SOT (Fig. 6). The role of topology in AFM and AFM based heterostructures^{40} in generating SOT, however, still requires a better theoretical understanding. Most of the studies related to topology and Hall effect initially were focused on non-collinear antiferromagnets.^{151–153} However, recent discovery of *altermagnetism*^{154} has stimulated a growing attention in the study of topological features in collinear antiferromagnets.^{155}

### E. Future perspective

Inclusion of topology has added a new dimension to spintronics, which has made a very prominent appearance in transport properties as well as in SOT.^{90} This creates an immense opportunity for both theoretical and experimental studies with a scope for direct practical application. This has stimulated the discovery of a large number of new materials, namely, *quantum materials*,^{156} with new topological feature. While simplified model based calculations have been quite effective in understanding, a more complex *ab initio* calculation is necessary for understanding the true nature of the complex interfaces. This requires further development of the state-of-the-art first-principle framework, which can deal with large system and yet be able to provide intricate details of the system.^{157} While a wave function based approach is preferred for the homogeneous system, Green’s function based approach such as KKR^{158} is well suited for dealing with inhomogeneous system and impurity. For more rigorous results, one has to use a suitable NEQ Green’s function method with suitable diagrammatic corrections, which can also be implemented within the *ab initio* framework.^{80,81} At the same time, it requires systematic scanning of more materials, which requires a large scale high-throughput calculation.^{159} Such theoretical calculation is also beneficial for a systematic machine learning approach, which can be utilized to predict their non-equilibrium behavior^{160} or to discover new material.^{161–163} From theoretical perspective, an in-depth analysis of different non-equilibrium properties as well as their topological characterization is strongly needed at this moment. This requires a systematic study of the interfacial phenomena^{16,164} specifically for topologically nontrivial materials.^{14} This can also help with finding suitable magnetic material to interface with a suitable material with nontrivial topology without altering their chemical or topological properties, which still remains an open challenge. Such material can also be a rich source of spin pumping.^{112,114} In this section, we do not focus on spin pumping since it is mainly driven by FL torque and does not share any direct connection with the topology of the system. In this section, we keep our discussion within uniform magnetic configuration. Materials with nontrivial real space configuration are discussed in Sec. V. Apart from FM and AFM, recent discovery of *Altermagnets*^{154} has open new possibilities in this direction. Another interesting prospect is exploring the possibilities of two-dimensional van der Waals material^{165} and janus materials,^{68} which shows a high level of tunability.^{166} Recent discoveries in Heusler alloys^{167} also demonstrate promising features and can be a suitable candidate for device application. With the state-of-the-art theoretical framework, there is enormous potential to discover more novel materials and new mechanism to realize high performance and energy efficient SOT devices for the next generation computers.

## II. SPIN ORBIT TORQUE WITH 3D TOPOLOGICAL INSULATORS AND MAGNETIC IMPURITIES

Magnetic doping of topological insulators provides a rich playground to explore the physics of spin-momentum locked topological electrons and their influence on spin orbit torque. In this section, we introduce the semi-classical Boltzmann theory of transport and discuss the effect of impurity scattering on the spin orbit torque with a special emphasis on topological insulators.

### A. Semi-classical Boltzmann theory of transport

^{168–171}This approach combines the quantum mechanical scattering process of electrons at defects with semi-classical equations of motion for the trajectory of the electrons in between the scattering processes. It is valid whenever the electron’s mean free path is large compared to the interatomic distances and is, thus, ideally suited to treat the limit of dilute disorder as it is, for instance, the case in magnetically doped topological insulators. In the linear response regime, the non-equilibrium distribution function $ f n( k)$ is assumed to be separable in the equilibrium Fermi–Dirac distribution $ f n 0( \epsilon n k)$ and the deviation from that $ g n( k)= f n( k)\u2212 f n 0( \epsilon n k)$, where $n$ is the band index, $ k$ is the wave vector, and $ \epsilon n k$ is the dispersion relation. In the steady state, the Boltzmann equation relates the response of the distribution function to an external electric field to the scattering off impurities,

^{169}

^{172}from the incoming Bloch wave $ \Psi k( r)$ and the scattered wave $ \Psi k imp( r)$, which are coupled through the difference the impurity potential introduces $\Delta V= V imp\u2212 V host$

*Ansatz*

*constant relaxation time approximation*that neglects the $ k$-dependence completely.

^{169}

### B. *Ab initio* calculation of impurity scattering

^{170}specific defects in crystals generally are more complex.

^{172–174}In order to be able to solve the complicated impurity scattering problem, impurities that break the translational invariance need to be embedded into a crystalline solid. This can be achieved from the knowledge of the Green’s function of the host crystal into which an impurity can be embedded by solving the Dyson equation

^{158}

*ab initio*impurity embedding which makes this type of simulations routinely accessible.

^{175}

*Ab initio* impurity embedding, thus, gives access to the complicated impurity scattering that generally depends on the host’s electronic structure (i.e., the symmetries and orbital character of the incoming Bloch waves) as well as the defect (e.g., due to orbital character of the impurity states, the strength of the scattering potential), which can lead to strongly anisotropic scattering off magnetic and non-magnetic defects.^{173,174}

### C. Applications of the semi-classical Boltzmann transport theory

The semi-classical Boltzmann approach is a very common tool utilized in the theoretical modeling of transport phenomena. This section gives some examples of literature where the Boltzmann approach to transport is used. This list should, however, not be considered to be a comprehensive. Early on, it was realized that a proper description of the extrinsic contribution of defects plays a crucial role in properly describing the electrical resistivity in metals.^{176–178} For example, Mertig *et al.*^{177} found deviations to the Matthiessen rule associated with impurity scattering in ferromagnetic hosts, which is also captured in the need to use spin-dependent values in the relaxation time approximation used by Mochesky *et al.*^{179} to explain electrical transport in Fe and Fe/Au multilayers. The Boltzmann approach to transport was also extensively used to reveal the role of impurities in the study of the spin and anomalous Hall effect where, for instance, the role of skew-scattering,^{180} the effect of short range order,^{181} or the extrinsic contributions to the spin-Hall effect of certain defects in $\beta $-W have been studied.^{182} Another important research field is the study of thermoelectric transport where the Boltzmann formalism in constant relaxation time approximation,^{183–186} or extrinsic contributions of impurity scattering to the Nernst effect^{187} have been investigated. Here, the effect of phonon scattering also plays an important role and can be included via the Boltzmann approach.^{188,189} This was, for example, employed to study even the thermal and electrical resistivity of Fe at Earth core conditions where the Boltzmann formalism allows to include the effect of electron–phonon scattering.^{190} For further reading, we recommend the dedicated review of *ab initio* approaches to spin-caloric transport including a comparison of Kubo, Boltzmann, and Landauer–Büttiker approaches has been compiled by Popescu *et al.*^{191} Finally, the Boltzmann approach to transport also contributed significantly to the understanding of spin–orbit torque.^{50} It was used within the 2D Rashba ferromagnet model to study SOT in the weak disorder limit,^{192} Haney *et al.* showed that damping-like torque can be derived from bulk spin Hall effect,^{9} and Géranton *et al.* employed the Boltzmann approach to study the SOT in FePt/Pt, Co/Cu and $ Ag 2 Bi / Fe$(110) thin films from first-principles.^{169,170} These examples from different applications in the description of transport phenomena highlight the power of the Boltzmann approach to include scattering explicitly, which allows to gain insights that are complementary to other approaches like the Kubo formula.

### D. Spin orbit torque on magnetic impurities on a topological insulator

Impurities in topological insulators have been both a utensil in understanding the physical properties of topological insulators e.g., by means of quasiparticle interference,^{173,193} and they are used to design other topological phases (i.e., quantum anomalous Hall insulators) and induce new functionality. In the dilute limit, magnetic Mn dopants deposited on the surface of the strong topological insulator $ Bi 2 Te 3$ exhibit ferromagnetic coupling for concentrations of a few percent of a monolayer.^{194} The bulk band inversion in $ Bi 2 Te 3$ make it a strong topological insulator (TI), which leads to the existence of a single topological surface state (TSS) when in contact to a topologically trivial insulator (like the vacuum). For $ Bi 2 Te 3$, this is shown in Fig. 7(a), where the influence of the crystal’s ligand field, that leads to a deviation of the TSS dispersion from the perfectly isotropic Dirac cone in the form of a hexagonal warping, is seen. The strong spin–orbit interaction in typical TI materials like $ Bi 2 Te 3$ or $ Sb 2 Te 3$ leads to spin-momentum locking of the TSS electrons which is illustrated in Fig. 7(b). Under an external electrical field, only electrons from the TSS will contribute to the transport if the TI is bulk insulating. Thus, magnetic impurities at the surface of the TI will be under the influence of a spin-polarized current, which will naturally result in a torque on the direction of the impurity moments.

To evaluate the torque on the impurity, one simply replaces the host wavefunction $ \Psi k( r)$ in Eqs. (17)–(19) by the impurity wavefunction $ \Psi k imp( r)$ of Eq. (11), which gives access to the spin accumulation, spin flux, and torque on the impurity magnetic moment.^{171} Even in the dilute limit of a few percent of magnetic impurities, the average impurity–impurity distance is in the order of five in-plane lattice constants of the $ Bi 2 Te 3$ host crystal. Hence, within one mean-free path of the TSS electrons lie several randomly placed impurities and multiple-scattering effects between the defects cannot be neglected.^{171} This is illustrated in Figs. 7(d)–7(g), where the torque on randomly placed Mn impurities [panels (d) and (e)] and Fe impurities [(f) and (g)] is shown for an external field that is applied along the $ \Gamma \xaf\u2212 K \xaf$ direction. While for Mn impurities the spread in the torque due to randomly placed impurities is only sizable for nearest-neighbor dimers [e.g., most torque vectors are aligned except for the pair around $ k\u2248(\u22126,\u22124) \xc5 \u2212 1$], for Fe the spread is generally much larger. This can be understood from the nature of impurity potential that is shown in terms of the impurity density of states (DOS) in Fig. 7(h). While Mn has a half-filled $d$-shell with a minimum around $ E F$ in the impurity DOS, there is a resonance in the minority DOS for Fe at $ E F$. This makes Fe a resonant scatterer where the TSS electrons interact strongly with the impurity potential, which leads to a strong spin-lattice coupling for Fe.^{171} Overall, for Fe the average torque value vanishes [Fig. 7(g)] and no collective switching of the magnetization is expected.

The example of impurity torque on Fe and Mn at the surface of a TI illustrates (i) how transport properties can be calculated based on the semi-classical Boltzmann formalism and (ii) how details of the impurity scattering process can determine transport response coefficients, in particular, the torque on the impurity moment.

### E. Future perspectives

As we have discussed in Sec. II D, scattering in large real-space structures can be included in transport simulations using the Boltzmann approach. This is a prerequisite to properly take the effect of multiple scattering into account, which can drastically change the scattering properties due to a changing local ordering of nearby scatterers. Going from random impurity distributions to ordered superstructure might, therefore, be a way to engineer the local environment of the magnetic atoms that is in contact to the TI surface state. Such short-range order may be a way to control the multiple-scattering contribution to the impurity scattering and result in an enhancement of the torque. Random impurity distributions, on the other hand, tend to average out the damping like torque, as discussed in Sec. II D and which was also found in metallic Co/Pt.^{60} Furthermore, the anisotropy in the topological surface state that arises from its hexagonal warping [Fig. 7(b)] could be exploited together with atomic-scale superstructures that harness the resulting different scattering probabilities for different crystallographic directions.

Another road might be to control the impurity scattering by using other magnetic dopants like $f$-impurities like Eu,^{195} which typically have a much larger magnetic moment than $3d$ defects. Describing $f$-electron systems in DFT is, however, challenging due to their strong localization, which leads to stronger correlation effects that require the use of approaches like LDA+U to not underestimate electron correlations with standard DFT. Furthermore, appropriate co-doping with other (non-magnetic or magnetic) defects could be a way to control the magnetic order, the effective strength of spin–orbit interaction or the impurity properties by additional charge transfer between the magnetic impurity and other co-dopants. These possibilities lead to a huge combinatoric complexity and an efficient screening of the vast material space can only be achieved via high-throughput approaches. Such high-throughput approaches are, however, in reach as exemplified in Fig. 8 where a high-throughput-generated AiiDA graph structure together with properties of a large number of single impurities embedded in a topological insulator are shown.^{175,196} Obviously, chemical trends are clearly captured by these high-throughput calculations, which can serve as guidelines to future material exploration. Together with micromagnetic spin-dynamics simulations for magnetic ordering,^{197,198} high-throughput screening of the SOT and physics informed machine learning may be employed to predict future candidate materials for SOT devices in topological materials.

Other material classes like magnetic van der Waals materials^{199} interfaced with topological insulator materials could also be explored in the future to exploit the magnetism of materials like $ Fe 5 GeTe 2$ or $ Fe 3 GeTe 2$^{200,201} and spin–orbit locked electrical transport in the topological surface state of TIs. Furthermore, other topological matter apart from TIs can be a promising route toward strong SOT materials. The inversion-symmetry-broken Weyls semimetals $ MoTe 2$ and $ WTe 2$ have a very anisotropic Fermi surface, which is shown in Fig. 9. The combination of strong SOC and broken inversion symmetry also results in spin-polarized sheets of the Fermi surface. Naturally, this leads to strongly anisotropic scattering, both in the bulk and also on the surface where the spin-polarized Fermi arcs contribute additionally.^{202,203} These features could be exploited in future SOT applications with Weyls semimetal materials where the combination of strong SOC, topology, and crystal symmetry provides a rich playground to optimize SOT effects.

## III. GEOMETRICAL AND TOPOLOGICAL MEANING OF SOT

^{26}This is a gauge invariant quantity which plays a significant role in different physical observables and is the key to understand their geometrical as well as topological nature. The Berry curvature of a system that smoothly depends on a general parameter $\lambda $ can be expressed as

*mixed Berry curvature*.

One way to change the topology and generate monopoles in the extended space is to manipulate the magnetization direction $ M ^$ in a 2D ferromagnet where the magnetization vectors characterized by a single angle $\theta $ with respect to, e.g., the $z$-axis. The “topological partner” of the AHE in this case, which is driven by the mixed component of the Berry curvature tensor $ \Omega k \theta $, is the anti-damping spin–orbit torque (SOT), which is believed to play a key role in switching of magnetization. Within linear response theory, the anti-damping SOT $ T$ due to an electric field $ E$ is given by the torkance tensor $\tau $, i.e., $ T=\tau E$.^{92}

^{205}

*magnetic anisotropy torque*, since it corresponds to the change in the sum of the eigenvalues of the system in response to the change in the magnetization direction. The second part gives, on the other hand, rise to the so-called

*anomalous torque*, which is expressed in terms of the mixed Berry curvature $ \Omega M ^ k n$. Note that the mixed Berry curvature tensor is in general not an antisymmetric matrix and therefore it cannot be represented as a vector $\Omega $ in all cases. Therefore, the form of Eq. (31) is applicable only when polycrystalline magnetic bilayers are considered, where the symmetry leads to an antisymmetric mixed Berry curvature tensor. A generalized form of Eq. (31) which uses the full mixed Berry curvature tensor is given in Eq. (20) in Ref. 96. The Berry phase nature of the anti-damping SOT, thus, manifests in the fact that the tensor elements of the torkance tensor $ \tau i j$ are proportional to the mixed Berry curvature of all occupied states,

^{206,207}which is crucial for the emergence of chiral domain walls and chiral skyrmions and can be quantified by the so-called spiralization tensor $D$ reflecting the change of the free energy $F$ due to chiral perturbations $ \u2202 j M ^$ according to $F= \u2211 i j D i j e ^ i\u22c5( M ^\xd7 \u2202 j M ^)$. By referring to the Berry phase arguments which take into account the phase space energetics,

^{208}the spiralization is obtained as

*mixed*Weyl points in the electronic structure of the system in $( k x, k y,\theta )$-space generates a flux of the Berry curvature around them, which is determined by the Berry index. In addition to leading to enhanced values of $ \Omega k$ in their proximity, it also results in large values of $ \Omega k \theta $. Let us consider a system which exhibits only isolated mixed Weyl points at the Fermi energy in its electronic structure in $( k x, k y,\theta )$-space as a

*mixed*Weyl semimetal (MWS).

^{135}The Berry index of a mixed Weyl point can be probed by the change in the

*mixed*Chern number, defined by

^{135}Analyzing the evolution of the mixed Chern number $ Z$ as a function of $ k y$ in Fig. 10(g) for the latter system, we detect two magnetic monopoles of opposite charge that emerge at the transition points between the topologically distinct phases with $ Z=\u22121$ and $ Z=0$. Alternatively, these crossing points and the monopole charges in the composite phase space could be identified by monitoring the variation of the momentum-space Chern number with magnetization direction. These monopoles, acting as sources of the general curvature, occur at generic points near the valley $K$ for $\theta = 43 \xb0$ [see Fig. 10(c)] and in the vicinity of the $ K \u2032$-point for $\theta = 137 \xb0$, respectively. For an out-of-plane magnetization, the complex nature of the electronic structure in momentum space manifests in the quantization of $ C$ to $+1$, Fig. 10(h), which is primarily due to the pronounced positive contributions near $K$. Calculations of the energy dependence of the torkance and spiralization in the system, shown in Figs. 10(i) and 10(j), reveal the extraordinary magnitudes of these phenomena of the order of $1.1e a 0$ for $ \tau y x$ and $50 meV a 0$/uc for $ D y x$, exceeding the typical magnitudes of these effects in magnetic metallic materials.

In mixed Weyl semimetals, the magnetization switching by anti-damping SOTs can be utilized^{136,209} to induce topological phase transitions from a CI to a trivial magnetic insulator via complex interplay between magnetization direction and momentum-space topology. Combining an exceptional electric-field response with a sizeable bandgap lays out extremely promising vistas in room-temperature applications of magnetoelectric coupling phenomena for low-dissipation magnetization control. The associated changes in the DMI with doping in such a situation could be particularly valuable for the stabilization of chiral magnetic structures such as skyrmions in insulating ferromagnets. The large values of the anti-damping SOT arising in these systems would open exciting perspectives in manipulation and dynamical properties of chiral objects associated with minimal energy consumption by magnetoelectric coupling effects. At the end, we would like to remark that magnetic monopoles in the composite phase space, which we discuss here, do not only govern the electric-field response in insulating magnets but are also relevant in metals, where they appear on the background of metallic bands.

## IV. SPIN ORBIT TORQUE IN NON-COLLINEAR MAGNETS

Several cases of noncollinear magnetism are of great interest for SOT applications. In all of these cases, the noncollinearity is expected to contribute additional SOT mechanisms. The first case are magnetic multilayers in which two magnets are separated by a normal metal spacer and in which the magnetizations in the magnetic layers point into different directions^{210–212} [see Fig. 11(a)]. The second case are smooth magnetization gradients, which occur in domain walls and in skyrmions [see Fig. 11(b)]. Third, instead of using the SHE of a non-magnetic heavy metal such as Pt one may use noncollinear antiferromagnets such as $ IrMn 3$ to generate the spin current [see Fig. 11(c)].^{213} Fourth, in CoFeBTa/Pt it is possible to switch only the atomic layers close to the interface by the SOT, whereby a noncollinear state is formed in CoFeBTa, which has been shown by using resonant x-ray reflectivity measurements [see Fig. 11(d)].

In order to compute the SOT from first principles in the geometries of Figs. 11(a), 11(c), and 11(d), it is not necessary to modify the standard Kubo formalism for the SOT described in detail in Refs. 92 and 204. The main difference in the workflow of the calculation is the use of a mode for noncollinear magnetism, which is available in many DFT codes. Here, the wavefunctions are spinors, with spin-up and spin-down components. Since the wavefunctions are already spinors in SOT calculations of collinear magnets due to SOI, typically little modifications are required in the code implementation for this type of noncollinear magnetism. One conceptual novelty in the case of several magnetic layers is that one may investigate if the SOT is even or odd under magnetization reversal of the magnetization in a particular layer. However, strictly speaking one investigates different systems which are not related by symmetry, when one inverts the magnetization only in one layer while fixing it in all the others. Therefore, these additional options to define even and odd torques are valid only approximately. Even in the noncollinear case the definition of even and odd torque can be used, provided magnetization reversal is performed for all magnetic moments in all layers.^{212}

The fixed in-plane magnetization $ M bottom$ of the bottom Co layer in Co/Cu/Co magnetic trilayers [Fig. 11(a)] can be used to control the SOT on the top ferromagnet and to make it anisotropic.^{212} Due to the interrelations between DMI and SOT,^{96} the DMI exhibits analogous properties in Co/Cu/Co magnetic trilayers: It is tunable by the magnetization direction of the bottom layer and the wavelength of DMI-induced spin spirals in the top layer depends on their propagation direction in relation to the in-plane magnetization direction of the bottom Co layer.

Several additional mechanisms for SOT have been identified in the geometry of Fig. (a): Magnetic spin Hall effect,^{214} spin current from the AHE and AMR of the bottom magnet,^{215} spin–orbit filtering, and spin–orbit precession.^{15,216,217} These additional mechanisms have in common that they may be understood by considering the spin current generated in the bottom magnet or the spin current generated at the interface between the bottom magnet and the normal metal spacer, i.e., the noncollinearity aspect is not required to understand the main mechanisms, which are related to additional types of spin currents in this geometry. However, since $ M top$ and $ M bottom$ point into different directions, the *ab initio* study of the entire unit cell requires the use of a noncollinear magnetism mode. Similarly, the SOT in Fig. 11(c) may be understood from the spin current generated in $ IrMn 3$ and injected into Py. However, one contribution to the SHE of $ IrMn 3$ is associated with its triangular chiral magnetic structure.^{213} Therefore, the calculation of the SHE requires the noncollinear magnetism mode in this case.

In order to compute the SOT in the geometry of Fig. 11(b) from first principles, one may use, in principle, the same approach as discussed above for the geometries of Figs. 11(a), 11(c), and 11(d). However, in practice the computational unit cell may be too large, i.e., the computational time requirements may be forbiddingly high, e.g., when skyrmions with a large radius or wide domain walls are studied. The computer time requirement to obtain the electronic structure is proportional to the third power of the unit cell volume in many DFT codes. However, since atomistic spin dynamics equations to describe the motion of domain walls and skyrmions require the determination of the torque locally everywhere in space, also the evaluation of the Kubo formula for the SOT has to be done everywhere in space, i.e., the number of linear response calculations increases proportionally to the unit cell volume, where the complexity and number of the Bloch wavefunctions increases as well with the unit cell volume. Therefore, it may be advantageous to use a multiscale approach instead. Here, one considers the magnetization and the gradients of the magnetization at a given point in space and formulates the SOT at this point in space in terms of the local magnetization and its gradients. The first term of such a gradient expansion depends only on the local magnetization direction. The next term contains a gradient correction, which takes into account the local magnetization gradient. Using only the first term one may already find interesting conclusions concerning the role of the anisotropy of the SOT.^{218}

^{219}i.e., to compute the coefficient $ \chi i j k l CIT 2$ in

^{204},

^{219}

Figure 11(d) suggests that finding an exact reciprocity between the direct SOT and the inverse SOT in experiments may be hindered in some cases if the SOT induces noncollinearity: If the inverse SOT is measured in FMR experiments, the magnetization may be collinear in the FMR experiment.

Interestingly, it is found that the spatial gradients of the current-induced DMI need to be considered in the theory of SOT in noncollinear magnets. This finding can be understood as an analogy to the necessity to subtract the curl of the magnetization from the electric current computed within the Kubo formalism in order to obtain the observable transport current: An applied electric field may induce DMI^{220,221} and in a noncollinear magnet this induced DMI depends on the local magnetization direction so that its gradients correspond to a torque which contributes to the SOT. In the particular case of the Rashba model, there is no net current-induced DMI linear in the electric field, i.e., the integral of the linear current-induced DMI along the period of a spin-spiral is zero. However, locally there is a linear current-induced DMI the gradients of which need to be included into the theory of SOT in order to satisfy the Onsager reciprocity relations.^{219}

^{223}

^{96}

^{224}Similarly, the mixed Berry curvature dipole Eq. (46) may be employed to express the nonlinear SOT. Thus, Eqs. (41), (42), (44), and (46) have in common that they may be used to describe effects arising from a change $\delta f k n$ of the equilibrium occupancies due to a shift of the Fermi sphere when an electric field is applied. Such a change of the occupancies can also be induced by magnetization dynamics and it is given by (Ref. 204)

^{224}Similarly, we may write

^{222}

## V. TOPOLOGICAL HALL EFFECT AND SPIN–ORBIT TORQUE ON MAGNETIC SKYRMIONS

^{225}

^{,}

^{226}as they behave like physical particles that can be formed, detected, and transported.

^{227}Fert

*et al.*

^{228}proposed the magnetic skyrmions as new types of magnetic states that introduce a new era in non-volatile memories, because they can be utilized as carriers of information in the future skyrmion-based racetrack memory devices.

^{229}

An electric current caused by the application of an external electric field results in the detection or the manipulation of these complex magnetic states. Hence, magnetic skyrmions are closely related to the field of the “electrically controlled spintronics.”^{230} An auspicious phenomenon that could play a crucial role in magnetic skyrmions detection is the Topological Hall effect (THE).^{231–238} This effect is caused by the deflection of the moving electrons off the non-collinear skyrmions magnetic texture when a lateral current is applied along a layered magnetic system with skyrmions, as depicted in Fig. 12. It can be considered an additional contribution to the ordinary Hall effect (OHE) and anomalous Hall effect (AHE),^{239} as a consequence of the non-collinear magnetism.

An additional special property of skyrmions is that they can be controlled with respect to their motion by spin–orbit torque (SOT) effect.^{50} In response to an applied current, the magnetic moments of skyrmions precess in such a way that the center of the skyrmion shifts, allowing for its movement via the SOT.

The density-functional theory simulations of spin-transport phenomena in non-collinear, chiral topological magnetic structures, such as the skyrmions, require a high degree of complexity, due to the position-dependent magnetization direction of the skyrmion system. The investigation of spin–orbit torque phenomenon and the Topological Hall effect induced by magnetic skyrmions can be achieved on a density-functional theory level within first-principles calculations based on the KKR Green function method combined with transport theory within Boltzmann formalism. In this way, spin-transport phenomena in realistic materials that can be controlled with respect to their motion can be explored.

### A. *Ab initio* calculation of THE and SOT on non-collinear magnetic structures

The full potential KKR Green function method can be applied to perform non-collinear spin DFT calculations for the formation of stable magnetic skyrmions of a few nanometers in diameter in thin film ferromagnetic layers deposited on heavy metal substrates. Within this approach, the skyrmion is considered a real space impurity cluster of a few hundreds of atoms embedded into the ferromagnetic host system.^{240}

Concerning potential technological applications, the formation of stable two-dimensional skyrmions has been focused on thin ferromagnetic layers deposited on heavy metal substrates. In Fig. 13, the spin structure of a stable single magnetic skyrmion of 0.5 nm radius formed in the ferromagnetic Fe layer of Pd/Fe/Ir thin film^{240–244} is depicted.

In the next step, within this method the scattering problem of the surface electrons off skyrmion can be treated in order to study spin-transport phenomena. The self-consistent calculation of the Dyson equation allows us to compute the scattering matrix elements (10) for the Fermi surface states in the non-collinear skyrmions structure. In this way, one can proceed with the solution of the linearized Boltzmann transport equation, as described in Sec. II, and to carry out spin-transport calculations, based on quantities obtained from *ab initio* calculations.

By the solution of the Boltzmann transport equation combined with KKR method, the response tensors of the spin accumulation, torque and spin flux on magnetic impurity atoms, including non-collinear magnetization, can also be found in correspondence with Eq. (16). The exchange correlation field vector is generalized to have an arbitrary direction in Eq. (18). Hence, the torque is determined including the magnetization direction.^{245} As a result, the current-induced spin–orbit torque on magnetic skyrmions can be computed.

### B. Future perspectives

The investigation of THE caused by magnetic skyrmions is of crucial importance due to its application in designing skyrmion based racetrack memory. This phenomenon holds great potential for providing an efficient method for the readout of logical states. Moreover, gaining an understanding of skyrmion transport by SOT effect becomes a crucial aspect for the design of a skyrmion based memory. In this Section, this has been explained in terms of the KKR method and the semi-classical Boltzmann theory. One can also extend more rigorous Keldysh formalism described in Sec. IV to study such system which can be computationally quite expensive. The theoretical framework presented in this section, on the other hand, can provide a useful tool to study large scale skyrmions in realistic systems such as Pd/Fe/Ir(111) heterostructure. An additional challenge in this regard is to consider the effect of impurity within suitable scattering formalism, which can have a significant impact on the THE. This can be achieved by the methods described in Sec. II. Another great challenge is to consider the motion of skyrmion in real time, which can be treated within *ab initio* spin-dynamics simulations. Together, such theoretical framework can provide a more complete picture of the SOT in non-trivial magnetic texture such as skyrmions and also provide a better understanding of the role of real space and reciprocal space topology in resulting SOT.

## VI. CONCLUSIONS

In this Perspective, we present some of the most crucial aspects of spin orbit torque. These aspects pursued independently often fail to provide a complete picture of the underlying physics. Therefore, it is imperative that one considers their interconnections, which reveal their geometric as well as topological nature. This Perspective has systematically demonstrated these subtle connections, which provides a unified picture of spin orbit torque and topology in the context of both reciprocal and real space spin textures. It presents the state of the art of different branches and also directs toward the open challenges, which can be tackled through an interdisciplinary approach. The perspective, therefore, not only provides a complete overview of the different branches of modern spintronics, but also unifies them for better understanding of the complex phenomena, which would be crucial for future theoretical and experimental investigation of new exotic phenomena in this field.

## ACKNOWLEDGMENTS

S.G. would like to acknowledge helpful discussion with Kirill Belashchenko. We would like to thank Phivos Mavropoulos for fruitful discussion on KKR method. S.G., Y.M., and F.F. acknowledge support from Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—No. TRR 173/2-268565370 (project A11), No. TRR 288-422213477 (project B06), and the Sino-German Research project DISTOMAT (No. MO 1731/10-1). P.R. and A.K. acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy–Cluster of Excellence Matter and Light for Quantum Computing (ML4Q) EXC 2004/1-390534769. AK acknowledges financial support by the Hellenic Foundation for Research and Innovation (HFRI) under the HFRI PhD Fellowship grant (No. 1314). P.R. thanks the Bavarian Ministry of Economic Affairs, Regional Development and Energy for financial support within High-Tech Agenda Project “Bausteine für das Quantencomputing auf Basis topologischer Materialien mit experimentellen und theoretischen Ansätzen.”

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Sumit Ghosh:** Writing – original draft (equal). **Philipp Rüßmann:** Writing – original draft (equal). **Yuriy Mokrousov:** Writing – original draft (equal). **Frank Freimuth:** Writing – original draft (equal). **Adamantia Kosma:** Writing – original draft (equal).

## DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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