# Symmetry and topology in antiferromagnetic spintronics

###### Abstract

Antiferromagnetic spintronics focuses on investigating and using antiferromagnets as active elements in spintronics structures. Last decade advances in relativistic spintronics led to the discovery of the staggered, current-induced field in antiferromagnets. The corresponding Néel spin-orbit torque allowed for efficient electrical switching of antiferromagnetic moments and, in combination with electrical readout, for the demonstration of experimental antiferromagnetic memory devices. In parallel, the anomalous Hall effect was predicted and subsequently observed in antiferromagnets. A new field of spintronics based on antiferromagnets has emerged. We will focus here on the introduction into the most significant discoveries which shaped the field together with a more recent spin-off focusing on combining antiferromagnetic spintronics with topological effects, such as antiferromagnetic topological semimetals and insulators, and the interplay of antiferromagnetism, topology, and superconductivity in heterostructures.

## 1 Introduction

The phase of matter can be characterized by symmetry and topology Hasan2010 (); Haldane2017 (). For certain effects, symmetry provides the basic condition to occur while topology can add exceptional robustness. A text-book example is the Hall effect enabled by the broken time-reversal symmetry () in the applied magnetic field. The exact discrete resistance values in the quantum Hall effect (QHE), unperturbed by disorder, are then a consequence of the topological Landau level form of the electronic structure in a strong quantizing magnetic field Hansson2017 (). In this chapter, we show how the fundamental concepts of symmetry and topology apply to antiferromagnetic spintronics Jungwirth2016 ().

We start our chapter by briefly illustrating the symmetry and topology principles on three key functionalities of spintronic memory devices, namely the retention, reading, and writing of magnetic information Chappert2007 (); Kent2015 (). Side by side we compare in this introductory section how the principles apply when considering the more conventional ferromagnetic and the emerging antiferromagnetic spintronic devices.

Ferromagnetism can (and often does) lower the symmetry of the crystal, depending on the direction of magnetic moments. For example, a rotation along a certain crystal axis remains a symmetry operation when moments are aligned with the axis but the symmetry is broken when the moments are perpendicular to the rotation axis. It implies that reorientation of ferromagnetic moments can, in the presence of spin-orbit coupling (SOC), change the electronic structure and by this the total energy. This is the origin of the magnetocrystalline anisotropy energy (MAE) barrier that supports the non-volatile storage in spintronic memories Chappert2007 (). The same symmetry principle and corresponding magnetic storage functionality apply equally to antiferromagnets (AFs) Jungwirth2016 (). On top of that, the lack of a net magnetic moment and suppressed dipolar fields make the storage in AFs less sensitive to magnetic fields and allow for denser integration of memory bits than in ferromagnets. Apart from binary storage, AFs naturally host series of different stable multi-domain reconfigurations which is suitable for integrated memory-logic or neuromorphic computation devices Wadley2016 (); Kriegner2016 (); Olejnik2017 (); Borders2017 ().

Anisotropic resistance, i.e. the sensitivity of electronic transport to the current direction, requires broken cubic symmetry. Ferromagnetism where spins align with a specific crystal axis always breaks cubic symmetry. This implies that ferromagnets can have the anisotropic resistance. Here typically the leading dependence of the resistance on current direction is when measured with respect to the magnetization axis. The effect called (spontaneous) anisotropic magnetoresistance (AMR) is known for more than 150 years Thomson1856 () and provides arguably the most straightforward means for electrically detecting different directions of ferromagnetic moments. AMR was used, e.g., in the first generation of magnetoresistive field sensors for hard-disk read-heads or for electrical readout in the first generation of magnetic random access memories (MRAMs) Daughton1992 (). The same symmetry argument applies to the AMR in AFs where it has been used to demonstrate electrical readout in experimental memory devices, as shown in Fig. 1 Park2011b (); Marti2014 (); Wadley2016 (); Kriegner2016 (); Olejnik2017 (). On the other hand, AMR in AFs is not suitable for external magnetic field sensing because of the lack of the net magnetic moment.

Another transport effect that can be used to internally detect magnetic moments of a conductor is the anomalous Hall effect (AHE). It can occur in crystals with magnetic space groups whose symmetry allows for the presence of a net magnetic moment. Remarkably, this symmetry argument holds independently of whether the system indeed has a ferromagnetic moment or is in a fully compensated antiferromagnetic state. While the AHE in ferromagnets was discovered more than 100 years ago Hall1881 (), its experimental demonstration in AFs, shown in Fig. 2, is one of the most recent developments in spintronics Nakatsuji2015 ().

Unlike time-reversal, the symmetry operation of space-inversion () does not rotate the axial magnetic moment vector which implies that ferromagnets cannot have a combined -symmetry. This removes the Kramers -symmetry protection of the spin-up/spin-down degeneracy of electronic bands. As a result, electrons moving in the unequal spin-up and spin-down bands have different resistivities. In ferromagnetic bilayers this leads to different resistance states for parallel and antiparallel alignments of moments in the two layers and the corresponding giant/tunnelling magnetoresistance (GMR/TMR) effects Chappert2007 (). These phenomena, that tend be stronger than the AMR, are used in modern hard disk read-heads and MRAMs.

Different resistivities in spin-up and spin-down transport channels in a ferromagnet can be also used to filter an unpolarised current passing through the ferromagnetic layer by suppressing one spin-component of the electrical current. The resulting spin-polarized current filtered through such a ferromagnetic polarizer can exert a spin transfer torque (STT) on the adjacent ferromagnetic layer and switch its magnetic moment Ralph2008 (). This reversible electrical writing method is used in the latest generation of MRAMs.

There is no equivalent counterpart of the two-spin-channel GMR/TMR or STT phenomena in AFs with equal spin-up and spin-down bands. On the other hand, AFs can have the combined -symmetry which opens a possibility of the Dirac crossing of two doubly-degenerate bands, as shown in Fig. 3(a) Tang2016 (); Smejkal2016 (). The topological protection of these Dirac points can be turned on and off by changing other symmetries of the antiferromagnetic crystal, e.g., via changing the direction of the antiferromagnetic Néel vector. This could enable very large topological AMR effects in AFs and remedy the absence of the two-spin-channel GMR/TMR.

In time-reversal symmetric paramagnets, a broken space-inversion symmetry leads to the Kramers degeneracy of states with opposite spins and opposite crystal momenta, while the states at a given crystal momentum can be spin split. As a result, the crystal can develop a net spin polarization in a non-equilibrium, current-carrying state. When these spin Hall or Edelstein effects occur at an inversion-asymmetric interface between a paramagnet and a ferromagnet, or inside a ferromagnetic crystal that lacks inversion symmetry, they can induce a spin-orbit torque (SOT) in the ferromagnet Sinova2015 (); Hellman2016 (). The charge to spin conversion efficiency driving the SOT can outperform that of the STT and is explored as a prospect writing mechanism for future fast MRAMs. A particularly large charge to spin conversion efficiency is expected to occur in time-reversal symmetric topological insulators (TIs) whose 2D surfaces host Dirac cones with the spins locked perpendicular to the 2D momenta Mellnik2014 (); Fan2014a (); Fan2016 (). The ultimate charge to spin conversion efficiency would then occur in 1D surface states of 2D TIs, the so called quantum spin Hall states Roth2009 (), with opposite electron spins locked to the opposite crystal momenta at a given 1D edge.

A direct counterpart of the SOT is observed in AFs that break the symmetry and the symmetry but have the combined symmetry. In these antiferromagnetic crystals, a global electrical current induces a staggered non-equilibrium spin polarization that is commensurate with the staggered equilibrium Néel order Zelezny2014 (). The phenomenon was used to demonstrate, in combination with AMR, experimental antiferromagnetic memory devices with electrical writing and readout Wadley2016 (). Because of the THz-scale antiferromagnetic resonance, compared to the GHz-scale ferromagnetic resonance, SOT switching in antiferromagnetic memory devices was demonstrated with writing current pulses as short as 1 ps Olejnik2017a (). Moreover, since it is again the -symmetry that enables the antiferromagnetic SOT, it is potentially compatible with the large topological AMR Smejkal2016 ().

Another class of AFs, with no symmetry counterparts in ferromagnets, has the combined -symmetry, where is the translation by a half of the magnetic unit cell. This symmetry allows in principle for TIs with spin-momentum locked surface states, despite the breaking of the symmetry by the magnetic order, as illustrated in Fig. 3(b) Mong2010 (). Overall, an antiferromagnetic order can occur in 1421 magnetic space groups out of which only 275 allow also for the ferromagnetic order. Similarly, the 122 magnetic point groups are all compatible with the antiferromagnetic order out of which only 31 also support ferromagnetic states. This not only underlines why antiferromagnetism is more common than ferromagnetism and spans the whole range of materials from insulators to superconductors but also highlights how much is the symmetry and topology playground enlarged by including AFs. We are just beginning to unravel what new spintronics phenomena and functionalities this may offer. In Section 2 we give a brief overview of symmetry and topology concepts in condensed matter systems. In Section 3 we discuss in more detail antiferromagnetic topological semimetals, Chern insulators, and TIs. Topological spintronic phenomena in AFs are reviewed in Section 4 and Section 5 gives a brief summary of the Chapter.

## 2 Antiferromagnets: symmetry and topology

The phases of matter can be classified by Landau symmetry breaking mechanism. In an AF, Néel vector breaks the rotational symmetry present in the paramagnetic state. The order parameter is the Néel vector and SOC determines the particular direction(s) of the magnetic moments in crystal, the so-called easy axes, corresponding to the lowest MAE. While certain AFs tends to be to a very good approximation isotropic Heisenberg magnets, e.g. MnAu AF can have in-plane MAE at the level of 10 eV per formula unit Shick2010 (); Bodnar2017 (), noncollinear MnSn AF was reported to have MAE of 0.1 eV per formula unit Sandratskii1996 (); Duan2015 (), and a giant MAE of 10 meV per formula unit was predicted in noncollinear IrMn Szunyogh2009 ().

The discoveries in the 1980s in the theory of superfluid vortices, QHE, or dislocation defects revealed an additional label of the phases different from the symmetry, based on topology david1998topological (). The phases can be characterized by an integer topological index which does not change upon continuous transformations of the Hamiltonian and thus supports the relative robustness of a topological phase. For instance, the topological invariants in TIs are the indices, which in simple centrosymmetric non-magnetic TIs are related to counting the number of parities at time-reversal invariant crystal momenta Hasan2010 (). Dirac semimetals (DSMs), such as graphene, are protected by the vorticity around the Dirac point bernevig2013topological (). The DSMs protected by crystalline symmetry can be assigned a topological index by subtracting this crystalline symmetry eigenvalues of the conduction and valence band along the line in Brillouin zone (BZ) invariant under this symmetry. Red and blue lines in Fig. 3(a) illustrate the symmetry eigenvalues with an opposite sign.

We note that TIs and semimetals represent a symmetry protected topological order Hermanns2017 (). For instance, AF DSMs or TIs require the presence of the or symmetry as illustrated in Fig. 3(a,b). In contrast, Weyl semimetals (WSM) and Chern insulators are protected by Chern numbers. WSM can be realized in system with broken symmetry, while Chern insulators materialize in systems with broken symmetry as we explain further. The slices of constant wavevector component between two Weyl points in simple model WSM Vafek2013 () (with symmetry broken by breaking symmetry, Fig. 4(a)) can be thought of as Chern insulators, whose examples are QHE or quantum anomalous Hall effect (QAHE). If we use the Bloch ansatz, the Chern number of a =const. plane reads,

(1) |

where the Berry curvature has a meaning of an emergent magnetic field b in the crystal momentum space and quantifies the underlying topology of the wavefunction bernevig2013topological ():

(2) |

The topological index of a Weyl point can be defined as a Chern number of a closed surface surrounding the Weyl point, which can be calculated due to the Gauss theorem as a difference between two Chern numbers along the line connecting the Weyl points:

(3) |

Here is a small sphere surrounding the Weyl point at , n is the surface normal vector, and is the Chern number of the plane slightly below and above the Weyl point . Thus the Chern number is nonzero along the between the two Weyl points and zero outside as marked in Fig. 4(a).

The Berry curvature acts as a source and sink at the Weyl points as we illustrate in Fig. 4(a) and consequently the two Weyl points along have , and .

Topological phases in the crystal momentum space are very often accompanied by quantized or almost quantized and low dissipation transport properties and nontrivial surface states Hasan2010 (); Wang2017b (). In Fig. 4(b) we show the quantized Hall plateaus in EuMnBi AF Masuda2016 (). The Chern insulator exhibits quantized Hall conductivity: On the other hand, quantum spin Hall effect (QSHE) in the 2D TI shows a quantized spin Hall conductivity In a Chern insulator, chiral edge states arise (single spin polarized electrons), which are in fact 1D Weyl fermions of a given chirality Armitage2017 (), as we show in Fig. 4(c). In 2D TIs, helical edge states are observed (two counter-propagating perfectly polarized currents with opposite spins) as we illustrate in Fig. 4(d). In contrast, a WSM exhibits surface BZ Fermi arcs for the constant in-between Weyl points as we show in Fig. 4(a). The stacked Fermi arcs from chiral edge states of QAHE subsystems yield almost quantized Hall conductivity: where is the distance between the Weyl points in the BZ. In Section 3 we will discuss in detail possible realizations of topological semimetals, Chern insulators, and TIs in AFs.

### 2.1 Magnetic symmetry and spintronics effects

The potential presence of long-range ordered antiferromagnetic textures and spintronics effects in AFs is determined by magnetic symmetries. Magnetic point groups (MPGs) bradley2010mathematical () are obtained from the ordinary point groups by adding an additional antiunitary operation whose application reverses the direction of magnetic moments. Antiferromagnetic order can be in principle found in all 122 MPGs. The colorless MPGs, the so called category I bradley2010mathematical () (see an example of a MPG I in Fig. 5), are those which do not contain the operation at all and there are 32 of them which is the same number as the number of nonmagnetic classical point groups. The grey magnetic point groups (category II bradley2010mathematical ()) contains as an element of the magnetic symmetry group. There is also 32 of them obtained from the category I by adding the operation. Antiferromagnetic order may appear in this category since the point groups can be obtained from the magnetic space groups by removing all nontrivial unit cell translations. Thus antiferromagnetic sublattices connected by a combination of nontrivial translation and time reversal fall into this category and we show in Fig. 5(MPG II) exemplar FeSe AF structure with the operation. Finally, the category III black and white MPGs contain only in a combination with another point group symmetry (mirror, or rotation). There are 58 of them and we show in Fig. 5(MPG III) three antiferromagnetic examples (from left): A DSM AF crystalSmejkal2016 (), a WSM AF crystalWan2011 () and a WSM AF crystal with a nonzero AHE Yang2016c ().

The form of spintronics linear response tensors is obtained by the application of magnetic symmetries. The Neumann principle states that any physical observable of a system must exhibit symmetry of the point group of the system bradley2010mathematical (). A special role is played by and symmetries which define the basic transformation properties of tensors as shown in Tab. 1.

Tensor rank | Even (scalar, matrix) | Odd (vector, 3rd rank) | ||||||
---|---|---|---|---|---|---|---|---|

Time reversal | + | - | + | - | ||||

Spatial inversion | + (polar) | - (axial) | + (polar) | - (axial) | - (polar) | + (axial) | - (polar) | + (axial) |

+ | - | - | + | - | + | + | - | |

Exemplar tensor | AMR | b |

For the conductivity and spin Hall conductivity analysis, it is sufficient to use the magnetic Laue group Seemann2015 (). and do not change sign under spatial inversion and thus this symmetry can be omitted leading to only 32 magnetic Laue groups to investigate Kleiner1966 (). In contrast, the SOT torkance tensor changes sign under spatial inversion and thus non-centrosymmetric lattice sites are required and all 122 MPGs have to be considered Wimmer2016a (); Zelezny2017 (). This procedure leads to the conclusion that in the MPG from category I, and III there are in total 31 MPGs which allow for uncompensated moments, ferrimagnetism, ferromagnetism and also a nonzero AHE conductivity . In Fig. 5(MPG III) - right panel we show a corresponding example of the non-collinear AF structure of MnGe.

The torkance tensor is defined by Freimuth2014a (), where t is a torque generated by an applied electric field E. The Onsager reciprocal for SOT is the inverse SOT obtained by interchanging the perturbation and response and both effects can be desribed by the torkance tensor Freimuth2015 (). SOT can be decomposed into an even part and an odd part : . There are 21 non-centrosymmetric point groups which allow for a global or local current-induced nonequilibrium spin polarisation, and thus for SOT, in the lowest order that is independent of magnetization Ciccarelli2016 (). Spin-orbit fields generating the SOT can be decomposed into a combination of Rashba, Dresselhaus, and Weyl symmetry as shown in Fig. 6(a). The AF variants of SOT can be found in AFs with non-centrosymmetric magnetic sublattices connected by crystalline symmetries as we illustrate on the example of the CuMnAs AF in Fig. 6(b) Zelezny2014 (); Wadley2016 (); Smejkal2016 (); Bodnar2017 (). Here the dominating SOT is driven by a staggered effective current-induced field of Rashba symmetry and the magnetic sublattices are connected via symmetry Smejkal2016 (). Wimmer et al. Wimmer2016a () list forms of torkance tensors for all 122 MPGs.

There are also 21 MPGs which contain combined symmetry and can host Dirac quasiparticles as we explain in Section 3. Other effects can be treated analogically. For instance, the magneto-optical Kerr effect Feng2015 () and anomalous Nernst effect Ikhlas2017 () were predicted and observed in non-collinear AFs as well. Also, this scheme can be applied to the layer-resolved quantities in heterostructures, e.g. the layer-resolved conductivity Seemann2015 ().

Finally, AF order can coexist with superconductivity as we illustrate in Fig. 5 by the grey shaded ellipse and examples are the iron-based superconductors Wang2016e (). More complicated magnetic structures such as spin spirals, spin density waves, and skyrmions may not allow to be described completely in the classical MPG framework. Gopalan and Litvin Gopalan2011 () have shown the possibility of new hidden symmetries, an example being local roto-inversion. This operation does not rotate the whole crystal but just a finite subset while unchanging its MPG. The novel magnetic counterpart symmetries might be also relevant for the complete description of antiferromagnetic structures.

### 2.2 Electronic structure and band touchings

Antiferromagnetic exchange interactions arise as a result of the complex interplay among the electrons. Different types of exchange interactions are possible, e.g., direct, indirect, superexchange, or itinerant exchange. The electronic structure of AFs is thus very often complicated and requires the inclusion of correlation and many particle effects. A realistic insight into the electronic structure and the existence of the antiferromagnetic phase can be determined by the density functional theory (DFT). In DFT, the interacting many-particle problem is mapped onto non-interacting electrons in an effective Kohn-Sham potential Kohn1965 (). Hohenberg and Kohn have shown that the ground-state properties of the effective electronic gas are uniquely determined by the electronic density Hohenberg1964 (). The reformulation of the problem as a variational one tremendously decreases the macroscopic 10 degrees of freedom to just 3 - the spatial coordinates of the electronic density. In a magnetic system with a strong relativistic SOC, the magnetic relativistic ”spin-only” (neglected diamagnetic effects strange1998relativistic ()) Kohn-Sham-Dirac Hamiltonian reads strange1998relativistic ():

(4) |

where is a momentum, and is the spin independent part of the potential and the exchange-correlation magnetic field, are 4x4 Dirac matrices, m is spin density, and the electronic density is obtained from strange1998relativistic (). The Kohn-Sham potential is not known exactly and has to be approximated, e.g., by the local density approximation (LDA) or generalized gradient approximation (GGA) prasad2013electronic (). The set of equations for the electronic wavefunctions and Kohn-Sham potential is solved iteratively. The procedure yields ground state wavefunctions and Hamiltonian from which other quantities, e.g., linear response coefficients can be calculated. For instance in Fig. 8(b) we show electronic bands of the orthorhombic AF CuMnAs as calculated within GGA Smejkal2016 ().

AF systems are often correlated and disordered. Electronic correlations can be treated within DFT+Hubbard U, or DFT+dynamical mean field theory (DMFT). We show the generalized band structure of an AF BaFeAs calculated by DFT+DMFT in Fig. 7(b). The effects of disorder can be captured by the supercell technique prasad2013electronic () or coherent potential approximation prasad2013electronic () as was demonstrated, for instance, for disordered MnAu AF Bodnar2017 ().

Symmetries impose constraints on the electronic spectrum, including the existence and protection of band touchings. The palette of quasiparticles in solids is more rich than the three types of high energy physics excitations: Weyl, Dirac and Majorana fermions Vafek2013 (). This is because of the more complex crystalline symmetries that are not present in the high energy vacuum Wieder2016 (); Bradlyn2016 (). In the next Section we illustrate how the effective Hamiltonian arises for Dirac and Weyl quasiparticles in an AF.

## 3 Topological antiferromagnetic phases

Topological magnetic phases can be found in heterostructures with antiferromagnetic elements as well as in bulk AFs. In Tab. 2, we list promising topological AFs for spintronics together with the status of theoretical predictions and experimental observations of topological state or spintronics effects.

AF | Phase | T | Space group | Representative effect |

\svhline | ||||

FeSe | QSHE Wang2016e () | P4/nmm | Superconductivity | |

GdPtBi | TIMong2010 (); Li2011 (); Li2015 ()/Weyl Hirschberger2016 () | 9 Suzuki2016 () | F3m | Large thermopower Hirschberger2016 (), AHE Suzuki2016 () |

\svhline | ||||

SrMnBi | Dirac metalPark2011a () | 290 Park2011a () | I4/mmm | Angular dependent magnetoresistance Wang2011e () |

CaMnBi | Dirac metal | 300 Guo2014 () | P4/nmm | Dirac fermions coupled to magnetism Guo2014 () |

EuMnBi | Dirac metal | 22* Masuda2016 () | I4/mmm | QHE controlled by magnetism Masuda2016 () |

BaFeAs | 2D Dirac metalChen2017 () | 143 Huang2008 () | I4/mmm | Superconductivity |

CuMnAs | Dirac semimetalMaca2012 (); Tang2016 (); Smejkal2016 () | 400 Maca2012 () | Pnma | TopoMIT, TopoAMR Smejkal2016 () |

X Ir O | Weyl semimetalWan2011 (); Sushkov2015 () | Fdm | TopoMIT Tian2015 (), wealth of topo. phases Wan2011 (); Kondo2015 () | |

MnSn | Weyl (semi)metal Yang2016c (); Kuroda2017 () | 430 Nakatsuji2015 () | P63/mmc | AHE controlled by magnetic fieldNakatsuji2015 () |

Italic font marks theoretical prediction. Normal font marks existing experimental signatures.

### 3.1 Low dimensional Dirac antiferromagnets and superconductors

Introducing magnetism into TIs is known to modify the spin texture of the Dirac quasiparticles Xu2012 (). Magnetism can couple to TIs either by creating the magnetically doped TI (MTI) Xu2012 (), or by proximity coupling between the TI and magnetic order in heterostructures Katmis2016 (). AF order was shown to increase the critical temperature of the adjacent MTIs in the MTI/AF CrSb/MTI heterostructure He2016 (). Improved performance of the SOT in terms of larger spin Hall angles and lower critical currents was achieved in the TI/ferrimagnetic CoTb alloys Han2017 () with the AF coupled Finley2016 () Co and Tb sublattices.

The first TI AF was predicted in systems with combined symmetry Mong2010 (). GdPtBi AF was suggested as a possible candidate, as we show in Fig. 3(b). GdPtBi was up to date not confirmed as a TI presumably due to the low resolution of data obtained by the angle resolved photoemission spectroscopy (ARPES) Liu2011f (). However, signatures of the coexistence of a 2D TI (Fig. 4(c)), and a superconducting state in hole-doped and electron-doped antiferromagnetic monolayers of FeSe were demonstrated Wang2016e (). FeSe belongs to the metallic building block of the iron-based high-T superconductors. Remarkably, the combined effects of SOC, substrate strain, and electronic correlations can induce band inversion and QSHE edge states, as we show in Fig. 7(a) Wang2016e (). Creating a p-n junction across FeSe and attaching two ferromagnetic electrodes can generate Majorana zero modes at the interfaces Tsai2016 (). Majorana states are considered for a possible use in quantum computing Beenakker2016 ().

One of the first systems explored for observing Dirac quasiparticles in condensed matter beyond graphene were the SrMnBi type AFs. The electronic structure of these systems is governed by the quasi-2D square Bi planes. The Bi states create close to the Fermi level massive Dirac quasiparticles. High mobilities, and Fermi velocities, and pseudospin structure of wavefunctions are reminiscent of graphene properties. In contrast to graphene, however, the quasiparticles are highly anisotropic with anisotropy factor of 8 Wang2011e (). Several of these types of AFs were reported in recent years including SrMnBi Wang2011e (), or CaMnBi Guo2014 (). The systems belong to the 112-type pnictides where the antiferromagnetism and Dirac quasiparticles might coexist also with superconductivity. Related systems, e.g. YbMnBi, were inconclusively Armitage2017 () reported to be either WSM Borisenko2015 (); Chinotti2016 () or DSM Wang2016k (); Chaudhuri2017 (). Most likely the collinear AF order cants in, e.g., the (Sr,Yb)MnBi alloy Liu2017 () where the double band degeneracy breaks and Weyl points might emerge. Despite many recent studies, more accurate and detailed measurements are needed to reveal the detailed nature of Dirac quasiparticles in these systems. Finally, in the sister compound, EuMnBi AF, the half-integer QHE was reported controllable by the strength of an external magnetic field, as we showed in Fig. 4(b). EuMnBi contains at very low temperatures two antiferromagnetic sublattices. The presence of QHE was linked to the confinement of the massive Dirac quasiparticles by the spin-flop at the Eu AF sites Masuda2016 ().

Recently, the high temperature superconducting 122-type pnictide XFeAs (X=Ba, or Sr) AFs showed signatures of topological Dirac quasiparticles in the infrared spectra in high magnetic fields. In Fig. 7(b) we show the state-of-the-art DFT+DMFT calculation of quasi-2D Dirac cones close to the Fermi level which are consistent with the observed Landau level spectra and density of states Chen2017 (). When the Fermi states are dominated by Dirac quasiparticles, the topological semimetal is achieved. Electron filling enforced semimetals with a single Dirac cone were predicted theoretically in 2D model AFs Young2016 (). In Fig. 7(c) we show the single Dirac cone at the point in the BZ of the monolayer of the FeSe AF with a stripe order. The quasi-low dimensional systems and heterostructures, however, suffer from fragile magnetism and low critical temperatures. In two following subsection, we describe possible room temperature 3D Dirac and Weyl semimetal AFs.

### 3.2 3D Dirac semimetal antiferromagnets

Dirac quasiparticles are allowed in doubly-degenerate bands bernevig2013topological () realized in systems invariant under symmetry. {svgraybox} The low energy Hamiltonian might maintain an effective Dirac form Vafek2013 (); Yang2014a (); Smejkal2017 (), corresponding to the Dirac Hamiltonian (4) Burkov2016 (); Smejkal2017 (); Armitage2017 ():

(5) |

Here is the Fermi velocity, is the crystal momentum measured from the Dirac point at , is the mass (in units of energy), and is the vector of Pauli matrices. The energy dispersion gives massive Dirac cones,

(6) |

The mass term can be removed by the presence of an additional crystalline symmetry, and in this case describes the four-fold degenerate band touching Burkov2016 (); Smejkal2017 () of a 3D DSM as we show in Fig. 8(b) on the band dispersion of an antiferromagnetic DSM orthorhombic CuMnAs with a high Néel temperature of 400 K Maca2012 ().

The 3D DSM state cannot occur in ferromagnets because -symmetry breaking prevents double band degeneracy. However, a topological crystalline 3D DSM was predicted in an AF, namely in the orthorhombic phase of CuMnAs Tang2016 (); Smejkal2016 (). The unit cell of the orthorhombic CuMnAs contains four Mn sublattices that are connected in pairs by the symmetry Tang2016 (); Smejkal2016 () as we show in Fig. 8(a). Although individually the and symmetries are broken, the preserved combined symmetry ensures the double band degeneracy over the whole BZ. In the calculation with a switched-off SOC (see Fig. 8(b)), we observe three Dirac points at the Fermi level along the , , and lines which are part of the nodal line protected by the symmetry. The Dirac quasiparticles are 3D as can be seen from the quadratically vanishing DOS at the Fermi level/Dirac point as we show in Fig. 8(c). In the presence of SOC and for the Néel order along the [001] axis, the nodal lines become gapped except for the two Dirac points along the line, as we show in Fig. 8(d). The Dirac points are protected by the non-symmorphic screw axis symmetry Tang2016 (); Smejkal2016 () and are connected via nontrivial surface states Tang2016 (). The topological invariants and surface states can be linked to the crystalline symmetry protecting the degeneracy and even in the non-magnetic DSM, the surface states are in general less stable than in WSM and strongly depend on the crystalline orientation at the surface termination Yang2014a (); Kargarian2016 (). The easy axis in orthorhombic CuMnAs tends to be along [100] according to ab initio calculation Smejkal2016 (), however, we will discuss in the next section the possibility of reorienting the Néel vector.

The orthorhombic CuMnAs AF is an attractive hydrogen atom for magnetic DSMs induced by band inversion since only a single pair of Dirac points appears near the Fermi level of the ab initio band structure. However, presumably, the correlation and disorder effects prevented the observation of Dirac quasiparticles in non-stoichiometric CuMnAs to date Emmanouilidou2017 (); Zhang2017e ().

### 3.3 Weyl semimetal antiferromagnets

In solids, quite often at least one of the or symmetries (and also ) is broken and thus the double band degeneracy is lifted. When the two non-degenerate bands are touching close to the Fermi level a 3D WSM can be formed. {svgraybox} WSM is described by the generalized two-band Weyl Hamiltonian Burkov2016 (); Smejkal2017 ():

(7) |

where the first term corresponds to the tilting of the Weyl cone and . Weyl points always come in pairs with opposite topological charges. Dimensionality is important here. Because the Weyl points are 3D objects in the BZ, the effective Hamiltonian uses all three Pauli matrices. Thus any small perturbation expressed without loss of generality as a linear combination of these three Pauli matrices just shifts but not gaps the Weyl point. We illustrate this on the dispersion around the Weyl points for a perturbation of a form ,

(8) |

WSM states can be found in nonmagnetic, ferromagnetic Xu2011 (); Wang2016c () or antiferromagnetic solids, where the symmetry is broken. A WSM state was observed in non-centrosymmetric non-magnetic mono-pnictides of the TaAs type Xu2015b (); Lv2015 (); Yang2015d (). TaAs is well described by the DFT single quasiparticle picture. Despite numerous predictions, the true magnetic WSM remained for a long time experimentally elusive Kuroda2017 (). The reason is that the magnetic system is very often also strongly correlated, disordered, and the symmetry breaking is provided by the complex collective phenomenon - magnetism. Antiferromagnetic candidates include pyrochlore irridates Wan2011 () like the EuIrO Sushkov2015 (), or YbMnBi AF which were suggested to be either the WSM Borisenko2015 (); Chinotti2016 () or the DSM Wang2016k (); Chaudhuri2017 ().

Weyl fermions were proposed also in non-collinear AFs MnGe and MnSn Kubler2014 (); Yang2016c () (see Fig. 2(a)) Yang2016c (). These AFs are potentially appealing for spintronics due to the measured large AHE, established magnetic structure, and Néel temperatures reaching 365-420 K. The structure of MnGe and MnSn crystals is built from stacked kagome planes along the [001] axis, as we show in the right panel in Fig. 5(MPG III). These AFs have a relatively weak magnetic anisotropy, reaching 0.1meV per formula unit for MnSn Sandratskii1996 (); Duan2015 () due to the vanishing second and fourth order MAE of the inverted triangular AF structure on the kagome lattice Nagamiya1982 (); Tomiyoshi1982 (). The magnetic order can be thus reoriented by low external magnetic fields. Ref. Kuroda2017 () reports reorientation fields of 200 Gauss. The net magnetic moment reaches 0.005 B per unit cell Tomiyoshi1982 (). In spite of the weak anisotropy of the inverted chiral structure, the materials show relatively high stability against thermal fluctuations. Also, a possibility to influence the in-plane chiral AF magnetic structure by a spin-filtering effect was reported Fujita2017 ().

MnGe and MnSn were predicted to exhibit several different types of Weyl points in their metallic bandstructure coexisting with trivial bands close to the Fermi level Yang2016c (). The Weyl points found by tracking the Berry curvature in the BZ are tilted, thus of the so-called type-II Soluyanov2015 (). The Fermi arc surface states - the hallmark of a WSM - were predicted by first-principles calculations of the local density of states (LDOS) as we show in Fig. 9(a) Yang2016c (). The tilting of the Weyl points does not influence the Berry curvature, however, the electron and hole pockets due to the tilting influence the transport effects, particularly they can renormalize the almost perfectly quantized AHE Zyuzin2016a (). Signatures of these band crossings were reported recently in an ARPES study of MnSn AF Kuroda2017 (), as we show in Fig. 9(b). Weyl band crossings were found along or lines presumably depending on the orientation of the triangular magnetic texture stabilized by an external magnetic field Kuroda2017 (). A comparison between DFT and ARPES points towards strong electron correlations in the Mn 3d bands as seen in the strong renormalization of the bands in ARPES in Fig. 9(b). In the next Section, we describe in detail the AHE in MnGe and MnSn and its relation to Weyl points.

## 4 Topological antiferromagnetic spintronics effects

Topological variants of common magnetotransport effects and their novel cousins can potentially offer large signal to noise ratios important for reading signals in spintronics nanodevices. Again, we will demonstrate that very often the unique antiferromagnetic symmetries are of vital importance for certain topological spintronics effects to occur, such as topological AMR or AHE.

### 4.1 Large magnetoresistance and chiral anomaly

Application of a magnetic field perpendicular to the current flow results in a positive magnetoresistance, defined as

(9) |

where is the resistivity in a magnetic field , as commonly observed in metals, semimetals, and semiconductors. However, in a topological semimetal with Weyl quasiparticles, a negative magnetoresistance might occur, attributed to a chiral anomaly. In the original proposal for a consensed matter realization of the chiral anomaly, Nielsen and Ninomiya Nielsen1983 () considered a chiral Weyl linear dispersion of the zero-th Landau levels. Application of an electric field parallel to the magnetic field generates an imbalance between the zero Landau levels of opposite chiralities. This axial current leads to a positive magnetoconductivity Armitage2017 ():

(10) |

where is the density of states at the Fermi level. Remarkably, this expression can be derived both in the quantum limit or in the semi-classical framework without introducing Landau levels Armitage2017 (). We illustrate the chiral anomaly with Weyl fermions in Fig. 10(a).

The negative magnetoresistance became accepted as the signature of the presence of linearly dispersing topological quasiparticles, and was possibly observed for instance in GdPtBi Hirschberger2016 (), which is a quadratic gapless semiconductor. In the magnetic field, the fourfold degenerate band-touching splits and pairs of Weyl points are created as we illustrate in schematics in Fig. 10(b). Rotating the external magnetic field from out-of-plane to in-plane (see Fig. 10(c)) changes the magnetoresistance from positive to negative. Positive non-saturating magnetoconductance was observed recently also in the MnSn AF Kuroda2017 (). Alternative sources of positive magnetoconductance such as current jetting and weak localization were carefully ruled out in this study. The positive magnetoconductance in MnSn is linear in a magnetic field which was attributed to the type-II Weyl fermions Kuroda2017 () in contrast to the quadratic magnetoconductance observed in type-I (non-tilted) Weyl semimetals Zyuzin2017 (). The detailed role of topological quasiparticles in the negative and large non-saturating magnetoresistance remains to be clarified Ali2014 (); Pletikosic2014 (); Soluyanov2015 (); Khouri2016 (). Here the first step was made by ab initio Kim2017e () and transport studies Zhang2017f () of Weyl points in strong magnetic fields signaling the importance of linear dispersion for the observation of negative magnetoresistance. In strong magnetic fields, the negative magnetoresistance disappears which was attributed to the gapping of Weyl points by the Zeeman splitting.

### 4.2 Topological phase transitions and anisotropic magnetoresistance in antiferromagnetic systems

Topological phase transitions were experimentally demonstrated in heterostructures with TIs and AFs He2016 (); He2016a () or systems with an artificially engineered AF coupling Mogi2017 (). AF CrSb/TI (Bi,Sb)Te /AF CrSb heterostructure shows spikes in the magnetoresistance which were attributed to a topological phase transition of Dirac quasiparticles at the interfaces He2016a (). A MTI/TI/MTI heterostrucutre was reported for a presumed topological phase transition between QAHE state and axion insulator (quantized topological magnetoelectric effect) by switching the magnetic order in the MTI from ferromagnetic to antiferromagnetic by an external magnetic field or electric gating Mogi2017 (); Tokura2017 (). Albeit at mK temperatures, the phase transition yields very large magnetoresistance or electroresitance changes corresponding to switching on and off the quantized conductivity plateous Mogi2017 (); Tokura2017 (). The QAHE effect was to date observed only at mK temperatures Wang2017b (). Searching for novel mechanism and material candidates with a more robust, controllable and room temperature QAHE states represents an important direction of future research in topological spintronics. Here for instance, QAHE induced by electrical gating was predicted in SrFeOsO AF Dong2016 ().

The prediction of tuning the Néel order parameter by the Néel SOT or gating in antiferromagnetic DSMs opens the possibility of using the topological metal-insulator transition (TopoMIT) in a bulk AF Smejkal2016 (); Smejkal2017a (). The origin of these effects is in the sensitivity of Dirac crossing hybridations on the orientation of the Néel vector in the orthorhombic magnetic crystalline symmetry, as shown in Fig. 8(d). In the presence of SOC, only for the Néel vector along the [001] axis, protected Dirac fermions emerge. All the other Néel vector orientations lead to a gapped spectrum at the Fermi level Smejkal2016 (). The transport counterpart of the topoMIT was predicted to be the topological anisotropic magnetoresistance (topoAMR) Smejkal2016 (). The topoAMR can be extremely large and is understood as a limiting case of the crystalline AMR. In Fig. 11(a) we show the tetragonal lattice of a minimal model of the DSM AF Smejkal2016 (). The corresponding Hamiltonian reads , where first two terms are first and second neighbor hoppings, third term is a staggered SOC, last term is AF s-d type exchange and , and are Pauli matrices corresponding to orbital and spin-degree of freedom, respectively. The longitudinal conductivity is calculated from the Boltzmann formula in the limit of a small spectral broadening :

(11) |

where is the size of the system, and is the angle between [100] axis and magnetization. The AMR is defined as:

(12) |

where is the average conductivity within the plane. In Fig.11(a) we show the angular dependence of the AMR in this model YamamotoSmejkal2017 (). In Fig. 11(b) we show the band structure of the model. For the Néel vector orientations [100] and [010], preserving the glide mirror planes of the system Smejkal2016 (), the Dirac points are gapless and conduct. Once the Néel vector is rotated away from these high symmetry axes, the crystalline symmetries are broken, Dirac bands hybridize and gap opens. Consequently the conductivity decreases exponentially. The sharp peaks in the angular dependence are very different when comparing to the standard harmonic AMR dependence in ferromagnetic alloys. Also the origin is very distinct. The topoAMR originates in Fermi surface topology changes instead of scattering effects responsible for the standard AMR. The difference in conductivity between the [100] and [010] direction originates in the anisotropy of the Dirac cones. The orthorhombic CuMnAs was predicted as the realistic material candidate based on ab initio calculations Smejkal2016 (). The interplay of the Dirac points and topoAMR with disorder, interaction effects, and nonequilibrium currents needs to be carefully addressed to potentially make the effect relevant for real spintronics device applications. Foreseen applications include topological transistors or memories Smejkal2017a (). Since the control of the Néel vector can be achieved either by the Néel SOT due to the applied current or due to the tuning of the MAE by electric gating, the effect is presumably more favorable for spintronics than the MIT manipulated by external magnetic field in pyrochlores Tian2015 () or AF topological semimetal candidate NdSb Wakeham2016 (). Although this topoAMR due to the MIT was not experimentally discovered yet, analogical effects controlled by external magnetic field were observed in WSMs.

In GdPtBi, Weyl point positions are sensitive to the orientation of the applied magnetic field Hirschberger2016 (). This leads in turn to a pronounced angular dependence of the magnetoconductance, as we show in Fig. 11(c). The changes in magnetoconductance are attributed to the varying angle between the crystalline axis and the Zeeman field what is in contrast to the behavior predicted for the AF DSM CuMnAs. The spikes in magnetoconductance have been measured also in the correlated WSM AF MnSn as we illustrate in Fig. 11(d) Kuroda2017 (). Here the magnetic order is controlled by the relatively weak external magnetic field. The reorientation of the moments changes the local symmetry and possibly redistributes the Weyl points close to the Fermi level Kuroda2017 (). Importantly, the spikes in magnetoconductance persist to temperatures 100 K, despite the WSM is highly correlated and disordered. These temperatures are much higher temperatures than the reported QAHE critical temperatures of 10 mK.

### 4.3 Anomalous Hall effect in noncollinear antiferromagnets

AHE refers to the transversal electric current generation in the magnet subjected to an applied longitudinal electric field. The anomalous Hall conductivity is for the magnetization along axis the antisymmetric part of the conductivity tensor:

(13) |

For a long time, the AHE was considered to scale with the magnetization: , where the first part corresponds to the ordinary Hall effect due to the external magnetic field , and the second term is the AHE due to the symmetry breaking due to the magnetization , and , and are ordinary and (spontaneous) anomalous Hall coefficients, respectively. AHE was traditionally attributed to the simultaneous presence of symmetry breaking by the ferromagnetism and SOC. Thus, naively, one would expect that the AHE must vanish in AFs because of the compensation moments of the opposite sublattices. Indeed, this picture is valid in simple AFs where the combination of symmetry with another crystalline symmetry forces the AHE to vanish. Typical examples include collinear AFs with or symmetries which we discussed in the context of the Dirac quasiparticles and TIs. Remarkably, the AHE was observed in systems with a negligible net magnetization and without the necessity for SOC. Interestingly, already Haldane pointed out in 1988 Haldane1988 () the possibility of the quantized AHE in honeycomb lattice with complex hoppings with a staggered potential and Shindou et al. Shindou2001 () later demonstrated the nonzero AHE in a model calculation in AHE induced by distorting the FCC lattice with 3Q AF order. More recently Hua Chen et al. Chen2014 () and Kübler and Felser Kubler2014 () predicted a large AHE in noncollinear AFs with a negligible net moment. Presumably, the large MAE in MnIr AF makes it impossible to orient the magnetic domains and thus prevents the experimental detection of the AHE in this compound.

However, MnGe and MnSn have a much smaller MAE. AHE from the AF texture, , was indeed observed in these compounds by carefully substracting the Hall effect originating from the external field, , and from the small ferromagnetic moment, Nakatsuji2015 (); Nayak2016 ():

(14) |

where is the net magnetization. The experimental value of the AHE in MnGe is 380 Kiyohara2015 () while the ab initio calculation from Berry curvature gives 330 Zhang2016d (). The noncollinear AF order on the kagome lattice breaks the time-reversal symmetry as we show in Fig. 12(a) and 12(b). For the AF structure in Fig. 12(a) Tomiyoshi1982 (); Kiyohara2015 () there is an effective time reversal symmetry ( is the mirror (100) plane symmetry) which gives , and the emergent magnetic field lies along [010], and only . In constrast, for the chiral texture in Fig. 12 (b) Tomiyoshi1982 (); Kiyohara2015 (), the effective symmetry ( is the glide mirror plane ) implies , the emergent magnetic field points along [100], and only . Furthermore, independent on the in-plane orientation, there is an effective time reversal symmetry ( is the mirror (001) plane symmetry) making the component . The symmetry analysis is consistent with the experimental data measured on MnSn and MnGe and presented in Fig. 2(b) and 12(c). In conclusion, the spin-orbit entangled bands generate a large fictitious magnetic field in the crystal momentum space parallel to the direction of the field stabilizing the triangular order, and the AHE takes place in the plane perpendicular to the field.

The emergent magnetic field was estimated to be very large, of the order of 200 T in MnGe Kiyohara2015 (). Although the MnGe and MnSn were predicted to host Weyl points close to the Fermi level, the ab initio calculation of the AHE shows that the largest contributions come from BZ regions not related to any identified Weyl points, but rather from spin-orbit entangled avoided crossings Zhang2016d (). A recent study by Kübler and Felser Felser2017 (), however, demonstrates the possibility of propagation of Fermi sea Weyl points in MnGe and MnSn to the Fermi surface-quasiparticle transport Haldane2004 (); Gos2015 ().

### 4.4 Topological Hall effects and antiferromagnetic skyrmions

In the topological Hall effect, the role of SOC is overtaken by the spin chirality. We show in Fig.12(d) the spin chirality generating a nonzero Berry curvature. Spin chirality is nonzero in non-coplanar spins, in contrast, it vanishes in coplanar non-collinear antiferromagnetic structures of, e.g., MnGe and MnSn. The spin chirality generates a fictitious magnetic field (see red arrow in Fig. 12(d)), . This field acting on the Bloch electrons generates a Hall response. The topological Hall effect and the AHE can be possible to experimentally disentangled by analyzing the disorder dependence Kanazawa2011 (). The topological Hall effect was initially reported in antiferromagnetic pyrochlore iridates (see Fig.12(e)) Machida2010 () and later in MnSi chiral antiferromagnetic alloys Surgers2014 (); Surgers2016 (). We note that the effect does not imply in this case a correspondence to a topological invariant, in sharp contrast to the topological Hall effect in skyrmions. However, in the quantum topological Hall effect proposed for the non-coplanar AF KRhO Zhou2016 (), with its magnetic sublattices shown schematically in Fig.12(d), the topological charge occurs in the momentum space as in the QAHE.

The topological Hall effect from a skyrmion spin texture is associated with a topological winding number of the skyrmion Tokura2017 (). It is important to distinguish it from the skyrmion Hall effect which refers to the deflection of skyrmion center due to the Magnus force. The Magnus force is according to micromagnetic calculations not present in AF skyrmions, which implies that AF skyrmions might move in straight lines Barker2016 (). This is favorable when considering skyrmions for storing information in racetracks. Finally, the topological Hall effect in AF skyrmions with a compensating sublattices vanishes, while the topological spin Hall effect can be still sizable and can be used to detect an AF skyrmion Buhl2017 (); Gobel2017 ().

## 5 Summary

Antiferromagnetic spintronics has been recently established as a new branch of magnetism MacDonald2011 (); Jungwirth2016 (); Smejkal2017 (); Sander2017a (). In parallel, last few years have seen progress in coupling magnetism with topological states of matter, giving rise to a new spin-off: topological spintronics Fan2016b (). We have shown that AF order might play an important role in topological spintronics due to the unique AF symmetries Smejkal2017a (). While signatures of correlated AF WSM were already observed Kuroda2017 (), other topological AF phases remain to be discovered. The large signal to noise ratio was reported in magnetoresistance signals of topological semimetals Smejkal2016 (); Kuroda2017 (). Further theoretical and experimental progress will possibly lead to topological spintronics effects improving the reading and writing signals in AFs Smejkal2017a (). The progress in writing efficiency due to the nontrivial topologies is in its infancy, although an increase of the spin Hall angle in TI/FMMellnik2014 (), TI/MTIFan2014a (), or TI/ferrimagnetic systems Han2017 () was already reported and novel mechanisms for dissipationless SOT were suggested Hanke2017a (). Here we focused on the state-of-the-art effects which were predicted, and some of them already experimentally confirmed, in antiferromagnetic systems. The unique AF symmetries and the abundancy of AF allow for other research directions to emerge such as topological superconductivity in AFs and spintronics based on antiferromagnetic skyrmions Smejkal2017a ().

###### Acknowledgements.

We acknowledge support from the Ministry of Education of the Czech Republic Grants LM2015087 and LNSM-LNSpin, the Grant Agency of the Czech Republic Grant No. 14-37427, and the EU FET Open RIA Grant No. 766566.## Bibliography

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