Intrinsic magnetic topological materials

Abstract

Topological states of matter possess bulk electronic structures categorized by topological invariants and edge/surface states due to the bulk-boundary correspondence. Topological materials hold great potential in the development of dissipationless spintronics, information storage and quantum computation, particularly if combined with magnetic order intrinsically or extrinsically. Here, we review the recent progress in the exploration of intrinsic magnetic topological materials, including but not limited to magnetic topological insulators, magnetic topological metals, and magnetic Weyl semimetals. We pay special attention to their characteristic band features such as the gap of topological surface state, gapped Dirac cone induced by magnetization (either bulk or surface), Weyl nodal point/line and Fermi arc, as well as the exotic transport responses resulting from such band features. We conclude with a brief envision for experimental explorations of new physics or effects by incorporating other orders in intrinsic magnetic topological materials.

Graphical abstract

Keywords

intrinsic magnetic topological insulator / magnetic topological metals / magnetic Weyl semimetal / topological surface states / magnetic gap

1 Introduction

The first two decades of the new millennium have witnessed the surge of topological states of matter, whose electronic structures can be categorized by topological invariants. The prediction [1-6] and realization [7-11] of two-dimensional (2D) and three-dimensional (3D) topological insulators (TIs) have caused a paradigm shift to predict, understand and make use of quantum materials based on the topology of their band structures. Started with the 2D TI, quantum spin Hall (QSH) effect realized based on HgTe/CdTe quantum well [6,7] has revealed to the world the fundamental novelty and potential application of topological materials. QSH state is insulating in the bulk but has a pair of one-dimensional (1D) conducting edge states protected by time-reversal symmetry. Electrons in the 1D edge states move without elastic backscattering by nonmagnetic impurities, holding potential for dissipationless spintronics. Likely, a 3D strong TI is also insulating in the bulk and has 2D gapless topological surface states (TSSs). 3D TI was first realized based on Bi−Sb alloys [8] and then on Bi₂Se₃ family [5,9-11]. The robustness of topological protection to the TSSs from nonmagnetic perturbations has been experimentally demonstrated [12-14], pointing to feasible electronic and spintronic applications. Importantly, based on magnetically doped Bi₂Te₃ films, quantum anomalous Hall (QAH) effect was realized [15,16], a milestone towards low-power-consumption electronics without the need for applied magnetic field.

Besides insulators, quantum materials can also be metals and semimetals according to the detailed band structure around the Fermi level. After TIs, topological semimetals emerged as novel states of matter with degenerate band crossing close to which the band dispersion can be described by the massless 3D Weyl and Dirac equations [17-23].In a Dirac semimetal (DSM), the conduction and valence bands touch at discrete (Dirac) points with linear dispersion, forming bulk (3D) Dirac fermions. Given broken time-reversal or inversion symmetry, 3D Dirac fermion can be separated in the momentum space into two Weyl fermions (chiral massless fermions as a description of neutrinos with neglected mass in high-energy physics), resulting into topological Weyl semimetals (WSMs). Such nontrivial electronic features could bring into novel electrical and thermal transport behaviors such as anomalous Hall effect (AHE), anomalous Nernst effect, chiral anomaly signified by negative magnetoresistance and non-saturating magnetoresistance (see Refs. [17,24-26] for comprehensive reviews). Such properties come from the enhanced Berry curvature hosted in the Dirac/Weyl type band structure, which exhibits extreme responses to external stimuli such as magnetic field, voltage or current bias, temperature gradient and optical excitation.

There have been hundreds of materials predicted as 3D strong TI [27-29] and dozens of them have been experimentally verified, usually through direct observation of their TSS Dirac cones by angle-resolved photoemission spectroscopy (ARPES) [30-32]. By comparison, magnetic TIs, especially intrinsic magnetic TIs, are limited in the material candidates. So far there is only (MnBi₂Te₄)·(Bi₂Te₃)_n (n=0,1 ,2,3) family which has been intensively studied as an intrinsic magnetic TI [33-39]. There are also many materials predicted and demonstrated as DSMs and WSMs, most of which are time-reversal invariant and only few materials have been studied as magnetic WSMs [17,24-26]. Recently, there appeared several layered material families with hexagonal/Kagome lattices which host Dirac cones gapped by ferromagnetic (FM)/antiferromagnetic (AFM) order, such as Fe₃Sn₂ family [40]. In the 2D limit, these systems with gapped Dirac cones can be viewed as Chern insulating phase with quantized anomalous Hall conductance [41], given the Fermi level is positioned in the Dirac gap. In this sense, these materials share the same topological characters (the Chern number C) as intrinsic magnetic TIs. However, for 3D materials, the band structure is complicated by the coexisting trivial bands which locate at the same energy region as the Dirac gap, rendering such materials in metal phase with coexisting trivial and nontrivial conduction. Consequently, we feel it more appropriate to term such materials as magnetic topological metals. While quantized transport response from the edge conduction can be realized in intrinsic magnetic TIs by tuning the Fermi level in the gap of both bulk and surface bands, there is always transport contribution from the trivial bands in magnetic topological metals no matter where the Fermi level is. It is noted that there is no strict theoretical picture describing topological metals since the metallicity does not come from band topology but trivial bands. We choose this term only to emphasize its distinction from intrinsic magnetic TIs and topological SMs.

In this review, we focus on the recent progress in the exploration of these various kinds of intrinsic magnetic topological materials, categorized mainly into three groups: intrinsic magnetic (TIs), magnetic Weyl/Dirac semimetals and other magnetic topological metals. We will present representative materials for these novel topological states of matter, pay special attention to their characteristic band features such as the gap of topological surface state Dirac cone, gapped bulk Dirac cone, Weyl nodal point/line and Fermi arc, as well as the exotic transport responses resulting from such band features. There are also other intrinsic magnetic topological states of matter which have been proposed theoretically, yet lacking affirmative experimental evidence, such as topological Möbius insulators [42-44]. We briefly discuss the opportunities to explore new states of matter and novel physical properties based on intrinsic magnetic topological materials.

2 Intrinsic magnetic topological insulator

Intrinsic magnetic TIs provide an excellent platform for the study of exotic quantum states, such as QAH states, chiral Majorana fermions, and axion states [33-39], arising from the interplay between band topology and magnetism. Among them, QAH effect is of fundamental importance in the field of spintronics due to its non-dissipative properties in transport. One approach to realize it is to find a 2D TI that comprises long-range magnetic order. Introducing magnetism into the 2D TI can break the time-reversal symmetry, such that one direction of spin channels will be canceled. Although QAH effect has been proposed theoretically in the last century [41], it is until 2013 when quantized edge resistance (h/ e^2) was experimentally observed on Cr-doped (Bi, Sb)₂Te₃ thin films [15]. The chemical doping results into inhomogeneity in the band structure (gap, carrier density) and consequently extremely low quantization temperature. Therefore, intrinsic magnetic states of matter with uniform long-range magnetic order are highly desired.

As first discussed in the theoretical proposal of AFM TIs in 2010 [52], both time-reversal symmetry \mathrm{\Theta } and fractional translation T_{1/2} are broken but the combination S=\mathrm{\Theta } T_{1/2} is preserved in AFM TI, leading to a topologically nontrivial phase which shares with 3D strong TI similar topological Z_2 invariant and quantized magnetoelectric effect. The material realization of an intrinsic AFM TI was not initiated until 2017. “Magnetic extension” picture proposed that by inserting MnTe bilayer into the quintuple layer of Bi₂Te₃, the system tends to form septuple layers of MnBi₂Te₄, hosting a robust QAH state [53,54]. The material was first experimentally realized by molecular beam epitaxy (MBE) [55]. Subsequent theoretical works revealed its colorful physics and properties [56-59]. Since the successful preparation of single crystal MnBi₂Te₄, the surge of intrinsic magnetic TIs based on MnBi₂Te₄ family started. Following the discovery of MnBi₂Te₄, a series of superlattices of this family were discovered, denoted as MnBi₂Te₄·(Bi₂Te₃)_n (n = 1, 2, 3) [60-62]. In addition, we will briefly introduce other intrinsic magnetic TI candidates such as MnSb₂Te₄·(Bi₂Te₃)_n (n = 1, 2) and EuSn₂As₂ families.

2.1 MnBi₂Te₄·(Bi₂Te₃)_n

In 2013, Lee et al. [63] synthesized the polycrystalline powder of MnBi₂Te₄ by the flux-method. In 2017, from MBE growth of heterostructure composed of MnSe and Bi₂Se₃, it was found that the topological surface state of this structure is located on the surface of the whole system, rather than at the interface of the two materials like other topological heterostructures. It was realized that the layered structure of MnSe and Bi₂Se₃ is a new type of single crystal, MnBi₂Se₄. Such transformation is also applicable to MnBi₂Te₄[53,54,64,65], and it is MnBi₂Te₄ which is the focus of intrinsic magnetic TI study due to its desirable magnetic, electronic, and structural properties.

The structure of MnBi₂Te₄ was refined to be in the hexagonal space group R\overline{3}m (No. 166) [60]. Its minimum structural unit is composed of seven atomic layers with stacking order Te−Bi−Te−Mn−Te−Bi−Te, which is called a septuple-layer (SL) and the adjacent layers are bonded by van der Waals force, as shown in Fig.2(a). The unit cell of MnBi₂Te₄ is composed of three SLs stacked in the −A−B−C− fashion, and its lattice constant c is about 4.07 nm. Its Neel temperature T_N\left(124\right) \approx 24.4 \mathrm{K} [66], above which the AFM order is transformed into paramagnetic (PM) order [Fig.2(b)]. Neutron diffraction experiments point out that the ground state magnetic structure of MnBi₂Te₄ is the A-type AFM phase [66,67]. The magnetic moment of each SL points out of plane, and the magnetic moments of adjacent layers are opposite. Of course, if Bi₂Te₃ quintuple-layers (QLs) is inserted between SLs, we can get MnBi₄Te₇, MnBi₆Te₁₀, and MnBi₈Te₁₃ superlattices [60-62,68]. MnBi₄Te₇ can be regarded as a sandwich structure formed by inserting one QL into each SL. Similarly, MnBi₆Te₁₀ and MnBi₈Te₁₃ are formed by inserting two or three QLs in each SL respectively. Note that the space group of MnBi₂Te₄, MnBi₆Te₁₀, and MnBi₈Te₁₃ is R\overline{3}m, but the space group of MnBi₄Te₇ is P \overline{3} m1. Since the distance between two SLs in MnBi₄Te₇ and MnBi₆Te₁₀ is larger than that in MnBi₂Te₄, their interlayer AFM coupling is weaker. The results of magnetic transport measurement show that the AFM-PM transition temperature of MnBi₄Te₇ is T_N\approx 12\mathrm{K} [Fig.2(c)] and that of MnBi₆Te₁₀ is T_N\approx 10.7\mathrm{K} [Fig.2(d)]. More interestingly, with further increasing SL spacing, the compound of MnBi₈Te₁₃ has become the first intrinsic FM TI withT_C\approx 10.5\mathrm{K} [Fig.2(e)]. The lattice constants and magnetic transition temperatures of these different compounds are also summarized in Fig.2(f).

The band structure of MnBi₂Te₄, as the first intrinsic magnetic TI, has been intensively studied [59,69,73-76] and the TSS inside the bulk gap is the focus of attention. At the early stage, a sizable gap was found for the TSS Dirac cone with temperature-independent behavior [59,77,78]. However, subsequent ARPES works with systematic photon-energy-dependent measurement and higher energy and momentum resolution have revealed the nearly gapless behavior of TSS [69,73-76,79-83], showing sample and location dependence [Fig.3(a)]. Here we use the term “nearly gapless” to describe the experimental observation that the size of Dirac gap varying from being vanishing to dozens of millielectronvolts, being much smaller than expected by theoretical calculation [56-59]. Such behaviors suggest much reduced effective magnetic moments felt by the TSS, which may arise from surface magnetic reconstruction or TSS redistribution (extension to the bulk). Currently there are several proposed mechanisms which may lead to one of these two phenomena yet none of them has been experimentally validated. Please refer to our recent review for more details [33].

Since the SLs and QLs in the heterostructure members of this family (MnBi₄Te₇, MnBi₆Te₁₀, MnBi₈Te₁₃) are combined by van der Waals forces, there are different terminations after cleaving the sample. As shown in Fig.3(b−d), the band structure on SL-termination is very similar to that of MnBi₂Te₄, and the band structure on QL- and double QL-terminations show hybridization features between the TSS and certain bulk bands [70,71,79,84-92]. Again, no signature of sizable magnetic gap can be found for the TSS from all the different terminations of AFM members. The sizable magnetic gap of TSS was realized based on the SL-termination of FM MnBi₈Te₁₃, with the gap size decreasing monotonically with increasing temperature and closing right at the Curie temperature [72].

Although the lack of sizable magnetic gap of TSS obscures the realization of topological quantized transport at high temperature (say, at the level of AFM transition temperature), QAH effect has indeed been realized at low temperature [1.4 K, Fig.3(e)] based on 5 QLs films of MnBi₂Te₄, key evidence of a 2D Chern insulator [93]. The characteristics of an axion insulator state were also observed at zero magnetic field based on 6 SLs [45]. Under a perpendicular magnetic field (15 \mathrm{T}), characteristics of high-Chern-number quantum Hall effect without Landau levels and contributed by dissipationless chiral edge states are observed, indicating a high Chern number Chern insulator with C=2 (9, 10 SLs) [94]. The A-type AFM configuration exhibits layer Hall effect in which electrons from the top and bottom layers deflect in opposite directions due to the layer-locked Berry curvature, resulting in the characteristic of the axion insulator state (6 SLs) [95,96]. We envision that half quantized Hall transport at the level of 10 \mathrm{K} can be realized based on the SL-termination of FM MnBi₈Te₁₃ with sizable magnetic TSS gap [72].

2.2 MnSb₂Te₄·(Sb₂Te₃)_n

Since the successful synthesis of MnBi₂Te₄·(Bi₂Te₃)_n single crystals, elemental substitutions have been explored in order to manipulate its magnetic and electronic properties. It turns out the Bi site can be completely substituted by Sb atoms. The resulting MnSb₂Te₄·(Sb₂Te₃)_n family of materials are currently under intensive investigation. Theoretically, this family (n=0,1,2) is also predicted to host similar AFM ground state and AFM TI phase [98,99], yet there lacks consistency between/among experiments and calculations on the exact magnetic ground state and band topology of MnSb₂Te₄ [100-106]. Notably, ARPES results reveal significant hole doping for all the members studied so far, leaving the detailed TSS Dirac cone structure not straightforward to study [105,107].

The crystal structure of MnSb₄Te₇ adapts a space group of P\overline{3}m1. The Mn layer constitutes a long-range magnetic order with moments along the c direction [Fig.4(a, b)] [107] (A-type AFM with T_N=13.5\mathrm{K}). ARPES measurement also reveals hole doping for the band structure with expected Dirac cone located at 180\mathrm{m}\mathrm{e}\mathrm{V} above the Fermi level [Fig.4(c)]. Pressure experiments and DFT calculations have revealed multiple topological phases corresponding to various magnetic structures and the emergence of superconductivity [Fig.4(d)] [98,106-110]. Similar hole doping and multiple magnetic topological phases have also been found in MnSb₆Te₁₀, an FM member of this family at its ground state [Fig.4(f, g)] [111]. Considering the universal electron doping behavior in MnBi₂Te₄·(Bi₂Te₃)_n family, it is natural to expect carrier tunability and magnetic manipulation based onthe mutual substitution of Sb and Bi in Mn(Bi, Sb)₂Te₄·((Bi, Sb)₂Te₃)_n series. In fact, a tunable TSS Dirac gap varying from being gapless to larger than 100 \mathrm{m}\mathrm{e}\mathrm{V} has been reported in Sb doped MnBi₂Te₄, with its gap size proportional to the doping level [112].

Except MnBi₂Te₄·(Bi₂Te₃)_n and MnSb₂Te₄·(Sb₂Te₃)_n families, it is noted that MnBi₂Se₄ in the R\overline{3}m space group shares the same magnetic and topological properties of MnBi₂Te₄. This phase turns out to be unstable in the bulk crystal form. Recent efforts have succeeded in synthesizing ultrathin films of MnBi₂Se₄ using nonequilibrium MBE [113]. Its magnetic structure, however, deviates from the expected A-type AFMz structure and the response of TSS Dirac cone to the magnetic order remains to be investigated.

2.3 EuM₂X₂ (M = metal; and X = Group 14 or 15 element)

EuSn₂As₂ belongs to the group of compounds with formula AM₂X₂ (A = alkali, alkaline earth, or rare earth cation; M = metal; and X = Group 14 or 15 element). Here we focus on the A = Eu compounds with intrinsic AFM order. The M site can be occupied by various types of metals such as Mg, In and Sn. EuSn₂As₂, as an important member in intrinsic magnetic TI family, crystallizes in the hexagonal space group R \overline{3} m. The Eu atoms are triangularly distributed and sandwiched by two honeycomb SnAs layers to form a layered structure [Fig.5(a)]. The magnetic moment provided by Eu atom forms an A-type AFM configuration with T_N= 25\mathrm{K} [74,114] [Fig.5(b)]. ARPES measurements have revealed a TSS Dirac cone locating ~ 0.4 eV above the Fermi level at the PM phase, suggesting a strong 3D TI phase [Fig.5(c)]. Yet no observable change of the TSS or carrier concentration can be found in the AFM state, indicating weak coupling between the Eu moments and low-energy bands [74,121]. Magnetic property and transport measurements report negative magnetoresistance and complicated magnetic transitions from an AFM state to a canted ferromagnetic state and then to a polarized FM state as the magnetic field increases [121,122]. Electrical resistance measurements under pressure reveal an insulator-metal-superconductor transition at low temperature around 5 and 15 GPa [Fig.5(d)]. A new C2/m phase appears when the pressure is higher than 14 GPa. As the pressure continues to increase, the superconductivity persists up to 30.8 GPa with T_C maintaining a constant value ~ 4 K [119]. It is also found that the pressure has an enhancement effect on the AFM transition temperature and negative magnetoresistance [123].

For EuMg₂Bi₂, it crystallizes into the tetragonal CaAl₂Si₂ structure type with space group P\overline{3}m1 (No. 164) [117] [Fig.5(e)]. Magnetic property measurements revealed AFM transition temperature T_N\sim 7 \mathrm{K} with slight anisotropy and positive Curie-Weiss temperature indicating FM interaction between Eu atoms (7.8 {\mu }_B) [Fig.5(f)]. Like Mn−Bi−Te family, AFM configuration between FM layers of Eu is established. The difference is that the moments point out-of-plane in Mn−Bi−Te but in-plane for EuMg₂Bi₂. ARPES measurements and DFT calculations have revealed Dirac surface state features and nontrivial band topology [Fig.5(g, h)], suggesting EuMg₂Bi₂ as a magnetic topological insulator candidate [115,117].

EuIn₂As₂ crystallizes into the hexagonal space group P{6}_3/mmc, containing layers of Eu²⁺ cations separated by In₂As₂²⁻ layers along the crystallographic c-axis [124] [Fig.5(i)]. Magnetic property and neutron diffraction measurements have determined a colinear AFM ground state with the moments lying in the ab-plane [118,120,124,125] [Fig.5(j)]. Furthermore, a complicated broken helix order is reported by neutron diffraction, tripling the unit cell along c-axis. EuIn₂As₂ was predicted as a high-order topological axion insulator candidate [120,126]protected by the magnetic crystalline symmetry. Such a state has gapless TSS Dirac cone at the symmetry-protected termination and gapped ones at other surfaces [Fig.5(k−l)]. However, like other AM₂X₂ compounds, its hole-doping nature as observed by ARPES [116,118] and scanning tunneling microscope (STM) [127] has prevented the detailed study on the TSS band structure, especially the gap behavior. Further chemical and band structure engineering are strongly called for to tune the chemical potential for access to the TSS Dirac point in this family.

There are also theoretical calculations which predict materials such as several Eu₅M₂X₆ (M = metal, X = pnictide) Zintl compounds [128,129], 2D EuCd₂Bi₂ [130], and NiTl₂S₄ [131] to be intrinsic magnetic TI candidates yet their growth, band structure, magnetic structure and band topology remain to be investigated.

3 Magnetic Weyl/Dirac semimetals

In a DSM, two doubly degenerate bands contact at discrete momentum points called Dirac points and disperse linearly along all directions around these points. The four-fold degenerate Dirac points need symmetries to ensure their existence, such as time-reversal symmetry T, inversion symmetry P, rotational symmetry and nonsymmorphic symmetry. In a DSM with TP symmetry, when either T or P is broken, each doubly degenerate band is lifted, so that the Dirac cones can split into multiple Weyl cones, giving birth to WSMs. However, in 3D systems with AFM order that breaks both T and P but respect their combination PT, four-fold degenerate Dirac points can still exist, resulting into AFM DSM [132]. Such consideration has also been generalized to 2D systems [133-135].

In magnetic WSMs, spin-polarized conduction and valence bands touch at finite number of nodes, forming pairs of Weyl nodes. In each pair, the quasiparticles carry opposite chirality and can be viewed as the “source” (“+” chirality) and the “sink” (“−” chirality) of the Berry curvature. Odd pairs of Weyl nodes with opposite chirality can be expected in systems withT symmetry breaking, such as Co₃Sn₂S₂ [136,137] and Mn₃X (X = Sn, Ge) [138-140]; while for systems with time-reversal symmetry T, the total number of Weyl nodes pairs must be even. Noncentrosymmetric WSMs belong to this category, such as TaAs family [141-144]. If P and T symmetries are both preserved, Weyl nodes with opposite chirality can merge at the same momentum and form a four-fold Dirac point (assisted by additional crystal symmetry), such as Na₃Bi [21,145] and Cd₃As₂ [23,146,147]. Due to non-zero Berry curvature, many novel physical properties such as giant AHE and giant anomalous Hall angle, chiral anomaly, anomalous Nernst effect will emerge in magnetic WSM, holding potential applications in spintronics field. In the early stage, several candidate materials were predicted, such as R₂Ir₂O₇ (R = Nd, Pr) [20], HgCr₂Se₄ [148]. Recent efforts have focused on Co₃Sn₂S₂ [136,137] and Mn₃X (X = Sn, Ge) [138-140] which clearly host the band structure and transport characters as expected by magnetic WSM. We will briefly introduce these magnetic materials.

3.1 FeSn

FeSn crystallizes in a hexagonal structure (P6/mmm) with the Fe atoms forming a Kagome lattice [150,151,153,154]. Like Fe₃Sn₂, FeSn is formed by interlacing Fe₃Sn layer and Sn layer. The difference is that there is only one Kagome layer (Fe₃Sn layer) in a unit cell [Fig.6(a)]. It is closer to the 2D limit than Fe₃Sn₂. Below T_N= 365\mathrm{K} [Fig.6(b)], the Fe spins form FM Kagome layers which are stacked antiferromagnetically along the c axis. The Dirac nodal line along the K−H line opens small energy gaps when SOC is considered, except at the H point where a gapless Dirac point (protected by PT and S_{2Z} symmetry) still exist, rendering FeSn as an AFM DSM. Such gapless Dirac cones have been directly observed by ARPES [150,151] [Fig.6(c, d)]. Besides, the flat band because of the Kagome layer has also been observed directly by ARPES [Fig.6(e)]. Furthermore, in a planar tunneling spectroscopy measurement [152], an anomalous enhancement in tunneling conductance within a finite energy range of FeSn has been observed in its Schottky heterointerface with Nb-doped SrTiO₃ [Fig.6(f)]. Such tunneling conductance peak is attributed to spin-polarized flat band localized at the FM Kagome layer at the Schottky interface.

3.2 Co₃Sn₂S₂

Co₃Sn₂S₂ crystallizes in the R\overline{3}m space group with a stacking order −Sn−S−Co₃Sn−S− from top to bottom. The central Co layer forms a 2D Kagome lattice with one Sn atom at the center of the hexagon, as shown in Fig.7(a). Co₃Sn₂S₂ is a ferromagnet with a curie temperature of 175 K and a magnetic moment of 0.3{\mu }_B/\mathrm{C}\mathrm{o}. In magnetization measurement, the saturation field along c axis is low (0.05 T) but along in-plane is extremely high (> 9 T), confirming that the easy magnetic axis is c-axis [48,155]. Combining theory and experiments, Co₃Sn₂S₂ is an ideal FM WSM with three pairs of Weyl points whose energies are only ~ 60 meV above the Fermi level [136,137,156-161]. The Weyl nodes have been observed by ARPES after doping alkaline metal [Fig.7(c)]. Three Fermi arcs form a triangular-like loop around the K′ point near Fermi surface. Meanwhile, the electronic structure does not undergo obvious dispersion along the k_z direction, suggesting the nature of TSSs [Fig.7(b)]. Termination-dependent surface band structures of Co₃Sn₂S₂ were observed by using STM [136]. Different surface potentials imposed by three different terminals will change the Fermi arc contour and Weyl node connectivity. On the Sn-termination, the Fermi arcs connect Weyl nodes within the same Brillouin zone, while on the Co-termination, the connectivity spans the two adjacent Brillouin zones. On S-termination, Fermi arcs overlap with the trivial surface-projected bulk bands. The topologically protected and unprotected electronic properties of WSMs Co₃Sn₂S₂ were verified.

According to first-principles calculation, the Weyl nodes in Co₃Sn₂S₂ locate close to the Fermi level and produce a giant anomalous Hall conductivity (AHC) (~1100 Ω⁻¹·cm⁻¹), which has been directly observed in transport measurement [Fig.7(d, e)] [48,161,162]. Besides, giant anomalous Hall angle also emerges in this material. As shown in Fig.7(e), with increasing temperature, a maximum value of nearly 20% is reached around 120 K, which is at least one order of magnitude higher than that of conventional magnetic materials. Negative magnetoresistance is found in Co₃Sn₂S₂, as shown in Fig.7, when the magnetic field is applied in the in-plane direction, the longitudinal resistance is negative, and when the external magnetic field is applied in the out-of-plane direction, the longitudinal resistance changes from negative to positive, showing evidence of chiral anomaly [48,155,161,162]. In Co₃Sn₂S₂ thin film, a maximum Nernst thermopower of 3 µV·K⁻¹ is achieved [50], demonstrating the possibility of application of hard magnetic topological semimetals for low-power thermoelectric devices.

3.3 Mn₃X (X = Sn, Ge)

Mn₃X (X = Sn, Ge) has a hexagonal Ni₃Sn-type structure and crystalizes in the P 6_3/mmc space group. One unit cell consists of two sets of Mn layers stacked along c-axis and each Mn layer forms a breathing-type Kagome lattice with one Sn atom at the center of the hexagon, as shown in Fig.8(a). Mn₃Sn and Mn₃Ge are both chiral antiferromagnets which means Mn moments are forming a 120° ordering with a negative vector chirality [Fig.8(b)] [164,165]. The AFM transition temperature of Mn₃Sn and Mn₃Ge is 430 \mathrm{K} and 372 \mathrm{K}, respectively. Because the electronic structures of Mn₃Sn and Mn₃Ge are quite similar and the study of Mn₃Sn is more comprehensive, we will mainly focus on Mn₃Sn. Mn₃Sn possesses non-collinear AFM spin texture and strong SOC effect, which produce multiple pairs of Weyl points close to the Fermi level, according to first-principal calculation [138,166,167] [Fig.8(c)]. However, ARPES spectra [138] measured on Mn₃Sn lacks clear features of quasiparticle bands, likely due to strong correlation effect of Mn 3d electrons [Fig.8(d)].

Novel transport properties governed by the topological nature can serve as evidence for Weyl fermions. In Mn₃Sn, strongly anisotropic magnetoconductance was observed. The sign of magnetoconductance changed when rotated the direction of magnetic field from being parallel to perpendicular to the current direction, serving as strong evidence of chiral anomaly [138,164]. The large AHE is also a key characteristic of magnetic WSM. In the traditional sense, because the magnetic configuration of Mn₃Sn is AFM, there is no net magnetic moment in this material and the AHE will not emerge. But many reports revealed that Mn₃Sn exhibits a large AHE [139,140,166,168-171]. Fig.8(e, f) show the temperature-dependence of zero-field Hall conductivity under different magnetic field and current directions [140]. We can see that when the magnetic field and the current are applied along the (01\overline{1}0) and (0001) direction, the {\sigma }_H will achieve a maximum value of nearly 130 Ω⁻¹·cm⁻¹ at 50 \mathrm{K}. In Mn₃Ge, by employing similar magnetic field and current direction, even higher AHC have been obtained [139,168]. The large AHE in Mn₃X is mainly caused by the non-zero Berry curvature produced by Weyl nodes [164]. Besides chiral anomaly and large AHE, many other exotic physical properties such as large anomalous Nernst effect [163,164,172-174], planer Hall effect [171,175,176], magnetic spin Hall effect and magnetic inverse spin Hall effect [177] are also observed in Mn₃X. Furthermore, as shown in Fig.8(g), anomalous Nernst voltage V_{\mathrm{A}\mathrm{N}\mathrm{E}} image mapped by scanning thermal gradient microscopy reveals the existence of magnetic domains. The orientation of these domains can be changed (written) by laser-induced local thermal gradient [163], offering a chance to study spintronics phenomena in non-collinear antiferromagnets with spatial resolution.

3.4 Co₂MnGa

A new family of magnetic WSM emerged among the magnetic Heusler alloys, i.e., the Heusler alloy WSMs [181,182]. It is an important family due to their rich transport properties and several superiorities. Firstly, the Curie temperatures of most Heusler compounds are above the room temperature [183,184]. Secondly, this kind of materials has a significant AHE and spin Hall effect arising from the large Berry curvature [179,181,182,185-187]. Thirdly, Heusler compounds are usually soft magnetic materials, which means that their magnetization direction can be tuned by a weak magnetic field. These properties facilitate spin manipulation and applications in spintronics, as a result, these Heusler alloy WSMs have been widely studied.

As full Heusler compounds, Co-based Heusler materials have the formula of Co₂XZ (X = IVB or VB; Z = IVA or IIIA), here we focus on Co₂MnGa and Co₂MnAl. Co₂MnGa (Co₂MnAl) crystalizes in a face-centered cubic Bravais lattice (space group Fm\overline{\text{3}}m, No. 225), as shown in Fig.9(a). The relevant symmetries are the three mirror planes and three C_4 rotation axes. The Curie temperature of Co₂MnGa and Co₂MnAl are known to be ~700 K [179] and 726 K [183], respectively. Transport experiments showed that Co₂MnAl has a giant room-temperature AHE with the Hall angle ({\mathrm{\Theta }}^{\mathrm{H}}) reaching a record value tan{\mathrm{\Theta }}^{\mathrm{H}} = 0.21 at the room temperature among magnetic conductors [178], as shown in Fig.9(b). This property results from the gapped nodal rings that generate large Berry curvature. Furthermore, for Co₂MnGa films, when the E_F is set in the magnetization-induced gap of the Weyl cones by the electronic doping, the highest anomalous Nernst thermopower of a record value 6.2 μV·K⁻¹ will be reached at room temperature [188].

The Hopf link is originally a mathematical concept which consists of two rings on the two perpendicular planes, each passing through the center of each other, as shown in Fig.9(c). The symmetry of Co₂MnGa can protect this band crossing associated with the unusual linking-number (knot theory) invariant, giving rise to a variety of new types of topological semimetals [179,180,185-187,189-192]. Systematic ARPES investigation of the electronic structure of Co₂MnGa has been carried out and directly revealed three intertwined degeneracy loops in the material’s three-torus bulk Brillouin zone [Fig.9(d−f)]. In addition, the Seifert boundary states protected by the bulk-linked loops have been predicted and observed, while the links and knots in the electronic structure and the accompanied exotic behaviors remain unexplored.

3.5 EuB₆

The EuB₆ crystallizes in a similar body-centered-cubic-like crystal structure with space group Pm\overline{3}m (No. 221) [Fig.10(a)]. EuB₆ is a soft FM semimetal which has a very small magnetic anisotropy energy so that the magnetization can be easily modulated by magnetic field [193,195-197]. Electronic transport and magnetic susceptibility measurements showed that the system undergoes a paramagnetic to FM phase transition at about 15.3 \mathrm{K} and a new FM phase manifests below about 12.5\mathrm{K} with moment oriented to the (111) direction [198-200]. The magnetotransport properties of EuB₆ have been widely studied around magnetic phase transition point, such as the metal-insulator transition, colossal magnetoresistance and quantum nematic phase [201-203].

It has been predicted that EuB₆ is a topological nodal-line semimetal when the magnetic moment is aligned along the (001) direction, and it turns out to be a WSM with three pairs of Weyl nodes when rotating the magnetic moment to (111) direction. Specifically, when the moment is in the (110) direction, a composite semimetal phase featuring the coexistence of a nodal line and Weyl points manifests [195]. The electronic structures on the two different cleavage planes in EuB₆, i.e., the Eu- and B-terminated surfaces, have been investigated [194,204]. For the B-termination, in the FM state, obvious Zeeman splitting occurs for both the conduction and valence bands, which gives rise to the overlap of subbands and thus the band inversion at the time-reversal point X of the Brillouin zone [Fig.10(b)]. In this case, EuB₆ enters a topological semimetal state with an ideal electronic structure near E_F. The topological properties can be investigated by measuring the magnetotransport properties due to the correlation between the band structure and the local moments. Fig.10(c) shows the intrinsic anomalous Hall conductivity as a function of magnetization with different directions at 2 K [193]. An intrinsic large anisotropic magnetoresistance of −18% at 0.2 T was observed and interpreted as the modification from the Berry curvature in a tilted Weyl cone [205]. The theoretical prediction that a large-Chern-number QAH effect could be realized in its (111)-oriented quantum-well structure [195] needs further investigations.

3.6 Fe₃GeTe₂

Fe₃GeTe₂ crystallizes in a hexagonal structure (P6₃/mmc, No. 194) in which the layered Fe₃Ge substructure are sandwiched by two layers of Te atoms [Fig.11(a)]. Fe₃GeTe₂ is FM with Fe moments along the c axis and a Curie temperature of 204−230 K [Fig.11(b)] [206, 211-213]. ARPES measurements have revealed two pockets around \mathrm{\Gamma } point and one at K point [Fig.11(c)]. Temperature-dependent ARPES spectra exhibits a massive spectral weight transfer in the FM state induced by exchange splitting [207]. Orbital-driven nodal line along K−H protected by crystalline symmetry has been predicted [Fig.11(f)]. Introducing SOC will gap the nodal line and generate large Berry curvature [210], an effective source of a large AHE in Fe₃GeTe₂. We note that Fe₃GeTe₂ is considered as a gapped nodal line semimetal with the Weyl point awaiting verification.

Fe₃GeTe₂ also contains very rich physical properties. Due to the gapped nodal line, negative magnetoresistance[214,215], anomalous Nernst effect [216] and AHE were observed [209,210,213]. Compared with other itinerant FM materials, Fe₃GeTe₂ has both large anomalous Hall factor and anomalous Hall angle [Fig.11(d, e)]. Due to the weak interlayer coupling, Fe₃GeTe₂ can be exfoliated into sheets with different number of layers. More importantly, its novel transport and magnetic properties show stability at room temperature and dependence on the number of layers, interlayer coupling and carrier density [208,209,217-223], holding potential in spintronics applications.

3.7 EuCd₂As₂

EuCd₂As₂ belongs to EuM₂X₂ (M = metal; and X = Group 14 or 15 element) family in which several members are studied as magnetic TI candidates (see Section 2).The exact band structure details and topological phase are sensitively related to the magnetic configuration. The crystal structure of EuCd₂As₂, with space group 164 (P\overline{3}m1), is shown in Fig.12. The Eu atoms form a simple hexagonal lattice at the 1a Wyckoff position. The As and Cd atoms at the 2b positions form the other four atomic layers with the sequence of −Cd−As−Eu−As−Cd− along the c axis [224,229,230]. Eu moments prefer an intralayer FM coupling and an interlayer AFM coupling along the c axis, i.e., an A-type AFM, which doubles the unit cell along the c direction. Fig.12 and (c) show two such magnetic configurations by showing Eu atoms with magnetic moment directions along c (A-type AFMc) and along a (A-type AFMa). A-type AFMc is proposed based on the anisotropic magnetic and transport properties [224,229]. A-type AFMa is proposed based on the resonant elastic X-ray scattering [225,231], first-principles calculations [232] and magnetostriction measurements [233]. Furthermore, neutron diffraction on isotopic ¹⁵³Eu and ¹¹⁶Cd revealed a \mathit{k}=(0, 0,0) FM order at zero field with the Eu moments pointing along the in-plane (210) direction with a \sim 30 ° out-of-plane canting [magnetic space group C2{}^{\prime }/m{}^{\prime }, Fig.12] [226].

According to the first-principles calculation and symmetry analysis, various topological phases emerge based on different magnetic configurations in EuCd₂As₂. For A-type AFMz, DSM phase exists with the gapless Dirac point protected by the PTL symmetry operation which is the product of inversion symmetry P, time reversal symmetry T and crystalline translation symmetry L [227,234]. For A-type AFMx, spin configuration breaks the C_3 symmetry in the AFM state of EuCd₂As₂ and leads to an axion insulator with a hybridization gap of ~1 meV. Massless Dirac surface states appear on some surfaces protected by the mirror or TL symmetries. For other surfaces without such symmetry, the surface states are gapped and the hinge states, associated with higher order TI states, emerge at the edges [126,235]. There are other calculations which predict EuCd₂As₂ as a WSM with a single pair of Weyl points very close to the Fermi level [226,230,236]. Such Weyl phase can be generated in EuCd₂As₂ by applying a magnetic field >1.5\mathrm{T} along the c axis [236] or alloying with Ba at the Eu site to stabilize the FM configuration [230]. In fact, the recently confirmed spin-canted structure as shown in Fig.12 naturally hosts such WSM phase [226]. Spectroscopically, ARPES measurements have observed linear band crossings at the Fermi level and especially an “M”-shaped feature around \mathrm{\Gamma } point [Fig.12], suggesting a nontrivial band inversion. Such features cannot distinguish between the semimetal and insulator phase as the gap is only ~1 meV, comparable to the thermal broadening effect at ~3 K. ARPES or scanning tunneling spectroscopy measurements at ultralow temperature are needed. Spin-resolved ARPES is also useful to examine the spin degeneracy of these linear bands and crossings.

Magnetic transport experiments have provided more information on the interplay between magnetism and band topology in EuCd₂As₂. Negative magnetoresistance [Fig.12], as signature of chiral anomaly is observed along with AHE [Fig.12] [225,226,228]. These transport results support as-grown EuCd₂As₂ in a semimetal phase, yet gate tunable transport is needed to verify the absence of gap close to the Fermi level. It was further reported that the Hall resistance shows a giant nonlinear behavior originating from a series of magnetic-field-induced Lifshitz transitions in the spin-dependent band structure [Fig.12] [228]. Combined with band structure calculation, these results suggest that in EuCd₂As₂, electronic structure is extremely sensitive to the spin canting angle, with the magnetic field causing band crossing and band inversion and introducing a band gap when oriented along specific directions, offering an ideal platform for Berry curvature engineering.

4 Other magnetic topological metals

As introduced in the previous section, intrinsic magnetic TIs have nontrivial bulk band topology featured by a global bulk gap and TSS residing inside the bulk gap. Chemical potential can be tuned into the bulk gap to eliminate the transport contribution from the bulk bands, a key prerequisite to realize quantized Hall transport. There exist other magnetic systems which lack a global bulk gap in the whole momentum space but possess a locally nontrivial bulk gap and TSS inside. Such systems always exhibit metallic transport behavior contributed by trivial bulk bands. AHE is generally expected from the coexisting net magnetic moment and locally nontrivial topology. We term such materials as magnetic topological metals. It is noted that there is no strict theoretical scheme describing magnetic topological metal since the metallicity does not only come from band-topology-induced TSS but rather the trivial bulk bands. We choose this term only to emphasize its distinction from intrinsic magnetic TIs and topological SMs.

4.1 Fe₃Sn₂

Fe₃Sn₂ is a layered Kagome compound with a space group of R\overline{3}m formed by interlacing two Fe₃Sn layers and one Sn layer. The Fe atoms in the Fe₃Sn layer form a Kagome structure, and the Sn atoms exhibit a honeycomb structure. The Sn atomic layer also exhibits a honeycomb distribution [Fig.13] [239]. Fe₃Sn₂ is FM in the ground state with a Curie temperature of T_C ~ 610 K [237,240-242]. Due to the weak binding force between layers, Fe₃Sn₂ produces three different cleavage planes, Fe₃Sn-1-termination, Fe₃Sn-2-termination, and Sn-termination [238,243]. The experimentally observed band structures mainly come from Fe₃Sn-1-termination. The shape of the Fermi surface confirms the trigonal structure of Fe₃Sn₂. ARPES measurements have revealed two Dirac cone features at the corner of Brillouin zone, which are gapped by the SOC effect [Fig.13]. Such strong SOC also couples the magnetic and electronic structure of Kagome lattice, exhibiting a magnetization-driven giant nematic (two-fold-symmetric) energy shift [244]. In the Kagome lattice, the destructive interference of the electron Bloch wave function can effectively localize the electrons to produce flat bands. Such flat bands are observed in Fe₃Sn₂, which are ~ 0.2 eV below the Fermi level [243].

The coexistence of nontrivial band topology and FM order in Fe₃Sn₂ produces giant AHE [40,238,245].The measured AHC is found to be temperature independent and persists above room temperature [Fig.13], which is suggestive of prominent Berry curvature from the time-reversal-symmetry-breaking electronic bands of the Kagome plane. Moreover, Fe₃Sn₂ shows complex magnetic bubbles and magnetic vortex structure like skyrmions[246-251]. These bubbles are 3D magnetic domains with complicated evolution of spin texture, which not only give rise to topological Hall transport response, but also show record-high temperature stability in magnetic racetrack memory devices [Fig.13].

4.2 RT₆X₆ (R = Rare earth metal; T = transition metal; X = Sn, Ge)

Layered Kagome compounds RT₆X₆ (R = rare earth metal, T = transition, alkali, alkaline earth metal, X=Sn or Ge) crystallize in the P6/mmm space group. As shown in Fig.14, T₃X is the Kagome layer of T ions with one X atom at the center of the hexagon. In RX layer, the R atom lies at the center of the hexagons surrounded by the X atoms. X layer is a hexagonal layer only consisting of X atoms and separating each unit cell. In this system, the 4f electrons in the R element interact with the 3d electrons in the transition metal element T to generate a rich magnetic structure. Many novel physical properties are also found in this system, such as flat band, giant AHE and Nernst effects. Recent published articles focus mostly on RMn₆Sn₆ and RV₆Sn₆. Therefore, the following content will discuss these two systems.

Since Mn is a well-known magnetic metal, there are many magnetic configurations emerged due to the interaction between Mn 3d magnetic moment and R 4f magnetic moments [Fig.14(b, e)] [252,255-258]. When R is a lanthanide element (R = Gd−Tm, Lu), its magnetic configuration varies from FM to AFM. For R = Gd to Ho, their magnetic configuration is ferrimagnetic, and for R = Er, Tm and Lu, they possess AFM ground state. The direction of the magnetic moment of the R element tends to be antiparallel to the magnetic moment of Mn, and the moment direction is variable for different R elements. GdMn₆Sn₆ moment is arranged in-plane, and TbMn₆Sn₆ moment is arranged out-of-plane. DyMn₆Sn₆ and HoMn₆Sn₆ possess a conical magnetic structure. When R is Er and Tm, the Mn and Er = Tm sublattices are independently ordered in an AFM manner because the strength of the magnetic coupling is weak. Since there is no 4f electrons in Lu and Y, they form in-plane FM and helical AFM along c-axis. For R = Gd to Ho, the Curie temperature of them is 435, 423, 393, and 376 K, respectively. For R = Er to Lu and Y, the Neel temperature of them is 352, 347, 353, and 333 K, respectively. In general, the electronic structure is closely related to magnetic configuration, when magnetic configuration change, the electronic structure will also change. However, for the RMn₆Sn₆ (R = Gd−Tm, Lu, Y) system, even for the different R, the band structure does not change significantly, indicating weak coupling between the low energy bands and magnetic moments.

Kagome lattice usually hosts three typical band features: flat band over the whole Brillouin zone, Dirac cones located at the Brillouin zone corners, and the saddle points located at the Brillouin zone boundary. Such features have indeed been observed in YMn₆Sn₆ and others by ARPES [259,260]. The strong correlation between magnetism and Kagome lattice can produce many novel physical properties. In TbMn₆Sn₆, its Kagome lattice features an out-of-plane magnetic ground state, so it is predicted to support the intrinsic Chern topological phase. In STM measurement, the Dirac cone with a Chern gap [Fig.14] and topological edge state are detected, implying its non-trivial topological nature [253].

The coexistence of nontrivial band topology and variation of magnetic structure results in novel transport behavior. In YMn₆Sn₆, a large room temperature anomalous transverse thermoelectric effect of ≈ 2 µV·K⁻¹ is realized, larger than all canted AFM material studied to date at the room temperature [261]. In addition, topological Hall effect is observed in the transverse conical spiral phase of YMn₆Sn₆ and ErMn₆Sn₆ with similar magnetic configuration [262-264]_. Large anomalous Hall conductivity is also observed in many RMn₆Sn₆ compounds such as LiMn₆Sn₆, TbMn₆Sn₆, DyMn₆Sn₆, and HoMn₆Sn₆, as shown in Fig.14(d, e) [252,254,258,262,264].

In isostructural RV₆Sn₆ compounds, V atoms have no magnetic moments, so that RV₆Sn₆ magnetic configuration is different from RMn₆Sn_6. The magnetic configuration is determined to be out-of-plane AFM for R = Tb−Ho and in-plane AFM for R = Er and Tm. because Lu and Y also possess no magnetic moment, so the compounds for R = Lu and Y are PM metals [265]. Typical band features such as Dirac cone, saddle point, and flat bands are also observed in this family [266]. Furthermore, TSS Dirac cones emerge from the nontrivial bulk band topology and can be tuned in binding energy via potassium deposition [267].

4.3 EuAs₃

EuAs₃ crystallizes in a monoclinic structure (space group C2/m, No. 12). As shown in Fig.15, the moments of Eu are oriented along with b axis [268]. The specific heat, electrical conductivity, susceptibility measurements [269], neutron diffraction [270], X-ray scattering technique [271,272] and μSR [273] studies showed that EuAs₃ orders in an incommensurate AFM state at T_N=11 \mathrm{K}, and goes through an incommensurate-commensurate lock-in phase transition at T_L= 10.3\mathrm{K}, reaching a collinear AFM ground state. Electrical transport studies showed an extremely anisotropic magnetoresistance related to the magnetic configuration [274].

Recently, the magnetism-induced topology of EuAs₃ has been demonstrated and the origin of extremely anisotropic magnetoresistance has been discussed [268]. An unsaturated extremely anisotropic magnetoresistance of 2 × 10⁵% at 1.8 \mathrm{K} and 28.3 \mathrm{T} has been observed, as shown in Fig.15. Meanwhile, through the DFT calculations and transport measurements, it is demonstrated that EuAs₃ is a magnetic topological massive Dirac metal at AFM ground state. ARPES results probed by different photon energies verify that EuAs₃ is a topological nodal semimetal in PM state [Fig.15(d, e)], this is related to the extremely anisotropic magnetoresistance. For 3 K ≤ T ≤ 30 K, the concentration of hole carriers is larger than that of electron carriers. Upon decreasing the temperature the concentration of electron carriers is suddenly enhanced, accompanied by a sharp increase in the mobility of hole carriers, indicating a possible Lifshitz transition [Fig.15].

5 Perspective

In this review, we have gone through several intrinsic magnetic topological states of matter by introducing their representing materials. The interaction between magnetic order and band topology in these materials brings forth characteristic band features such as Dirac gap, Weyl point, Fermi arc, hinge/corner state and so on, produces large Berry curvature and enables novel topological transport responses including quantum anomalous Hall effect, intrinsic anomalous Hall effect, anomalous Nernst effect, negative magnetoresistance as the signature of chiral anomaly and so on. Intrinsic magnetic topological insulators are of fundamental and practical importance because of the potential for the development of dissipationless spintronics, information storage and quantum computation. However, so far only Mn(Bi,Sb)₂Te₄·((Bi,Sb)₂Te₃)_n family is firmly verified as intrinsic magnetic topological insulator. For this family of materials, the lack of sizable magnetic gap hinders the realization of quantum anomalous Hall effect at the expected temperature. It is thus highly desired to search for new material systems hosting such topological state. Instead of incorporating magnetism into established topological systems like the way how Mn(Bi,Sb)₂Te₄·((Bi,Sb)₂Te₃)_n and magnetically doped Bi₂(Se,Te)₃ families are realized, we envision that looking for band topology based on known ferromagnets or antiferromagnets will be more efficient to realize intrinsic magnetic topological insulator. Recent high-throughput calculations and magnetic space group analyses [275-280] have predicted a large number of new magnetic topological materials which provide guidance for experiment.

While we are concentrating on the interplay between magnetism and band topology in these quantum states of matter, it is well known that magnets host many ordered phases such as spin density wave, charge density wave, superconductivity, nematicity and so on. The interplay between band topology and these orders could generate exotic states such as axionic charge-density wave [281], chiral Majorana fermions [282] and the unknown which deserved future theoretical and experimental investigation. Furthermore, besides ferromagnetism and antiferromagnetism, recently a third basic magnetic phase dubbed altermagnetism [282-284] has been developed to describe some supposed antiferromagnets with mysterious behaviors such as anomalous Hall effect [285-287], spin polarized bands [288] and spin splitting torque [289-292]. Novel topological states of matter based on altermagnets remains to be explored.

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