Abstract
Magnetic topological semimetals (TSMs) are topological quantum materials with broken timereversal symmetry (TRS) and isolated nodal points or lines near the Fermi level. Their topological properties would typically reveal from the bulkedge correspondence principle as nontrivial surface states such as Fermi arcs or drumhead states, etc. Depending on the degeneracies and distribution of the nodes in the crystal momentum space, TSMs are usually classified into Weyl semimetals (WSMs), Dirac semimetals (DSMs), nodalline semimetals (NLSMs), triplepoint semimetals (TPSMs), etc. In this review article, we present the recent advances of magnetic TSMs from a computational perspective. We first review the early predicted magnetic WSMs such as pyrochlore iridates and HgCr_{2}Se_{4}, as well as the recently proposed Heusler, Kagome layers, and honeycomb lattice WSMs. Then we discuss the recent developments of magnetic DSMs, especially CuMnAs in TypeIII and EuCd_{2}As_{2} in TypeIV magnetic space groups (MSGs). Then we introduce some magnetic NLSMs that are robust against spin–orbit coupling (SOC), namely Fe_{3}GeTe_{2} and LaCl (LaBr). Finally, we discuss the prospects of magnetic TSMs and the interesting directions for future research.
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Introduction
The classification of material phases and description of phase transitions in condensed matter physics have long been given by the Landau theory of spontaneous symmetry breaking, with different phases described by different local order parameters. People could understand, for example, the superconducting phase transition from the breaking of the U(1) gauge symmetry, the ferromagnetic phase transition from the breaking of the timereversal symmetry (TRS) and all sorts of structural phase transitions in crystals from the change of space group symmetries. Despite the great triumph of Landau theory, its limitations reveal when Klitzing discovered the quantum Hall effect (QHE) in a 2D electron gas (2DEG) under high magnetic fields.^{1} This remarkable discovery then opened a new field of study for the phase transitions of materials, i.e., the socalled topological phase transitions. QHE is beyond Landau theory because the transitions between electronic states holding different integer Hall conductances do not break any symmetry. Thouless et al.^{2} used the Kubo formula to calculate and interpret the integers and found their topological origin. These topological integers are known as the TKNN numbers in memory of their pioneering works and are now understood as the first Chern number in topological band theory.
The QHE in a 2DEG was later reproduced in the topological phases of a 2D lattice by Haldane, who proposed a honeycomb lattice model without applying any net magnetic fields.^{3} The TRS was broken by the staggered magnetic fluxes over the lattice with zero total flux. This was the first model for the quantum anomalous Hall effect (QAHE), where the quantized Hall conductance was characterized by the first Chern number but realized with no magnetic field applied. The first Chern number is calculated by the integral of the Berry curvature of the occupied bands over the first Brillouin zone (BZ) divided by 2π, and is stable against smooth perturbations of the system without closing the band gap. Such a topological invariant can only be defined for evendimensional systems and is only nonzero for magnetic systems where the TRS is broken.
Nearly 20 years later, Kane and Mele proposed a new topological invariant^{4} — the Z_{2} number. They also studied a 2D honeycomb lattice model but with TRS preserved and spinorbit coupling (SOC) considered. The model has vanishing QHE, although the quantum spin Hall effect (QSHE) can be realized. The Z_{2} topological number of the model is characterized by the difference of the Chern numbers of spinup and spindown states modulo 2, which can be either 0 (trivial) or 1 (nontrivial). Such a topological classification was then generalized to 3D systems to describe the nontrivial band insulators,^{5,6,7,8} which are known as the 3D topological insulators (TIs). These pioneering models and theoretical works inspired the prosperity of theoretical predictions and experimental realizations of the topologically nontrivial materials later on. The QSHE system of HgTe/CdTe quantum well was predicted^{9} and soon confirmed experimentally.^{10} 3D TIs were discovered in the Bi_{2}Se_{3} family,^{11,12,13} and QAHE was predicted and observed in the magnetically doped thin films of the Bi_{2}Te_{3} family.^{14,15}
The topological classification can be generalized to semimetals, known as topological semimetals (TSMs), where the lowest conduction band and highest valence band cross each other at isolated points (nodes) or lines (nodal lines) at the Fermi level.^{16,17,18,19} In the beginning, TSMs were mainly discussed as an intermediate phase between normal insulators (NIs) and TIs. When the inversion symmetry (IS) is broken, gapless points can appear in pairs during the NItoTI transition and move in the Brillouin zone under continuous changes of the model parameters until they meet and annihilate as the system reaches a TI phase.^{20,21,22} The intermediate gapless phase is called a Weyl semimetal (WSM) because the lowenergy excitations near a twofold degenerate point, called a Weyl point (node), are linearly dispersive and satisfy the Weyl equation that describes massless Weyl fermions in highenergy physics. Also, The WSM phase was modeling studied by alternately stacking thin films of magnetically doped TIs and NIs.^{23,24} At the same time, the singlecrystal WSM candidates such as pyrochlore iridates^{25} and HgCr_{2}Se_{4}^{26} were predicted. These early works of WSMs stimulated the research interest for TSMs greatly.
According to the Nielsen–Ninomiya theorem,^{27,28} Weyl nodes always appear in pairs. They are topologically stable because the Weyl Hamiltonian near a Weyl node
has used up all the three Pauli matrices σ_{1}, σ_{2}, σ_{3}. Perturbations can only move the Weyl node in the crystal momentum space but cannot annihilate it unless it meets with another Weyl node holding opposite chirality and opens a band gap. The chirality here is defined as the Chern number of the Bloch states on a 2D spherical surface enclosing the Weyl node. The result is given by
assuming the 3 × 3 matrix A has full rank so there is no nodalline direction in BZ. Hence, a Weyl node is like a magnetic monopole in the crystal momentum space and can behave like either a “source” (C = +1) or a “sink” (C = −1) of the Berry curvature.
Analogous to WSMs, we also have Dirac semimetals (DSMs) with a fourfold degenerate Dirac node and the lowenergy excitations near the node satisfy the fourcomponent massless Dirac equation
with 4 × 4 matrices γ^{0} = τ_{3} ⊗ I_{2×2} and γ^{j} = iτ_{2} ⊗ σ_{j}, j = 1, 2, 3 in the standard representation. Here we use two sets of Pauli matrices τ and σ to distinguish the directproduct spaces. The Dirac Hamiltonian can be rewritten as \(H(\vec k) = \hbar c\tau _1 \otimes \vec k \cdot \vec \sigma\), with the Pauli vector \(\vec \sigma = (\sigma _1,\sigma _2,\sigma _3)\). Since \(H(\vec k)\) commutes with the γ^{5}symmetry operator
with eigenvalues γ^{5} = ±1, the 4dimensional Hilbert space of ψ can be reduced into two uncoupled twodimensional subspaces of Weyl fermions with effective Hamiltonians \(H(\vec k) = \pm \hbar c\vec k \cdot \vec \sigma\), respectively. Hence, a Dirac node can be viewed as a fourfold degenerate “kissing” point of two Weyl nodes with opposite chiralities. It is not topologically stable against the mass term, which breaks the γ^{5}symmetry and couples the two Weyl subspaces to open a gap. In order to obtain a stable Dirac node, additional crystalline symmetries are necessary to protect the Dirac nodes on the highsymmetry points or lines in the first BZ.^{29} Such kind of DSM states have been theoretically predicted^{30,31} and experimentally confirmed in nonmagnetic materials Na_{3}Bi^{32} and Cd_{3}As_{2},^{33,34} etc.
Beyond the homologous particles such as the Weyl fermions and Dirac fermions, there are also other types of quasiparticles that are allowed in solids by the representation theory of crystalline symmetries, which are the socalled new fermions.^{35} As pointed out by Bradlyn et al., the irreducible representations of the little group at the highsymmetry BZ points in some specific space groups could suggest threefold,^{36,37,38} sixfold^{39} or eightfold^{40,41,42} band degeneracy. A threefold degenerate node can also be formed by one twofold degenerate band and one single band on the highsymmetry lines of BZ.^{43,44} The semimetals holding threefold degenerate nodes on the highsymmetry points or lines at the Fermi level are both called triplepoint semimetals (TPSMs). The semimetals with eightfold degenerate nodes are also called double Dirac semimetals (DDSMs), just like two overlapping Dirac nodes in the crystal momentum space.
If the valence and conduction bands are not touching at isolated points in the crystal momentum space, but at continuous onedimensional Fermi lines (including loops, chains and links) at the Fermi level, the semimetal is called a nodalline semimetal (NLSM).^{16,45,46,47,48} A nodal line can be viewed as a “kissing” line of Weyl nodes or Dirac nodes with a Chern monopole charge (for Weyl nodal lines) or a Z_{2} monopole charge (for Dirac nodal lines). Nodal lines are generally unstable against perturbations but can be protected by crystalline symmetries at highsymmetry planes (e.g., mirror planes) in the Brillouin zone.
Ever since the first theoretical prediction of pyrochlore iridates as candidates of WSMs, topological semimetals have become a highly attractive field of study. Currently, most of the TSMs calculated from first principles calculations and studied experimentally are nonmagnetic, i.e., the TRSpreserved semimetals, including the well known TaAs family,^{49,50,51} Cd_{3}As_{2},^{31,52} Na_{3}Bi,^{30,32} etc. In recent years, magnetic TSMs are receiving more and more attention, as they have several advantages over nonmagnetic TSMs. First, some magnetic WSMs can host only one pair of Weyl nodes, which are ideal for transport and chiral anomaly studies. Second, systems with broken TRS can have nonzero net Berry curvatures, which can induce unique properties such as intrinsic anomalous Hall effect, thermoelectric currents (anomalous Nernst effect), etc. Third, the halfmetallic feature of some magnetic TSMs such as HgCr_{2}Se_{4}, Heusler compounds and Co_{3}Se_{2}S_{2} makes them good for spin manipulations and spintronics applications. Finally, the magnetic materials are more varied and richer, and the magnetic space group (MSGs) are much larger and complex than space groups, which may derive some novel magnetic TSMs.
According to group theory,^{53} there are 1651 magnetic space groups (MSGs), which are divided into four types:
Here G is the unitary subgroup of M_{i}, i = 1, 2, 3, 4, which is an ordinary crystalline space group, \(\cal{T}\) is the antiunitary timereversal operator, R ∉ G is a Euclidean symmetry other than a pure translation, and τ ∉ G is a translation operator connecting the spinup and spindown sublattices. There are 230 ordinary crystalline space groups of Type I, 230 TRSpreserved space groups (i.e. the gray MSGs) of Type II, 674 MSGs of Type III and 517 MSGs of Type IV.
As we discussed above, the first theoretically predicted TSMs were magnetic pyrochlore iridates and HgCr_{2}Se_{4}. Recently, more and more magnetic WSMs have been proposed, including Heusler compounds,^{54,55,56} Kagome layers^{57,58,59} and honeycomblattice materials.^{60} The study of magnetic DSMs has also made great progress in the past several years. Candidates of DSMs in TypeIII and TypeIV MSGs were proposed, namely CuMnAs^{61} and EdCd_{2}As_{2}.^{62} Recently, SOCrobust magnetic NLSMs were predicted to emerge in the layered system Fe_{3}GeTe_{2}^{63} and LaCl (LaBr)^{64} from first principles calculations. Compared with these theoretical advances, the experimental studies of magnetic TSMs have been rarer and harder. The main difficulty comes from three aspects: (a) many magnetic TSMs proposed are metastable, which makes their highquality crystal samples difficult to synthesize, and (b) their topological properties can highly depend on their magnetic configuration and magnetic moment direction, which may get mispredicted sometimes by first principles calculations, and (c) their complicated domain walls often make their topological band structures difficult to measure and confirm using current experimental techniques such as angleresolved photoemission spectroscopy (ARPES).
In this review article, we will mainly focus on the recently proposed magnetic TSMs from first principles calculations. Section “Magnetic Weyl Semimetals” presents the proposed candidates of magnetic WSMs in chronological order. Section “Magnetic Dirac Semimetals” reviews the predicted magnetic DSMs CuMnAs and EuCd_{2}As_{2} with TypeIII and TypeIV MSG symmetries, respectively. Section “Magnetic Nodal Line Semimetals” reviews the magnetic NLSMs Fe_{3}GeTe_{2} and LaCl (LaBr), which are predicted to be robust against SOC on certain conditions. In the last section, we discuss the potential applications and possible future research directions of magnetic TSMs.
Magnetic Weyl Semimetals
Weyl semimetals (WSMs) are generally divided into two types: magnetic WSMs and noncentrosymmetric WSMs, which correspond to the breaking of the timereversal symmetry \(\cal{T}\) and the inversion symmetry I, respectively. If both symmetries \(\cal{T}\) and I are preserved, the two Weyl nodes with opposite chiralities will meet at the same \(\vec k\)point to form a Dirac node. So to create Weyl nodes, either \(\cal{T}\) or I needs to be broken. Historically, the first types of theoretically predicted topological semimetals were magnetic WSMs in pyrochlore iridates with strong spinorbit coupling and allin/allout (AIAO) magnetic configurations.^{25} Magnetic WSMs are not sufficiently studied at present because of the experimental difficulties due to their complex domain structures. However, magnetic WSMs are worth studying due to their unique properties such as large intrinsic anomalous Hall effect (AHE) and anomalous Nernst effect (ANE), which can be useful for building electronic devices. The AHE is related to the integral of the Berry curvature of the occupied bands in the BZ,^{65,66,67,68,69,70,71} and is only possible in magnetic materials. More explicitly, the intrinsic zerotemperature Hall conductivity at Fermi energy E_{F} is expressed as^{67}
where i, j, l = x, y, z, Θ is the step function and Ω_{l} is l component of the Berry curvature. The Berry curvature is highly enhanced near Weyl nodes, making large AHE possible in magnetic WSMs if the Fermi level is close to the Weyl nodes.^{23,72,73,74} Moreover, the carrier density is reduce to zero at the Weyl nodes, which suggests large anomalous Hall angle in those materials. The ANE is a nontrivial thermoelectric phenomena where a temperature gradient and a perpendicular magnetization can induce a transverse electric voltage.^{75,76,77} As the Berry curvature behaves like a magnetic field, like the AHE, the thermoelectric conductivity can also be calculated by an integral of the Berry curvature,^{67,71,78,79} which then gives rise to the Mott relation
where \(\sigma _{ij}^\prime (E_F)\) is the energy derivative of the intrinsic anomalous Hall conductivity. Thus one immediately expects that a giant ANE can also be generated in magnetic WSMs.^{80,81,82,83,84} We will review some typical magnetic WSMs in this section, such as pyrochlore iridates, HgCr_{2}Se_{4}, Heusler compounds, Kagome layers and honeycomb lattice GdSI.
Pyrochlore iridates
In 2011, Wan et al.^{25} first reported that the 5d transition metal oxides pyrochlore iridates A_{2}Ir_{2}O_{7} (A = Y or rareearth element) with AIAO magnetic order can be turned into the WSM phase. By the method of a “plus U” extension of density functional theory (DFT+U), they found 24 Weyl nodes in bulk and abundant Fermi arcs on surface at intermediate electronic correlation U ~ 1.5 eV.
The calculations show that the influence from the rareearth element on the bands near Fermi level is negligible in A_{2}Ir_{2}O_{7}, therefore, Wan et al. focus on Y_{2}Ir_{2}O_{7} to discuss the magnetic configuration and topological phase transition. In the pyrochlore iridates crystal, the cornersharing tetrahedra of Ir sublattice is largely geometrically frustrated, and the calculation gives the AIAO magnetic configuration, see Fig. 1a. Ir^{4+} is located at the tetrahedra corner with 5d^{5} outershell electrons halffilling the ten d levels. The surrounding oxygen octahedra provides a large crystalfield and causes the splitting between the doubly degenerated e_{g} and triply degenerated t_{2g} states. e_{g} bands are about 2 eV higher, hence the Fermi level is mainly dominated by t_{2g} bands. Due to the strong SOC of 5d transition metal element, the t_{2g} states further split to higher J = 1/2 doublet and lower J = 3/2 quadruplet. The latter is fully filled as Ir^{4+} has five delectrons, J = 1/2 doublet is halffilled and mainly dominate the low energy properties of band structure. Hence there are eight halffilling bands near Fermi level given the four Ir atoms in each unit cell. On the other hand, the electron correlation effect can not be ignored. considering the correlation U, local spin density approximation (LSDA) + SO + U calculations show the phase transition from normal metal at small U to WSM at intermediate U ~ 1.5 eV and Mott insulator phase at U above 2 eV.
In the weak correlation limit, the band structure of nonmagnetic phase calculated by LDA + SO method (without U) reveals that the eight levels near Fermi energy are in the sequence 2,4,2 of degeneracies, which must be metallic phase in the halffilling. On the contrary, experiments show that Y_{2}Ir_{2}O_{7} is an insulator.^{85,86,87} Considering U and other magnetic configuration still can not open the gap, but an insulation band structure can be obtained in strong correlation limit (U > 1.8 eV) and AIAO order, known as Mott insulator. At the intermediate correlation U ~ 1.5 eV, as shown in Fig. 1b, c, the band structure of AIAO magnetic order calculated by LSDA + SO + U demonstrates 24 Weyl nodes in BZ related by three fold rotation symmetry (same chirality) and IS (opposite chirality). Because of symmetry, all Weyl nodes are in the same energy. Adjusting U can move the Weyl nodes. With U increasing, Weyl nodes can move to meet at L point and annihilate, driving to a Mott insulator phase. With U decreasing to around 1 eV, two opposite Weyl nodes can annihilate at X point, and Wan et al. suggested axion insulator phase may appear. Unfortunately, the material will transform to a metallic phase around U ~ 1 eV before the the Weyl nodes annihilate according to the band structure calculation.
The AIAO groundstate magnetic configuration, which is originated from the nearestneighbor antiferromagnetic coupling and strong geometric frustration of the pyrochlore lattice, has been experimentally confirmed.^{88,89,90,91} The magnetic frustration, electronic correlation and strong SOC of the 5d orbitals in transition metal elements are crucial for understanding the origin of the WSM phase in pyrochlore iridates, and are also a treasury of other topological phenomena such as topological insulators, axion insulators and topological Mott insulators.^{92,93} WitczakKrempa et al.^{92,94,95} established a minimal model with the Hubbard Hamiltonian to capture the magnetic ground states and the topological phase by changing correlation U. Although the theoretical prediction of this magnetic WSM phase have not been directly confirmed by experiment, the study on pyrochlore iridates through varies of theoretical methods^{96,97,98,99,100} and indirect experimental signals^{101,102,103} is lasting to shed light on the Weyl nodes and their stabilities. For example, the discovery of the conducting magnetic domain walls in the insulating bulk pyrochlore iridates^{104,105} can be explained as the surviving midgap states at the domain wall.^{106}
HgCr_{2}Se_{4}
The Weyl nodes in pyrochlore iridates are subtle and sensitive to the finetuning of the electronic correlation U. Also there are many Weyl nodes in BZ, making it complicated to analyze the WSM phase. Nearly at the same time, Xu et al.^{26} proposed the ferromagnetic material HgCr_{2}Se_{4} with only one single pair of Weyl nodes with chirality ±2. HgCr_{2}Se_{4} is a ferromagnetic spinel exhibiting large coupling effects between electronic and magnetic properties.^{107} The spinel structure, with space group \(Fd\bar 3m\), can be related to the diamond structures by taking the small Cr_{2}Se_{4} cluster as a single pseudoatom (called X) located at the center of mass, see Fig. 2a, therefore Hg and X form two embedded diamond structure. The Cr_{2}Se_{4} cluster are connected by the corner sharing Cr atoms, hence each Cr atom is octahedrally coordinated by the 6 nearest Se atoms.
The first principles calculation confirms the ferromagnetic order with a total energy about 2.8 eV/f.u. lower than the nonmagnetic phase. The obtained magnetic moment (6.0μ_{B}/f.u.) agrees with experiments^{108,109} very well. Without SOC, it is suggested that the system can be approximately characterized as a “zerogap halfmetal”. It is a halfmetal because of the presence of a gap in the spinup channel and it is zerogap because of the bandtouching around the Γ point in the spindown channel. The Cr^{3+} 3d states are strongly spinpolarized, resulting in the configuration \(t_{2g}^{3 \uparrow }e_g^{0 \uparrow }t_{2g}^{0 \downarrow }e_g^{0 \downarrow }\). The octahedral crystal field surrounding the Cr atoms is strong and opens a gap between the \(t_{2g}^{3 \uparrow }\) and \(e_g^{0 \uparrow }\) subspaces. The top of the valence band from −6 to 0 eV is dominated by Se4p states. Due to the hybridization with Cr3d states, Se4p are slightly spinpolarized but with an opposite moment (about −0.08 μ_{B}/Se). The zerogap behavior in the down spin channel is the most important character, which suggests a band inversion around Γ, similar to the case in HgSe or HgTe.^{110,111}
The four low energy states (8 after considering spin) at the Γ point are the linear combinations P_{x}〉, P_{y}〉, P_{z}〉, S〉, with \(P_\alpha \rangle \approx \frac{1}{{\sqrt 8 }}\mathop {\sum}\nolimits_{i = 1}^8 {{\mathrm{}}p_\alpha ^i\rangle }\) and \(S\rangle \approx 0.4\mathop {\sum}\nolimits_{j = 1}^2 {s^j\rangle } + 0.24\mathop {\sum}\nolimits_{k = 1}^4 {d_{t_{2g}}^k\rangle }\), where α = x, y, z and i, j, k respect Se, Hg, Cr atoms, \(s\rangle ,p_{\alpha = x,y,z}\rangle ,d_{t_{2g} = xy,yz,zx}\rangle\) are corresponding atomic orbits of each atom. Taking these four states as bases, one can found the same situation as in HgSe and HgTe, the only difference is the presence of exchange splitting. The band inversion, where S, ↓〉 being lower than P, ↑〉 at Γ point, is due to the following two factors. Firstly, the Hg5d states are very shallow [located at about −7.0 eV] and its hybridization with Se4p states will push the antibonding Se4p states higher, similar to HgSe. Secondly, the hybridization between unoccupied Cr3d^{↓} and Hg6s^{↓} states will push the Hg6s^{↓} state lower in energy. Thus the S, ↓〉 is about 0.4 eV lower than the P, ↓〉 states, and further enhanced to be 0.55 eV in the presence of SOC. One should be aware of the correlation effect beyond GGA, because the higher the Cr3d^{↓} states, the weaker the hybridization with Hg6s^{↓}. It has been proved that the LDA + U calculations with effective U around 3.0 eV can describe the semiconducting CdCr_{2}S_{4} and CdCr_{2}Se_{4} very well.^{113,114} As for HgCr_{2}Se_{4}, The same LDA + U calculations shows that the band inversion remains unless the U is unreasonably large (>8.0 eV).
When considering SOC, the new lowenergy states at Γ are \(\left {\frac{3}{2}, \pm \frac{3}{2}} \right\rangle ,\left {\frac{3}{2}, \pm \frac{1}{2}} \right\rangle ,\left {\frac{1}{2}, \pm \frac{1}{2}} \right\rangle\), and \(\left {S, \pm \frac{1}{2}} \right\rangle\) contribute from P〉 and S〉 states. The exchange splitting energetically separates the eight bands, with the highest \(\left {\frac{3}{2},\frac{3}{2}} \right\rangle\) and lowest \(\left {S,  \frac{1}{2}} \right\rangle\) state. Several band crossings can be observed in the band inversion, as shown in Fig. 2b. Among them, however, only two kinds of band crossings (called A and B) are important for the states very close to the Fermi level. The crossing A gives two points located at \(k_z = \pm k_z^c\) along the Γ − Z line, the trajectory of crossing B is a closed loop surrounding the Γ point in the k_{z} = 0 plane, as schematically shown in Fig. 2c. Given a 2D plane with fixed k_{z} (k_{z} ≠ 0 and \(k_z \;\ne\; \pm k_z^c\)), the band structure are all gapped, hence one can calculate its Chern number C. It turns out that C = 0 for the planes with \(k_z \;<\;  k_z^c\) or \(k_{z} \;>\; k_{z}^{c}\), while C = 2 for the planes with \( k_z^c \;<\; k_z \;<\; k_z^c\) and k_{z} ≠ 0. Hence the crossing A points locate at the phase boundary between C = 2 and C = 0 planes are topologically unavoidable Weyl nodes. On the other hand, the crossing B points, i.e., the closed loop in the k_{z} = 0 plane is a Weyl nodal line due to the mirror symmetry. Therefore, HgCr_{2}Se_{4} is a material with coexisting Weyl points and Weyl nodal lines when the crystal mirror symmetry is preserved.
To capture the band inversion nature of \(\left {\frac{3}{2},\frac{3}{2}} \right\rangle\) and \(\left {S,  \frac{1}{2}} \right\rangle\) at Γ point, one can downfold the 8 × 8 k ⋅ p effective Hamiltonian to a 2 × 2 model:
where k_{±} = k_{x} ± ik_{y}, M = M_{0} − βk^{2} is the mass term expanded to the second order, and M_{0} > 0, β > 0 to ensure the band inversion. The two bases have opposite parity, hence the offdiagonal element has to be odd in k. \(k_ \pm ^2\) is to conserve the angular momentum along z direction. Thus, to the leading order, \(k_zk_ \pm ^2\) is the only possible form for the offdiagonal element. The energy eigenvalues \(E(k) = \pm \sqrt {M^2 + D^2k_z^2(k_x^2 + k_y^2)^2}\) suggest two gapless solutions: one is the degenerate points along Γ − Z line with \(k_z = \pm k_z^c = \pm \sqrt {M_0/\beta }\); the other is a circle around Γ point in the k_{z} = 0 plane determined by the equation \(k_x^2 + k_y^2 = M_0/\beta\). They are exactly consistent with the first principles calculation. The dispersion of two Weyl nodes are quadratic rather than linear, with their chirality are ±2 respectively, and the Chern number C = 2 for the planes with \( k_z^c \;< k_z \;< k_z^c\) and k_{z} ≠ 0. Two opposite Weyl nodes form a single pair of magnetic monopoles carrying the gauge flux, as shown in Fig. 2d. The nodal line in k_{z} = 0 plane is not topologically unavoidable; however, its existence requires that all gauge flux in the k_{z} = 0 plane (except the loop itself) must vanish.
The surface state of HgCr_{2}Se_{4}, i.e., Fermi arcs, are more stable than the accidental degeneracy in pyrochlore iridates,^{25} given that the band crossings of HgCr_{2}Se_{4} are topologically unavoidable. Another feature is that the fermi arcs are interrupted by the k_{z} = 0 plane, where the nodal line exists. HgCr_{2}Se_{4} is also a promising QAH material in its quantum well structure. When the well is thin enough, the band inversion in the bulk band structure will be removed entirely by the finite size effect. With increasing the thickness, finite size effect is getting weaker and the band inversion restores subsequently, leading to a quantized Hall coefficient σ_{xy} = 2e^{2}/h. In fact, the strong AHE in the bulk samples of HgCr_{2}Se_{4} has already been observed.^{115} On the contrary, the AHE in pyrochlore iridates should be vanishing because of the AF configuration.
Inspired by the doubleWeyl nodes in HgCr_{2}Se_{4}, Fang et al.^{36} classified the two band crossing in nfold rotational symmetric 3D system without TRS. By the k ⋅ p theory, they found that C_{4,6} symmetry can support doubleWeyl nodes on highsymmetry line, consistent with the above result in HgCr_{2}Se_{4}. Besides, the C_{6} symmetry can also support tripleWeyl nodes, which carry the ±3 monopole charges and disperse cubically in the offaxis plane. If one change the magnetization direction from (001) to (111), the C_{4} symmetry in HgCr_{2}Se_{4} is broken whereas the rotationreflection symmetry S_{6} along (111) direction arise. By calculating the \(C_{3(111)} = S_6^2\) eigenvalues and the twoband k ⋅ p theory, Fang et al. found that a doubleWeyl node on k_{z} axis with monopole charge −2 will evolve to a+1 Weyl node on (111) axis and three −1 Weyl nodes off the axis related by C_{3(111)} symmetry.
Although the experimental evidence has not been found yet, the prediction of a single pair of Weyl nodes in HgCr_{2}Se_{4} inspired a series of works about the magnetic and transport properties of this material^{116,117,118} and the quantum correction to the Hall conductance induced by electronelectron interaction.^{119} The transport studies on high quality HgCr_{2}Se_{4} single crystals^{116} confirmed the spinpolarized current in its sorbit conduction band, suggested its halfmetal nature.
Magnetic Heusler Compounds
In recent years, a series of papers predicted the Weyl nodes in Co_{2}based magnetic fullHeusler compounds. Wang and collaborators^{54} studied Co_{2}XZ Heusler compounds (X = IVB or VB; Z = IVA or IIIA) and found that the favorable magnetization direction is along the^{110} easy axis. In this configuration, there are at least two Weyl nodes close to the Fermi energy and largely separated in momentum space. Kübler and Felser^{55} found that the large anomalous Hall effect in Co_{2}MnAl is possibly linked to its two pair of Weyl nodes, and suggested the same WSM phase for Co_{2}MnGa. Soon after, Sakai et al.^{83} revealed the giant anomalous Nernst Effect in Co_{2}MnGa, and provided a guiding principle for increasing the intrinsic transverse thermoelectric conductivity. The Co_{2}MnGa compound is also predicted by Chang et al.^{120} to host the Hopf link protected by two perpendicular mirror plane, in which two nodal rings pass through the center of each other, and the Hopf link opens an extremely small gap (<1 mev) under the SOC. Chang et al.^{56} explored Co_{2}TiX (X = Si, Ge, or Sn) and found similar Weyl points in the^{110} and [001] magnetization ground state.
FullHeusler are magnetic intermetallic compounds with facecentered cubic crystal structure X_{2}YZ (space group \(Fm\bar 3m\), No.225), with transition metal elements X, Y, and maingroup element Z, with X the most electropositive.^{121} The proposed magnetic WSMs by Wang et al. are Co_{2}XZ Heusler compounds (X = IVB or VB; Z = IVA or IIIA) with valence electrons number N_{v} = 26, whose total spin magnetic moment m = N_{v} − 24, according to SlaterPauling rule. Without loss of generality, it is convenient to focus on the candidate Co_{2}ZrSn, which has been synthesized experimentally,^{122} to discuss the topological semimetal phase. The GGA + U without SOC calculated spinpolarized band structure (Fig. 3a) reveals its halfmetallic property, consistent with the experimental investigation of the spin resolved unoccupied DOS of the partner compound Co_{2}TiSn.^{123} the partial DOS suggests the states near Fermi level are dominated by Cod and Zrd electrons. The SOC only has little influence on the band structure (Fig. 3b) and halfmetallic property, because of the small SOC strength of both Co and Zr. The magnetization direction favors^{100,110} the former is slightly lower than latter energetically. Both magnetism demonstrate topological phase with Weyl nodes and nodal lines, in the following, the magnetization is chosen along.^{110}
In the absence of SOC, the energy bands show three nodal lines in the xy, yz, zx plane, protected by their mirror symmetry M_{z}, M_{x}, M_{y}, respectively, as shown in Fig. 3d. When considering SOC and in^{110} spin polarization, some spatial crystal symmetry including M_{z}, M_{x}, M_{y} are broken, leaving a magnetic space group generated by three elements: IS I, two fold rotation C_{2},^{110} and C_{2z}T the combination of time reversal and C_{2z}. The nodal lines are gapped, except a pair of Weyl nodes survived along,^{110} protected by C_{2},^{110} i.e, the crossing bands have different C_{2}^{110} eigenvalues ±i on the highsymmetry line. In addition, other two kinds of Weyl nodes can be found by carefully checking the nodal lines, as shown in Fig. 3e. Four Weyl nodes (W_{1}) in xy plane are related to each other by I and C_{2},^{110} and eight general Weyl nodes (W_{2}) are related by all the three generators of the magnetic group. In fact, the product of the IS eigenvalues of the occupied bands at the inversion symmetric points is −1, hinting the presence of an odd number of pairs of Weyl nodes.^{124} Those Weyl nodes position and topological charge and energy are presented in Table 1.
W_{2} Weyl nodes are removable by tuning SOC to move them to k_{z} axis and annihilate; W_{1} are locally stable in k_{z} = 0 plane due to C_{2z}T^{125} but the energy of W_{1} is very low; the Weyl nodes W, however, are topologically stable and can be tune to Fermi level by alloying. The 27electrons Cobased Heusler family such as Co_{2}NbSn, which have also been synthesized experimentally,^{122} contains one more electron per a unit cell than that of Co_{2}ZrSn. Therefore, by alloying Co_{2}ZrSn with Nb in the Zr site, one can expect the Weyl nodes more close to Fermi level with the main band topology unchanged. The band structure calculation^{54} for Co_{2}Zr_{1−x}Nb_{x}Sn (with x = 0.275) shows that, in this concentration the Weyl nodes are bring to the Fermi level. For the other experimental synthesized 27electron compound Co_{2}VSn,^{122} the alloy Co_{2}Ti_{1−x}V_{x}Sn (with x = 0.1) gives the same result.
When the magnetization parallel to^{100} direction, the remained magnetic group is generated by: I, C_{4x}, C_{2x}I, C_{2y}T, C_{2z}T. Two C_{4x} protected Weyl nodes with Chern number ±2 are found on k_{x} axis. Due to the mirror symmetry C_{2x}I, the nodal line in yz plane remains even with SOC. Also, C_{2y}T (C_{2z}T) allows the existence of Weyl points in xz plane (xy plane), as shown in Table 1. The similar Weyl nodes have also been found in Co_{2}TiX (X = Si, Ge,or Sn) and Co_{2}MnAl(Ga) Heusler compounds, their coordinations, topological charges and energy to Fermi level are summarized in Table 1.
Comparing to other Weyl materials, magnetic Heusler compounds are ferromagnetic halfmetal with Curie temperatures up to the room temperature,^{122} and their magnetism is “soft” and sensitive to external magnetic field. Chadov et al.^{126} studied the stability of the Weyl nodes in fullHeusler compounds, and found that number and coordinates of the Weyl nodes can be controlled by the magnetization direction. Moreover, the vast class of Heusler materials hints that one can tune those compounds across different compositions by alloying to get the desired properties. In summary, it is realistic to manipulate the spin and Weyl nodes in various of Heusler compounds, which provide a promising experimental platform to research spintronics and magnetic Weyl fermions.
Stacking Kagome Lattice
One of the most exotic properties of magnetic WSM is the large intrinsic anomalous Hall effect, Which, in turn, provides a clue for magnetic WSM materials searching. Very recently, several reports proposed the existence of Weyl nodes in layered Kagome lattice.^{57,58,59} Inspired by a series of first principles predictions^{127,128,129} and experimental discoveries^{130,131,132,133} of AHE and spin Hall effect (SHE) in Mn_{3}X (X = Sn, Ge and Ir), Yang et al.^{57} confirmed the Weyl nodes in chiral antiferromagnetic Mn_{3}Sn and Mn_{3}Ge with Kagome layers Mn atoms by ab initio calculation. On the other hand, 2D Kagome lattice with outofplane magnetization has become an excellent platform for AHE study.^{134,135} By stacking, it provides an effective way to realize magnetic WSMs.^{23,136} Following that guiding principle, two groups (Liu et al.^{58} and Wang et al.^{59}) individually claimed that outofplane magnetization Co_{3}Sn_{2}S_{2} with Kagome layers Co atoms is a magnetic WSM candidate. These theoretical and experimental works suggests a new direction to search and synthesize magnetic WSMs among the materials with large AHE. Moreover, they will deepen our understanding on the microscopic mechanisms of the arising of AHE. In the following, we will introduce the theoretical result of Weyl nodes in Mn_{3}Sn (Mn_{3}Ge) and Co_{3}Sn_{2}S_{2} in two subsubsections, respectively.
Mn_{3}Sn (Mn_{3}Ge)–In each layer of Mn_{3}Sn (Mn_{3}Ge) compound (space group P6_{3}/mmc, No.194), Mn atoms form a Kagome lattice with Ge(Sn) atoms located at the centers of each hexagons. In the ground magnetic states, Mn atom carries a magnetic moment of 3.2 μB in Mn_{3}Sn (2.7 μB in Mn_{3}Ge) and form a noncollinear AFM order. The magnetic moments lie inside the xy plane with 2π/3 angles between each two, as shown in Fig. 4c. Such a noncollinear magnetic ground state is originated from the interplay of the easyaxis anisotropy and the SOC induced significant Dzyaloshinskii–Moriya (DM) interactions in the strongly frustrated kagome lattice.^{137,138,139,140,141} This magnetic Kagome lattice has a nonsymmorphic symmetry M_{y}τ = {M_{y}0, 0, 1/2} and two magnetic mirror symmetries M_{x}T and M_{z}T.
Generally, the positions of Weyl nodes can be understood by symmetry analyzing. Time reversal operation will not change the chirality, while mirror reflection will reverse it. Hence, giving a Weyl node, other nodes related by M_{y}τ, M_{x}T and M_{z}T will be settle down. However, the symmetries are slightly broken due to the tiny net moment in real materials (~0.003 μ_{B} per unit cell). This weak symmetry broken is negligible for transport measurement, but will influence the band structure and induce a perturbation of the relationship of the Weyl nodes, for example, slightly shifting the positions of mirror partners, as shown in Table 2.
The bulk band structures with SOC of Mn_{3}Sn and Mn_{3}Ge exhibit similar dispersions, as shown in Fig. 3a, b. At first glance, there are two seemingly band crossing points below the Fermi level at Z and K. A tiny gap lifts the degeneracy and generates one pair of Weyl nodes near Z and K respectively. However, the Weyl node separations near Z and K are very small, and may generate negligible observable consequence in experiment. The physically interesting Weyl nodes are those general band crossings listed in the following.
In fact, Mn_{3}Sn and Mn_{3}Ge are metals with valence and conduction bands crossing many times near the Fermi level, leading to multiple pairs of Weyl nodes. Suppose the valence electron number is N_{v} and count in the crossing between the \(N_v^{th}\) and (N_{v} + 1)^{th} bands. In Mn_{3}Sn, there are 12 Weyl nodes classified into three groups (W_{1}, W_{2}, W_{3}, shown in Fig. 4d and Table 2, each one has three partners according to the symmetries). The Weyl nodes displayed in Mn_{3}Ge are more complicated, as shown in Fig. 4e and Table 2. There are nine groups of Weyl nodes with W_{1,2,7,9} in the k_{z} = 0 plane (W_{9} also on the k_{y} axis), W_{4} in the k_{x} = 0 plane, and W_{3,5,6,8} in generic positions. Therefore, W_{1,2,7,4} have other three partners, W_{9} has other one partner, while W_{3,5,6,8} have other seven partners according to the symmetries.
Right after the discovery of noncollinear magnetic WSM phase in Mn_{3}Sn and Mn_{3}Ge, Guo et al.^{142} studied the large AHE, ANE, as well as SHE and spin Nernst effect (SNE) in Mn_{3}X (X = Sn, Ge, Ga) through the ab initio calculation of the Berry phase. The large AHE and the giant ANE in the noncollinear antiferromagnetic materials Mn_{3}Sn and Mn_{3}Ge can also be understood by the revised linear response tensor^{143} and the cluster multipole extension method.^{144,145} The giant ANE has recently been experimentally confirmed by Ikhlas et al.,^{81} Li et al.,^{82} and Kuroda et al.^{146} in Mn_{3}Sn. Higo et al.^{147} recently observed the large magnetooptical Kerr effect (MOKE) in Mn_{3}Sn. The interplay between the MOKE and the Fermi arcs caused by the Weyl nodes is an interesting question to be answer. The AHE induced by the Fermi arcs in the magnetic domain walls has been observed in Mn_{3}Sn(Ge).^{148,149} Recently, the proposed dynamics of the textures in the noncollinear antiferromagnets provide a theoretical mechanism for driving domain walls in Mn_{3}Sn(Ge, Ir),^{150} which is a platform to study the interplay between the magnetic Weyl nodes and the domain walls.
Co_{3}Sn_{2}S_{2}–The structure of Co_{3}Sn_{2}S_{2} compound is shown in Fig. 5a, b, it is crystalized in a rhombohedral structure (space group \(R\bar 3m,No.166\)) with a quasi2D Co_{3}Sn layer sandwiched between sulfur atoms. The magnetic Co atoms form a perfect Kagome lattice in the xy plane with ferromagnetic order along the easy z axis (Curie temperature 177 K) and the magnetic moment is 0.29 μ_{B}/Co.^{151,152,153} The calculated band structure by Wang et al.^{59} with and without SOC reveals the halfmetallic feature with spin down gapped and spin up states crossing the Fermi level, consistent with the photoemission experimental measurements result.^{154}
When excluding SOC, there are linear band crossings along Γ − L and L − U line, as shown in Fig. 5c. In fact, they are just single points of the nodal line in the mirror plane protected by the mirror symmetry M_{y}. According to the C_{3z} and IS, there are six nodal lines in total in the BZ. Taking account the SOC, the mirror symmetry is broken. As a result, the nodal lines will be gapped as shown in Fig. 5d, except three pairs of Weyl nodes off the highsymmetry line survived. Those Weyl nodes also related by C_{3z} and IS, and contribute to the large intrinsic anomalous Hall effect in Co_{3}Sn_{2}S_{2}. Liu group^{58} also reported the Weyl nodes induced negative magnetoresistance and large anomalous Hall angle, claimed that this ferromagnetic Kagome lattice is the first material hosting both a large anomalous Hall conductivity and a giant anomalous Hall angle that originate from the Berry curvature.
The Weyl nodes near the Fermi level means that Co_{3}Sn_{2}S_{2} can host the large intrinsic transverse thermoelectric conductivity, and recently the giant ANE signal has been confirmed by Yang et al.^{155}
GdSI
Finding the systems exhibiting less pairs of Weyl nodes or other topological properties is a continuous mission. In 2017, Nie and corporators^{60} reported an IS broken honeycomb lattice model with promising topological phases and claimed that LnSI (Ln = Lu, Y, and Gd) satisfies this model. They predicted LuSI (YSI) as 3D strong TI, and GdSI can be an idea WSM with only two pairs of nodes.
LnSI crystal has the space group \(P\bar 6\),^{156} in which Ln atom and S atoms locate in the xy plane to form a honeycomb lattice with I atoms intercalated between two LnS layers, see Fig. 6a. The low energy bands near the Fermi level are dominated by the p_{z} orbits of S atoms and the d_{z2} orbits of the Ln atoms. Although in each unit cell, there are four S and four Ln atoms, only one pair of p_{z}type molecular orbital P_{2}〉 with j_{z} = ±1/2 and one pair of d_{z2}type molecular orbital D_{2}〉 with j_{z} = ±1/2 distribute to and invert at the Fermi level, owing to the chemical bonding and crystal field effects. For GdSI, the f orbits are partially occupied, hence GdSI is very likely to be stabilized in a magnetic phase. In fact, the GGA + U + SOC method comparing different magnetic configurations shows that the most stable one is noncollinear collinear AFM4, as shown in Fig. 6b, which breaks time reversal and the mirror symmetry M_{z}.
The calculated band structure reveals that GdSI is ideal WSM with two pairs of nodes (Fig. 6c, d). The band inversion occurs near Γ point and K(K′) point. Without SOC, due to the configuration II Rashba splitting, the Crossing bands belong to different eigenvalue of M_{z}, hence the crossings are stable and form nodal rings. However, SOC breaks M_{z} symmetry and destroys nodal rings except two pairs of Weyl nodes on the highsymmetry H − K(H′ − K′) line. They are protected by C_{3z} symmetry due to the decrease of the effective angular momentum of d_{z2} orbits at K, which can be understood as following: without loss of generality, suppose Gd atom carrying d_{z2} is located at (1/3, 2/3, 0) in the honeycomb lattice and choose (0, 0, 0) as the rotation center. The rotation can be defined as \(\hat R_3^z = e^{  i2\pi /3\hat J_z}\) with \(\hat J_z = \hat L_z + \hat S_z\). Then one can get \(\hat R_3^zd_{z^2}^{\{ 1/3,2/3,0\} },j_z\rangle _K = e^{  i2\pi /3j_z}d_{z^2}^{\{ 1/3,  1/3,0\} },j_z\rangle _K = e^{  i2\pi /3j_z}e^{i2\pi /3}d_{z^2}^{\{ 1/3,2/3,0\} },j_z\rangle _K = e^{  i2\pi /3(j_z  1)}d_{z^2}^{\{ 1/3,2/3,0\} },j_z\rangle _K\), where \(K = (  1/3,2/3,0)\) is defined with respect to the reciprocal lattice vectors. Therefore, the effective j_{z} for the \(d_{z^2}\) bands at K point will decrease by 1, becoming \( 1/2(d_{z^2} \uparrow \rangle )\) and \( 3/2(d_{z^2} \downarrow \rangle )\), respectively. However, the effective \(j_z^K\) of the p_{z} bands located at (0, 0, 0) site will not change.
The distribution of Weyl nodes in k_{z} > 0 BZ is summarized in will have their counterparts at the same k_{x}, k_{y} but opposite k_{z}, because the inverted bands are approximately symmetrical around K (K′), despite the M_{z} breaking in GdSI. The precise location of Weyl nodes given by the DFT calculation are (−1/3, 2/3, ±0.023) and (1/3, −2/3, ±0.021), where the small difference between K point and K′ point is induced by the TRS breaking.
Magnetic dirac semimetals
A Dirac node is a fourfold degenerate point where two spindegenerate bands cross. There are also some other Dirac nodes that we will not cover in this review, such as the doublerefraction Dirac nodes, in which case the bands near the fourfold degenerate point will split to four nondegenerate bands. Generally, a DSM needs Kramers degeneracy at every \(\vec k\)point to ensure double degeneracy everywhere in momentum space. In a nonmagnetic system, it needs the timereversal \(\cal{T}\) and inversionIsymmetries to be both preserved. In magnetic systems, \(\cal{T}\) is broken, thus one may need a “magnetic symmetry”, which is the product of a crystal symmetry with \(\cal{T}\) to realize the Kramers degeneracy everywhere. The “magnetic symmetry” is often chosen as \(I\cal{T}\) or \(I\tau \cal{T}\), where τ is a slip operation. A space group containing a “magnetic symmetry” (antiunitary generator) is called a magnetic space group (MSG).
The Dirac band crossing is not topologically stable. Generally, adding SOC can gap out the band crossing and change the Dirac node into the gapped dispersion relation of massive Dirac fermions. The material then becomes an insulator, which can be a topological insulator (TI), or a topological crystal insulator (TCI), etc. When the Dirac node is protected by crystal symmetry, e.g. if the two twofolddegenerate bands belong to different representations of some highsymmetry lines or points, the crossing is no longer avoided. On the other hand, compared with the nonmagnetic Dirac semimetals, the magnetic Dirac quasiparticles can be controlled by the Néel spinorbit torques and induce the topological metalinsulator transition, in which, the Néel vector orientation can switch on/off the symmetry that protect the Dirac band crossings.^{157,158} Hence the TRS breaking Dirac semimetals are promising for the spinorbitronics application.^{41,159,160,161} In the following, we review the prediction of magnetic DSMs CuMnAs and EuCd_{2}As_{2}, in which the “magnetic symmetry” causing Kramers degeneracy are \(I\cal{T}\) and \(I\tau \cal{T}\), and the Dirac nodes are protected by the screw axis \(\tilde C_{2z} = \left\{ {C_{2z}\frac{1}{2}0\frac{1}{2}} \right\}\) and threefold rotation C_{3z}, respectively.
CuMnAs
Magnetic DSM was firstly put forward by Peizhe Tang etc.^{61} and they proposed orthorhombic AFM CuMnAs as a candidate. Both TRS T and IS I are broken but their combination IT is respected in this antiferromagnetic system, and screw rotational symmetry \(\tilde C_{2z} = \left\{ {C_{2z}\frac{1}{2}0\frac{1}{2}} \right\}\) protected Dirac points are predicted to be robust on the highsymmetry XU line (k_{x} = π, k_{y} = 0). A rough analyzation can be taken as following. First, the combination IT symmetry gives the Kramers degeneracy everywhere. then to analyze the commuting relation between IT and \(\tilde C_{2z}\), one can denote \(\tilde C_{2z}\) as \(e^{i\pi \hat j_z}e^{ik_x/2 + ik_z/2}\), then we have \((IT)\tilde C_{2z}(IT)^{  1} = \tilde C_{2z}e^{  ik_x  ik_z}\), which becomes \((IT)\tilde C_{2z}(IT)^{  1} =  \tilde C_{2z}e^{  ik_z}\) given k_{x} = π. Hence the Kramers pair states on highsymmetry line will have the same \(\tilde C_{2z}\) eigenvalue, and if two pairs of bands crossing here have opposite \(\tilde C_{2z}\) eigenvalue, i.e., the different representation, the crossing is robust.
CuMnAs and CuMnP have already been confirmed experimentally as roomtemperature antiferromagnets,^{162,163} where nonzero magnetic moments of 3d electrons on Mn atoms order antiferromagnetically, see Fig. 7a. Their crystal structure has the nonsymmorphic space group D_{2h}(Pnma) with four formula units in the primitive unit cell. This space group has eight symmetry operations and can be generated by the IS I, and two nonsymmorphic symmetries: the gliding mirror reflection of the y plane \(R_y = \left\{ {M_y0\frac{1}{2}0} \right\}\), and the twofold screw rotation along the z axis \(\tilde C_{2z} = \left\{ {C_{2z}\frac{1}{2}0\frac{1}{2}} \right\}\). Considering the magnetic configuration will break some symmetries. In the most energyfavored AFM configuration in the orthorhombic phase, the magnetic moments on the inversionrelated Mn atoms are aligned along opposite directions, which breaks both T and I but preserves IT. If SOC is absent, the internal spin space is decoupled from real space, hence the spatial symmetries R_{y} and \(\tilde C_{2z}\) are kept. While when considering SOC, the residual symmetries will depend on the orientation of magnetic moments. For example, only \(\tilde C_{2z}\) can survive if magnetic moments are along the z direction, and protect Weyl nodes on highsymmetry line X − U, as shown in Fig. 7b.
The first principles calculated band structure are shown in Fig. 7c for a case where SOC is turned off in the antiferromagnetic system.One can find band crossings along highsymmetry lines, which are consistent with the previous report.^{162} Beyond these crossings, one can also find an entire elliptic Dirac nodal line (DNL) on the k_{y} = 0 plane around the Fermi level and centered at the X point. Examining the band dispersions under various perturbations shows that no gap opening along the nodal line as long as R_{y} is present. Nevertheless, the nodal structure is not protected by R_{y} because R_{y} and IT commute on the k_{y} = 0 plane which gives the fact that Kramers pair here have opposite R_{y} eigenvalue. By checking the orbital composition of the bands, one will confirm that the existence of such a DNL without SOC is associated with the behaviors of the underlying atomic orbits under R_{y}. For one of the crossing bands, it is composed by d_{xy} and d_{yz} orbits that are odd under the mirror reflection, while the the other band is composed by \(d_{xz},\,d_{z^2}\) and \(d_{{x^2}{y^2}}\) orbits that are even under the mirror reflection. The hopping terms between them must vanish, therefore the gapless DNL is strongly depends on the detailed electronic structures around the Fermi level. Corresponding to the DNL in the bulk, dispersive drumheadlike surface state will appear inside the projection of the DNL on the (010) surface. Such a nontrivial surface state can be measured as a clear signature of the DNL semimetal.^{164}^{,165}
If one still exclude SOC but break R_{y} and keep \(\tilde C_{2z}\) symmetry by, for example, applying the shear strain and shift the Mn atoms, the DNL will open a band gap except at four discrete points. One pair of them are located on the highsymmetry XU line (Fig. 7d), and the other pair is located in the interior of the Brillouin zone. The first pair of fourfold degenerate points are verified to be Dirac points and are guaranteed by the screw rotation symmetry \(\tilde C_{2z}\). Unlike R_{y}, \(\tilde C_{2z}\) and IT are anticommutative along the XU line, thus the doubly degenerate states at each k point along this line have the same \(\tilde C_{2z}\) eigenvalue. therefore, as long as the pair of doubly degenerate bands carry different \(\tilde C_{2z}\) eigenvalue, their crossing must be stable. Based on the ab initio results, the calculated \(\tilde C_{2z}\) eigenvalues of the bands near the Fermi level exactly match the symmetry argument. The other pair of Dirac points in the interior of BZ are enforced by Nielsen–Ninomiya theory.^{28} The argument is that a Dirac point without SOC is made up by two opposite Weyl points and each Weyl point have definite spin. For either spin components, the chirality of the Weyl points on the XU line are found to be the same. As a result, other two Weyl points carrying opposite chirality must exist in the BZ to vanish the total chirality.
When SOC is turned on, some crystalline symmetries can be broken by the magnetism, therefore the stability of the crossing points sensitively depends on the orientation of the Mn atoms’ local magnetic moments. If they are aligned along the z axis, only \(\tilde C_{2z}\) symmetry from the space group survives. As shown in Fig. 7e, in this case, the symmetry argument above for the robust crossing points on the X − U line still holds, hence the fourfold degenerate points here are intact under the protection of \(\tilde C_{2z}\), while the other pair of crossing points are fully gapped. If the magnetic moments are along other directions, \(\tilde C_{2z}\) is broken generally, and the Dirac fermions will obtain mass terms proportional to the strength of SOC. For orthorhombic CuMnAs and CuMnP considered here, the typical energy dependence on the magnetic moments orientation is relatively weak; therefore, to realize stable massless Dirac fermions here, several feasible methods, such as via proximity coupling,^{166} can be taken to pin the moments along the z axis even at finite temperatures.
Similar to nonmagnetic Dirac and WSMs, the nontrivial surface arc state and the orbital texture of Dirac cones could be the direct evidence for the magnetic Dirac fermions. And since the net magnetization in CuMnAs and CuMnP are zero, the arc state and orbital texture can be measured by ARPES.^{167,168} Large spin Hall effects could appear in the Dirac fermions system, in which these relativistic particles could contribute to electric control of local magnetization in IT invariant antiferromagnets. Although the magnetic configuration in the calculation is assumed to be frozen, in fact, AFM fluctuations are inevitably present in CuMnAs and CuMnP. In the massive Dirac fermions, the fluctuations act as the dynamical axion field and cause the exotic modulation of the electromagnetic field.^{169} All discussion in this subsection is based on the local moments totally along z axis. The moments along other direction, and the AFM fluctuation both may break the crystal symmetries that protect the band crossing, and lead to a massive Dirac fermion behavior in this system. The interplay between Dirac fermions, the AFM fluctuations and the symmetry breaking is still under research. Its exact description remains an open question.
EuCd_{2}As_{2}
Hua et al.^{62} exhaustively analyzed the DSMs in the magnetic space groups (MSGs), and proposed a candidate, the interlayer AFM EuCd_{2}As_{2}, as a DSM in centrosymmetric typeIV MSGs, where the group \(\cal{M}\) are defined as \(\cal{G} + T\tau \cal{G}\).
As shown in Fig. 8a, EuCd_{2}As_{2} crystallizes into the CaAl_{2}Si_{2}type structure (space group \(P\bar 3m1\), No.164)^{170,171} with Cd_{2}As_{2} layers separated by the trigonal Eu layers. Eu^{2+} has a halffilled 4f shell, and the interlayer AFM magnetic configuration is the most stable one. Figure 8b shows the projected band structures of the interlayer AFM EuCd_{2}As_{2}, where the low energy bands near the Fermi level are mainly contributed from the p orbits of As atoms and the s orbits of the Cd atoms. Around the Γ point, the doubly degenerate s − s bonding states of Cd atoms (even parity) invert with the p − p antibonding states of As atoms (odd parity), causing a Dirac band crossing along Γ − A line protected by C_{3z} symmetry.
A detailed symmetry analysis reveals that a nonsymmorphic TRS T′ = T ⊕ c, connecting the upspin momentum layer at z = 0 and the downspin momentum layer at z = c, exists in this interlayer AFM system. The MSGs of the interlayer AFM EuCd_{2}As_{2} can be expressed as \(D_{3d}^4 \oplus T^{\prime} D_{3d}^4\), generated by T′, IS I, rotation symmetry C_{3z} and twofold screw \(\tilde C_{2x} = C_{2x} \oplus c\). Combining T′ = T ⊕ c with I, the antiunitarity of PT′ would prohibit the hopping between the nonsymmorphic time reversal pair of states, such as 3/2, ±3/2〉 or 1/2, ±1/2〉, hence every energy state is doubly degenerate in such an interlayer AFM system.
Along Γ − A line, the little group can be described as C_{3v} ⊕ PT′C_{3v}. When SOC is included, the topology and band inversion of the system are dominated by the four states: 3/2, ±3/2〉^{−} from the p − p antibonding states of As and 1/2, ±1/2〉^{+} from the s − s bonding states of Cd. The 4 × 4 effective k ⋅ p Hamiltonian around Γ points, under symmetry restrictions, can be written as (in the order of 1/2, 1/2〉^{+}, 3/2, 3/2〉^{−}, 1/2, −1/2〉^{+}, 3/2, −3/2〉^{−})
where \(\varepsilon _0(k) = C_0 + C_1k_z^2 + C_2(k_x^2 + k_y^2)\), k_{±} = k_{x} ± ik_{y} and \(M(k) = M_0  M_1k_z^2  M_2(k_x^2 + k_y^2)\) with M_{0}, M_{1}, M_{2} < 0 to guarantee the band inversion. This effective model is very similar to the Hamiltonian in Na_{3}Bi, except that the off diagonal terms here is the leading order Bk_{±} rather than high order \(Bk_zk_ \pm ^2\) as in Na_{3}Bi. There are two double degenerate eigenvalues \(E_ \pm = \varepsilon _0 \pm {\mathrm{\Delta }}\) with \({\mathrm{\Delta }} = \sqrt {(A^2 + B^2)(k_x^2 + k_y^2) + M^2(k)}\), which tells two linear Dirac nodes at \(k_c = (0,0, \pm \sqrt {M_0/M_1} )\) along the Γ − A line. The Dirac nodes are confirmed by the calculated surface states and Fermi arcs shown in Fig. 8c, d based on the semiinfinite Green’s functions constructed by the maximally localized Wannier functions.^{172,173} The (001) surface states shown in Fig. 8d exhibit a clear band touching at the Γ point and Fermi level, where two Dirac nodes are projected to the same point. Moreover, a pair of Fermi arc states unambiguously connect the Dirac nodes on the (100) face as plotted in Fig. 8c. Even though the Fermi arcs appear to be closed, their Fermi velocities are discontinuous at the Dirac nodes.
Such a AFM DSM has its own uniqueness. Such a uniqueness is reflected by its derivatives, which makes it different from Na_{3}Bi and CuMnAs. When C_{3z} symmetry is broken, j_{z} is no longer a good quantum number, as a result, the hopping terms between j_{z} = ±1/2〉 and j_{z} = ±3/2〉 can be introduced, and the system will evolve into a strong TI phase due to the inverted band structure. However, due to the nonsymmorphic TRS T′ = T ⊕ c, the boundary states will be gapped on (001) surface, where T′ symmetry is broken. Therefore, a nontrivial AFM Z_{2} invariant protected by T′ can be defined, and the halfquantum Hall effect can be realized on the intrinsically gapped (001) face of such an AFM TI.^{174} On the other hand, when I symmetry is broken, instead of splitting into two pairs of ordinary Weyl points, the AFM DSM will split into two pairs of triple points protected by the small C_{3v} group. This is due to that the magnetic point group C_{3v} on Γ − A has one 2D irreducible representation E_{1/2} (±1/2〉) and two onedimensional irreducible representations E_{3/2} \(\left( {\frac{1}{{\sqrt 2 }}3/2\rangle \pm \frac{i}{{\sqrt 2 }}  3/2\rangle } \right)\). Hence, the degeneracy between ±3/2〉 states originally protected by PT′ is broken, while the degeneracy between ±1/2〉 remains, naturally leading to two pairs of triple points along the Γ − A line.
Magnetic nodal line semimetals
Nodal line semimetals (NLSMs) can be viewed as having a line of Weyl nodes or Dirac nodes with no dispersion along the nodal line and linear dispersion in perpendicular directions. Similar to the 1D Fermiarc surface states in WSMs, NLSMs have the nearly dispersionless 2D “drumhead” surface states embedded inside the band gap between the conduction and valence bands in the 2D projection of the nodal ring, and the drumhead states have infinite DOS. Like Dirac nodes, NLSMs are not topologically stable and need crystalline symmetries to protect the band crossing. When the protecting symmetry breaks, the nodal line can be either fully gapped or gapped into several nodes. So analyzing the evolution of the nodal line is helpful to predict new topological insulators or semimetals. For more details, one can read the recent review articles by Fang et al.^{16} and Yang et al.^{175} So far the research of NLSMs is mainly on timereversalpreserved systems with or without SOC. Several modeling works have studied the magnetic NLSMs via symmetry analysis,^{45,176,177} and generally the first principles predicted magnetic NLSMs emerge as the intermediate phase of a magnetic WSM and a magnetic DSM, as the nodal lines evolve into discrete nodes under SOC. For example, the band structure of GdSI without SOC shows nodal rings protected by M_{z}, which evolve into two pairs of Weyl nodes due to the breaking of M_{z} symmetry by SOC. There are also SOCimmune but impure (coexist with nodal points) magnetic NLSMs predicted. In the ferromagnetic material HgCr_{2}Se_{4},^{26} the mirror symmetry M_{z} protects the magnetic Weyl nodal line in the k_{z} = 0 plane, which coexists with the Weyl nodes on the k_{z}axis. In Co_{2}based Heusler compounds with (100) magnetization configuration,^{54,55,56} the nodal lines in the k_{y} = 0 and k_{z} = 0 planes are gapped into Weyl nodes while the nodal line in the k_{x} = 0 plane remains intact due to the preserved M_{x} symmetry under SOC. So far, finding pure magnetic NLSMs with stable nodal lines near the Fermi level is still on the way, especially in the presence of SOC. Recently, Kim et al.^{63} proposed Van der Waals material Fe_{3}GeTe_{2} and Nie et al.^{64} proposed the 3D layered LaX (X = Cl, Br) as candidates for ferromagnetic NLSMs that are robust against SOC, which will be reviewed in the rest of this section.
Fe_{3}GeTe_{2}
In 2018, Kim et al.^{63} predicted and ARPES detected the Van der Waals material Fe_{3}GeTe_{2} as a candidate ferromagnetic NLSM, which is stable in the case that orbital and spin angular momenta are perpendicular. The layered Fe_{3}GeTe_{2} is an itinerant electron ferromagnetic material with the Curie temperature high to T_{c} = 220 K.^{178,179} In the hexagonal crystal structure, as shown in Fig. 9a, the Fe_{3}Ge slabs are coupled via vdW interaction, and sandwiched by Te layers. The Fe^{I}Fe^{I} pairs across the center of the hexagonal plaquettes in the covalently bonded Fe^{II}Ge honeycomb lattice (Fig. 9b). The AB stacking configuration (Fig. 9c) of Fe^{II}Ge layers is essential for the fully spin polarized nodal line degeneracy because of its particular crystalline symmetry.
Before describing the AB stacking Fe^{II}Ge layer structure, one can first consider the Fe^{II}Ge bilayer as shown in Fig. 9c, which has the space group \(P\bar 31m\) (No.164) generated by C_{3z}, C_{2y} and the IS I. The highsymmetry K point is invariant under C_{3z}, C_{2y}, and IT, which allows 2D irreducible representation and therefore, double degeneracy in the absence of SOC. Given that the bands are fully spin polarized, the degeneracy comes from the orbital degree of freedom, as illustrated in Fig. 9d, e. On the contrary, in the AA stacking hypothetical bilayer (Fig. 9f), C_{2y} and I are broken and the new symmetry C_{2x} is not a relevant symmetry at the K point. As a result, the C_{3z} symmetry at K point only have 1D irreducible representation characterized by its eigenvalues, and hence the bands are nondegenerate here.
The 3D vdW layered Fe_{3}GeTe_{2} structure belongs to the space group P6_{3}/mmc (No.194), which is generated by \(\tilde C_{6z} = \left\{ {C_{6z}00\frac{1}{2}} \right\}\), C_{2y} and I. The highsymmetry point K (H) has little group generated by \(\tilde C_{6z}I\), C_{2y} and IT, which allows 2D irreducible representations. Without SOC, the calculated band structure confirms the crossing at K point and the typical Mexicanhat due to Rashba effect, as shown in Fig. 9g. The crossing bands come from the mixed 3d orbits with Fe^{I}Fe^{I} L_{z} = ±1 states and Fe^{II} L_{z} = ±2 states. More specifically, the eigenstates crossing at K can be denoted by \(\psi _{1,k} = L_z = + 1\rangle _U^I + L_z = + 2\rangle _U^{II}\) and \(\psi _{2,k} = L_z =  1\rangle _L^I + L_z =  2\rangle _L^{II}\). In this bases, the symmetry operators \(\tilde C_{6z}I\) and C_{2y} can be represented as \(\tilde C_{6z}I = {\mathrm{cos}}\frac{{2\pi }}{3} + i\,{\mathrm{sin}}\frac{{2\pi }}{3}\tau _z\) and C_{2y} = τ_{x} with the Pauli matrices τ_{x,y,z} denoting ψ_{1,k} and ψ_{2,k}. The effective k ⋅ p Hamiltonian can be written as
with the parameters ε_{0} = −0.03 eV, m_{xy} = 0.077 eV^{−1} Å^{−2}, m_{z} = 0.036 eV^{−1} Å^{−2} and α = 0.71 eV Å estimated from the band calculation. Straightforwardly, there is band degeneracy along K − H line, and also K′ − H′ line according to \(\tilde C_{6z}\) symmetry. The nodal lines are protected by C_{3z} and \(\tilde C_{6z}M_y\) (IT), which gives the 2D irreducible representations along K − H line.
Now consider the influence of SOC with the form H_{SO} = λ_{SO} L ⋅ S, which in the ferromagnetic spin configuration, can be treated as H_{SO} ≈ λ_{SO}L ⋅ 〈S〉. The crossing bands are composed of L_{z} = ±1〉^{I} and L_{z} = ±2〉^{II} states, hence in the bases of ψ_{1,k} and ψ_{2,k}, 〈L_{x}〉 = 〈L_{y}〉 = 0 and \(\langle L_z\rangle = \frac{4}{3}\tau _z\). As a result, the nodal lines are stable even with SOC in the situation of S∥x(y). On the other hand, SOC will open a gap of ~60 meV along the nodal line if S∥z. Generally, the SOC gap depends on the angle between S and z direction. Therefore, in this ferromagnetic NLSM Fe_{3}GeTe_{2}, the stability of the band crossing and its SOC gap can be tuned by the fully spin polarization direction.
Unfortunately, the real Fe_{3}GeTe_{2} magnetic configuration favors the spin polarization along z direction, thus the nodal lines will open a gap in the presence of SOC and produce a Berry flux along the line to induce the large AHE. Hence, to find a proper material with magnetic nodal lines which is robust even against SOC is still an urgent task in the topological semimetal field.
LaCl (LaBr)
In 2019, Nie et al.^{64} predicted the spinful nodal lines in 3D layered materials LaX (X = Cl, Br), which are constructed by stacking 2D Weyl materials. Generally, Weyl nodes can exist in 2D materials protected by crystal symmetry, for example, one pair of Weyl nodes protected by mirror symmetry M_{y}. When the 2D WSM is stacked into 3D layered system, three classes of topological semimetals can be obtained according to the symmetry and the interlayer coupling strength. Class 1 is two nodal lines extending through the BZ when the interlayer coupling is weak. Class 2 is nodal loops or nodal chains with strong interlayer coupling. Class 3 is 3D WSM if the symmetry on the stacking line is broken. Following the guideline, Nie et al. found the idea 2D WSMs LaX, and due to the weak interlayer coupling, the 3D layered LaX are ferromagnetic NLSMs with a pair of nodal lines extending through the BZ and protected by mirror symmetry.
The LaX crystal has the hexagonal layered structure (space group \(R\bar 3m\), No.166), as shown in Fig. 10a. It is built by stacking the tightly bound quadruple layer, which has XLaLaX sublayers made up by two hexagonal rareearthmetal La layers sandwiched between two hexagonal halogen (X) layers. The stacking pattern is ABCtype trilayer along the z axis with weak van der Waals interaction. The interlayer distance is around 10Å and the interlayer coupling is much weaker than most layered compounds, including graphene. As a result, it is easy to obtain the 2D singlelayer LaX through exfoliation methods. The calculated total energies of different magnetic configurations (see Fig. 10b, ferromagnetic FM1, FM2 and antiferromagnetic AFM1, AFM2) for 2D and 3D LaX suggest FM1 where the easy magnetization axis lies in the xy plane. Due to the almost negligible magnetocrystalline anisotropy (the energy is of the order of 0.0001 meV), the spin prefers to align along the y direction.
For the 2D singlelayer LaCl (LaBr will have the almost same result), the calculated band structures reveal a deep inversion at Γ point, as shown in Fig. 10c. Without SOC, the band inversion forms a nodal line around Γ point. With the consideration of SOC, the nodal line opens a gap except two Weyl nodes on the M − Γ − M′ line, which are protected by mirror symmetry M_{y}. When the idea 2D WSMs are stacked to build a 3D LaCl, due to the extremely weak interlayer coupling, one may obtain a 3D nodal line semimetal (class 1). In fact, the calculated band structures of 3D LaCl confirm this speculation, as shown in Fig. 10d–f. Without SOC, the nodal line from singlelayer LaCl forms a cylinder centered around the Γ point, as schematically shown in the lower inset of Fig. 10d. When considering SOC, the cylinder is gapped out everywhere except two nodal lines in the k_{x} − k_{z} plane crossing through the BZ (class 1). The crossing two bands have opposite eigenvalue of M_{y} as shown in the upper inset of Fig. 10d, hence the nodal lines are protected by M_{y} and robust to SOC.
Different from the ordinary nodal lines, the nodal lines in LaCl (LaBr) always appear in pairs because of the IS I. One pair of nodal lines can meet and annihilate in the momentum space without breaking the mirror symmetry M_{y}. On the other hand, the 3D LaX (X = Cl, Br) ferromagnetic NLSM with one pair of spinful nodal lines extending through the BZ is so far the only predicted magnetic NLSM candidate exactly robust against SOC. This discovery is meaningful to open a new path to realize and research the long pursued nodalline fermions, and its interplay with magnetization.
Discussion and outlook
Topological semimetals extend the topological classification of materials from insulators to metallic systems, and have become one of the most attractive fields of study in condensed matter physics in recent years. Nonmagnetic TSMs have been well studied theoretically and the mapping from the topological classification of TRSpreserved materials to the band representation of the space group at highsymmetry points in BZ has been well established.^{180,181,182,183,184} The database of topological materials including nonmagnetic TSMs has been published by Weng et al. (see http://materiae.iphy.ac.cn/). On the other hand, magnetic TSMs are a relatively new area that is still far from well developed. Although the topological classification of magnetic insulators has been performed based on the corepresentation theory and Khomology of magnetic point groups,^{185,186} the topological phases of magnetic TSMs based on magnetic space groups are awaiting further studies. Magnetic DSMs and NLSMs that are robust against SOC are presently quite rare. In the search of new magnetic TSMs, first principles calculations have encountered big challenges, as they often overestimate the energy gain of magnetization and end up predicting wrong magnetic configurations and direction. On the experimental side, conventional ARPES measurements are usually ineffective to study the band structures of magnetic TSMs because of the magnetic domain wall problem. Most experimentally reported magnetic WSMs are based on indirect evidences such as large AHE and anomalous Nernst effect. Nevertheless, magnetic TSMs have their unique advantages. Because their symmetries and electronic structures depend sensitively on their magnetic structures and direction, it becomes convenient and realistic to manipulate their topological properties and phase transition by applying an external magnetic field, which can be highly useful in the design of spintronic devices. Hence, more research works are necessary to find out robust and highquality magnetic TSMs with the degenerate points or lines close to Fermi level. More reliable experimental methods are needed to measure and confirm the topological properties of magnetic TSMs, such as the Weyl nodes, drumhead surface state etc.
“New fermions” such as those in TPSMs can also be realized in magnetic materials. In the antiferromagnetic EuCd_{2}As_{2}, when inversion symmetry is broken, a Dirac node will evolve to a pair of threefold degenerate nodes, each protected by the C_{3v}symmetry. Cheung et al.^{44} have checked all the magnetic symmorphic point groups to search for triple points protected on highsymmetry line and found that Dirac and triple points can coexist in particular systems. The nonsymmorphic antiferromagnetic CeSbTe is predicted to host various topological states including Dirac and Weyl as well as triple and eightfold degenerate points.^{41} A full classification and understanding of the topological properties in the MSGs is far from completion, and is still an open question waiting for answers, in which completely new topological states, beyond all the known topological states at present, may be discovered.
Finally, electron–electron correlations are usually very important in magnetic materials. The interplay of topological order with electron–electron correlations remains a widely open question.^{187,188}
References
Klitzing, Kv, Dorda, G. & Pepper, M. New method for highaccuracy determination of the finestructure constant based on quantized hall resistance. Phys. Rev. Lett. 45, 494–497 (1980).
Thouless, D. J., Kohmoto, M., Nightingale, M. P. & den Nijs, M. Quantized hall conductance in a twodimensional periodic potential. Phys. Rev. Lett. 49, 405–408 (1982).
Haldane, F. D. M. Model for a quantum hall effect without landau levels: condensedmatter realization of the “Parity Anomaly”. Phys. Rev. Lett. 61, 2015–2018 (1988).
Kane, C. L. & Mele, E. J. Z_{2} topological order and the quantum spin hall effect. Phys. Rev. Lett. 95, 146802 (2005).
Fu, L., Kane, C. L. & Mele, E. J. Topological insulators in three dimensions. Phys. Rev. Lett. 98, 106803 (2007).
Fu, L. Topological crystalline insulators. Phys. Rev. Lett. 106, 106802 (2011).
Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).
Qi, X.L. & Zhang, S.C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011).
Bernevig, B. A., Hughes, T. L. & Zhang, S.C. Quantum spin hall effect and topological phase transition in HgTe quantum wells. Science 314, 1757–1761 (2006).
Konig, M. et al. Quantum spin hall insulator state in HgTe quantum wells. Science 318, 766–770 (2007).
Zhang, H. et al. Topological insulators in Bi_{2}Se_{3}, Bi_{2}Te_{3} and Sb_{2}Te_{3} with a single Dirac cone on the surface. Nat. Phys. 5, 438 (2009).
Xia, Y. et al. Observation of a largegap topologicalinsulator class with a single Dirac cone on the surface. Nat. Phys. 5, 398 (2009).
Chen, Y. L. et al. Experimental realization of a threedimensional topological insulator, Bi_{2}Te_{3}. Science 325, 178–181 (2009).
Yu, R. et al. Quantized anomalous hall effect in magnetic topological insulators. Science 329, 61–64 (2010).
Chang, C.Z. et al. Experimental observation of the quantum anomalous hall effect in a magnetic topological insulator. Science 340, 167–170 (2013).
Fang, C., Weng, H., Dai, X. & Fang, Z. Topological nodal line semimetals. Chin. Phys. B 25, 117106 (2016).
Yan, B. & Felser, C. Topological materials: weyl semimetals. Annu. Rev. Condens. Matter Phys. 8, 337–354 (2017).
Burkov, A. A. Weyl metals. Annu. Rev. Condens. Matter Phys. 9, 359–378 (2018).
Armitage, N. P., Mele, E. J. & Vishwanath, A. Weyl and Dirac semimetals in threedimensional solids. Rev. Mod. Phys. 90, 015001 (2018).
Murakami, S., Iso, S., Avishai, Y., Onoda, M. & Nagaosa, N. Tuning phase transition between quantum spin Hall and ordinary insulating phases. Phys. Rev. B 76, 205304 (2007).
Murakami, S. Phase transition between the quantum spin Hall and insulator phases in 3D: emergence of a topological gapless phase. New J. Phys. 9, 356–356 (2007).
Murakami, S. & Kuga, S.i Universal phase diagrams for the quantum spin Hall systems. Phys. Rev. B 78, 165313 (2008).
Burkov, A. A. & Balents, L. Weyl semimetal in a topological insulator multilayer. Phys. Rev. Lett. 107, 127205 (2011).
Balents, L. Weyl electrons kiss. Physics 4 (2011).
Wan, X., Turner, A. M., Vishwanath, A., Savrasov, S. Y. Topological semimetal and Fermiarc surface states in the electronic structure of pyrochlore iridates. Phys. Rev. B 83, 205101 (2011).
Xu, G., Weng, H., Wang, Z., Dai, X. & Fang, Z. Chern semimetal and the quantized anomalous hall effect in HgCr_{2}Se_{4}. Phys. Rev. Lett. 107, 186806 (2011).
Nielsen, H. B. & Ninomiya, M. Absence of neutrinos on a lattice. Nucl. Phys. B 193, 173–194 (1981).
Nielsen, H. B. & Ninomiya, M. Absence of neutrinos on a lattice. Nucl. Phys. B 185, 20–40 (1981).
Yang, B.J., Nagaosa, N. Classification of stable threedimensional Dirac semimetals with nontrivial topology. Nat. Commun. 5, 4898 (2014).
Wang, Z. et al. Dirac semimetal and topological phase transitions in A_{3}Bi (A = Na, K, Rb). Phys. Rev. B 85, 195320 (2012).
Wang, Z., Weng, H., Wu, Q., Dai, X. & Fang, Z. Threedimensional Dirac semimetal and quantum transport in Cd_{3}As_{2}. Phys. Rev. B 88, 125427 (2013).
Liu, Z. K. et al. Discovery of a threedimensional topological dirac semimetal, Na_{3}Bi. Science 343, 864–867 (2014a).
Borisenko, S. et al. Experimental realization of a threedimensional dirac semimetal. Phys. Rev. Lett. 113, 027603 (2014).
Neupane, M. et al. Observation of a threedimensional topological Dirac semimetal phase in highmobility Cd_{3}As_{2}. Nat. Commun. 5, 3786 (2014).
Bradlyn, B., et al. Beyond Dirac and Weyl fermions: unconventional quasiparticles in conventional crystals. Science 353, aaf5037 (2016).
Fang, C., Gilbert, M. J., Dai, X. & Bernevig, B. A. Multiweyl topological semimetals stabilized by point group symmetry. Phys. Rev. Lett. 108, 266802 (2012).
Zhu, Z., Winkler, G. W., Wu, Q., Li, J. & Soluyanov, A. A. Triple point topological. Met. Phys. Rev. X 6, 031003 (2016).
Chang, G. et al. Nexus fermions in topological symmorphic crystalline metals. Sci. Rep. 7, 1688 (2017).
Chang, G. et al. Unconventional chiral fermions and large topological fermi arcs in RhSi. Phys. Rev. Lett. 119, 206401 (2017).
Wieder, B. J., Kim, Y., Rappe, A. M. & Kane, C. L. Double dirac semimetals in three dimensions. Phys. Rev. Lett. 116, 186402 (2016).
Schoop, L. M. et al. Tunable Weyl and Dirac states in the nonsymmorphic compound CeSbTe. Sci. Adv. 4, 2317 (2018).
Geilhufe, R. M., Borysov, S. S., Bouhon, A. & Balatsky, A. V. Data mining for threedimensional organic dirac materials: focus on space group 19. Sci. Rep. 7, 7298 (2017).
Weng, H., Fang, C., Fang, Z. & Dai, X. Topological semimetals with triply degenerate nodal points in θphase tantalum nitride. Phys. Rev. B 93, 241202 (2016).
Cheung, C.H. et al. Systematic analysis for triple points in all magnetic symmorphic systems and symmetryallowed coexistence of Dirac points and triple points. New J. Phys. 20, 123002 (2018).
Burkov, A. A., Hook, M. D. & Balents, L. Topological nodal semimetals. Phys. Rev. B 84, 235126 (2011).
Fang, C., Chen, Y., Kee, H.Y. & Fu, L. Topological nodal line semimetals with and without spinorbital coupling. Phys. Rev. B 92, 081201 (2015).
Bian, G. et al. Topological nodalline fermions in spinorbit metal PbTaSe_{2}. Nat. Commun. 7, 10556 (2016).
Hu, J. et al. Evidence of topological nodalline fermions in ZrSiSe and ZrSiTe. Phys. Rev. Lett. 117, 016602 (2016).
Huang, X. et al. Observation of the chiralanomalyinduced negative magnetoresistance in 3D Weyl semimetal TaAs. Phys. Rev. X 5, 031023 (2015).
Lv, B. Q. et al. Experimental discovery of weyl semimetal TaAs. Phys. Rev. X 5, 031013 (2015).
Shekhar, C. et al. Extremely large magnetoresistance and ultrahigh mobility in the topological Weyl semimetal candidate NbP. Nat. Phys. 11, 645 (2015).
Liu, Z. K. et al. A stable threedimensional topological Dirac semimetal Cd_{3}As_{2}. Nat. Mater. 13, 677 (2014).
Bradley, C. J., Cracknell, A. P. The mathematical theory of symmetry in solids. Oxford classic texts in the physical sciences. Clarendon Press, Oxford [u.a.], 1st edition (2010).
Wang, Z. et al. Timereversalbreaking weyl fermions in magnetic heusler alloys. Phys. Rev. Lett. 117, 236401 (2016).
Kübler, J. & Felser, C. Weyl points in the ferromagnetic Heusler compound Co_{2}MnAl. Europhys. Lett. 114, 47005 (2016).
Chang, G. et al. Roomtemperature magnetic topological Weyl fermion and nodal line semimetal states in halfmetallic Heusler Co_{2}TiX (X=Si, Ge, or Sn). Sci. Rep. 6, 38839 (2016).
Yang, H. et al. Topological Weyl semimetals in the chiral antiferromagnetic materials Mn_{3}Ge and Mn_{3}Sn. New J. Phys. 19, 015008 (2017).
Liu, E. et al. Giant anomalous Hall effect in a ferromagnetic kagomelattice semimetal. Nat. Phys. 14, 1125–1131 (2018).
Wang, Q. et al. Large intrinsic anomalous Hall effect in halfmetallic ferromagnet Co_{3}Sn_{2}S_{2} with magnetic Weyl fermions. Nat. Commun. 9, 3681 (2018).
Nie, S., Xu, G., Prinz, F. B. & Zhang, S.c. Topological semimetal in honeycomb lattice LnSI. Proc. Natl. Acad. Sci. USA 114, 10596–10600 (2017).
Tang, P., Zhou, Q., Xu, G., Zhang, S.C. Dirac fermions in an antiferromagnetic semimetal. Nat. Phys. 12, 1100–1104 (2016).
Hua, G. et al. Dirac semimetal in typeIV magnetic space groups. Phys. Rev. B 98, 201116 (2018).
Kim, K. et al. Large anomalous Hall current induced by topological nodal lines in a ferromagnetic van der Waals semimetal. Nat. Mater. 17, 794–799 (2018).
Nie, S., Weng, H. & Prinz, F. B. Topological nodalline semimetals in ferromagnetic rareearthmetal monohalides. Phys. Rev. B 99, 035125 (2019).
Fang, Z. et al. The anomalous hall effect and magnetic monopoles in momentum space. Science 302, 92–95 (2003).
Haldane, F. D. M. Berry curvature on the fermi surface: anomalous hall effect as a topological fermiliquid property. Phys. Rev. Lett. 93, 206602 (2004).
Xiao, D., Yao, Y., Fang, Z. & Niu, Q. Berryphase effect in anomalous thermoelectric transport. Phys. Rev. Lett. 97, 026603 (2006).
Wang, X., Vanderbilt, D., Yates, J. R. & Souza, I. Fermisurface calculation of the anomalous Hall conductivity. Phys. Rev. B 76, 195109 (2007).
Nagaosa, N., Sinova, J., Onoda, S., MacDonald, A. H. & Ong, N. P. Anomalous Hall effect. Rev. Mod. Phys. 82, 1539–1592 (2010).
Xiao, D., Chang, M.C. & Niu, Q. Berry phase effects on electronic properties. Rev. Mod. Phys. 82, 1959–2007 (2010).
Gradhand, M. et al. Firstprinciple calculations of the Berry curvature of Bloch states for charge and spin transport of electrons. J. Phys.: Condens. Matter 24, 213202 (2012).
Yang, K.Y., Lu, Y.M. & Ran, Y. Quantum Hall effects in a Weyl semimetal: possible application in pyrochlore iridates. Phys. Rev. B 84, 075129 (2011).
Chen, Y., Bergman, D. L. & Burkov, A. A. Weyl fermions and the anomalous Hall effect in metallic ferromagnets. Phys. Rev. B 88, 125110 (2013).
Burkov, A. A. Anomalous hall effect in weyl metals. Phys. Rev. Lett. 113, 187202 (2014).
Lee, W.L., Watauchi, S., Miller, V. L., Cava, R. J. & Ong, N. P. Anomalous hall heat current and nernst effect in the CuCr_{2}Se_{4−x}Br_{x} Ferromagnet. Phys. Rev. Lett. 93, 226601 (2004).
Miyasato, T. et al. Crossover behavior of the anomalous hall effect and anomalous nernst effect in itinerant ferromagnets. Phys. Rev. Lett. 99, 086602 (2007).
Pu, Y., Chiba, D., Matsukura, F., Ohno, H. & Shi, J. Mott relation for anomalous hall and nernst effects in Ga_{1−x}Mn_{x}As ferromagnetic semiconductors. Phys. Rev. Lett. 101, 117208 (2008).
Zhang, C., Tewari, S. & Das Sarma, S. Berryphasemediated topological thermoelectric transport in gapped single and bilayer graphene. Phys. Rev. B 79, 245424 (2009).
Dumitrescu, E., Zhang, C., Marinescu, D. C. & Tewari, S. Topological thermoelectric effects in spinorbit coupled electron and holedoped semiconductors. Phys. Rev. B 85, 245301 (2012).
Sharma, G., Goswami, P. & Tewari, S. Nernst and magnetothermal conductivity in a lattice model of Weyl fermions. Phys. Rev. B 93, 035116 (2016).
Ikhlas, M. et al. Large anomalous Nernst effect at room temperature in a chiral antiferromagnet. Nat. Phys. 13, 1085 (2017).
Li et al. Anomalous Nernst and RighiLeduc Effects in Mn_{3}Sn: berry curvature and entropy flow. Phys. Rev. Lett. 119, 056601 (2017).
Sakai, A. et al. Giant anomalous Nernst effect and quantumcritical scaling in a ferromagnetic semimetal. Nat. Phys. 14, 1119–1124 (2018).
Noky, J., Gayles, J., Felser, C. & Sun, Y. Strong anomalous Nernst effect in collinear magnetic Weyl semimetals without net magnetic moments. Phys. Rev. B 97, 220405 (2018).
Taira, N., Wakeshima, M. & Hinatsu, Y. Magnetic properties of iridium pyrochlores R_{2}Ir_{2}O_{7} (R = Y, Sm, Eu and Lu). J. Phys.: Condens. Matter 13, 5527 (2001).
Fukazawa, H. & Maeno, Y. Filling control of the pyrochlore oxide Y_{2}Ir_{2}O_{7}. J. Phys. Soc. Jpn 71, 2578–2579 (2002).
Soda, M., Aito, N., Kurahashi, Y., Kobayashi, Y. & Sato, M. Transport, thermal and magnetic properties of pyrochlore oxides Y_{2−x}Bi_{x}Ir_{2}O_{7}. Phys. B: Condens. Matter 329–333, 1071–1073 (2003).
Disseler, S. M. et al. Magnetic order in the pyrochlore iridates A _{2}Ir_{2}O_{7} (A = Y, Yb). Phys. Rev. B 86, 014428 (2012).
Tomiyasu, K. et al. Emergence of magnetic longrange order in frustrated pyrochlore Nd_{2}Ir_{2}O_{7} with metalinsulator transition. J. Phys. Soc. Jpn 81, 034709 (2012).
Disseler, S. M. Direct evidence for the allin/allout magnetic structure in the pyrochlore iridates from muon spin relaxation. Phys. Rev. B 89, 140413 (2014).
Lefrançois, E. et al. Anisotropytuned magnetic order in pyrochlore iridates. Phys. Rev. Lett. 114, 247202 (2015).
WitczakKrempa, W., Chen, G., Kim, Y. B. & Balents, L. Correlated quantum phenomena in the strong spinorbit regime. Annu. Rev. Condens. Matter Phys. 5, 57–82 (2014).
Du, Y. & Wan, X. The novel electronic and magnetic properties in 5d transition metal oxides system. Computational Mater. Sci. 112, 416–427 (2016).
WitczakKrempa, W. & Kim, Y. B. Topological and magnetic phases of interacting electrons in the pyrochlore iridates. Phys. Rev. B 85, 045124 (2012).
Go, A., WitczakKrempa, W., Jeon, G. S., Park, K. & Kim, Y. B. Correlation effects on 3D topological phases: from bulk to boundary. Phys. Rev. Lett. 109, 066401 (2012).
Chen, G. & Hermele, M. Magnetic orders and topological phases from fd exchange in pyrochlore iridates. Phys. Rev. B 86, 235129 (2012).
Moon, E.G., Xu, C., Kim, Y. B. & Balents, L. Nonfermiliquid and topological states with strong spinorbit coupling. Phys. Rev. Lett. 111, 206401 (2013).
Wang, R., Go, A. & Millis, A. J. Electron interactions, spinorbit coupling, and intersite correlations in pyrochlore iridates. Phys. Rev. B 95, 045133 (2017).
Wang, R., Go, A. & Millis, A. Weyl rings and enhanced susceptibilities in pyrochlore iridates: k⋅p analysis of cluster dynamical meanfield theory results. Phys. Rev. B 96, 195158 (2017).
Berke, C., Michetti, P. & Timm, C. Stability of the Weylsemimetal phase on the pyrochlore lattice. New J. Phys. 20, 043057 (2018).
Sushkov, A. B. et al. Optical evidence for a Weyl semimetal state in pyrochlore Eu_{2}Ir_{2}O_{7}. Phys. Rev. B 92, 241108 (2015).
Ueda, K., Fujioka, J. & Tokura, Y. Variation of optical conductivity spectra in the course of bandwidthcontrolled metalinsulator transitions in pyrochlore iridates. Phys. Rev. B 93, 245120 (2016).
Nakayama, M. et al. Slater to mott crossover in the metal to insulator transition of Nd_{2}Ir_{2}O_{7}. Phys. Rev. Lett. 117, 056403 (2016).
Moccia, M., Castaldi, G., Savo, S., Sato, Y. & Galdi, V. Independent manipulation of heat and electrical current via bifunctional metamaterials. Phys. Rev. X 4, 021025 (2014).
Ma, E. Y. et al. Mobile metallic domain walls in an allinallout magnetic insulator. Science 350, 538 (2015).
Yamaji, Y. & Imada, M. Metallic interface emerging at magnetic domain wall of antiferromagnetic insulator: fate of extinct weyl electrons. Phys. Rev. X 4, 021035 (2014).
Wojtowicz, P. Semiconducting ferromagnetic spinels. IEEE Trans. Magn. 5, 840–848 (1969).
Baltzer, P. K., Lehmann, H. W. & Robbins, M. Insulating ferromagnetic spinels. Phys. Rev. Lett. 15, 493–495 (1965).
Baltzer, P. K., Wojtowicz, P. J., Robbins, M. & Lopatin, E. Exchange interactions in ferromagnetic chromium chalcogenide spinels. Phys. Rev. 151, 367–377 (1966).
Delin, A. Firstprinciples calculations of the IIVI semiconductor β HgS: Metal or semiconductor. Phys. Rev. B 65, 153205 (2002).
Moon, C.Y. & Wei, S.H. Band gap of Hg chalcogenides: symmetryreductioninduced bandgap opening of materials with inverted band structures. Phys. Rev. B 74, 045205 (2006).
Weng, H., Dai, X. & Fang, Z. Topological semimetals predicted from firstprinciples calculations. J. Phys.: Condens. Matter 28, 303001 (2016).
Fennie, C. J. & Rabe, K. M. Polar phonons and intrinsic dielectric response of the ferromagnetic insulating spinel CdCr_{2}S_{4} from first principles. Phys. Rev. B 72, 214123 (2005).
Yaresko, A. N. Electronic band structure and exchange coupling constants in ACr_{2} X _{4} spinels (A = Zn, Cd, Hg; X = O, S, Se). Phys. Rev. B 77, 115106 (2008).
Solin, N. I. & Chebotaev, N. M. Magnetoresistance and Hall effect of the magnetic semiconductor HgCr_{2}Se_{4} in strong magnetic fields. Phys. Solid State 39, 754–758 (1997).
Guan, T. et al. Evidence for HalfMetallicity in ntype HgCr_{2}Se_{4}. Phys. Rev. Lett. 115, 087002 (2015).
Lin, C. et al. Spin correlations and colossal magnetoresistance in HgCr_{2}Se_{4}. Phys. Rev. B 94, 224404 (2016).
Lin, C.J., Shi, Y.G. & Li, Y.Q. Analytical descriptions of magnetic properties and magnetoresistance in nType HgCr 2 Se 4. Chin. Phys. Lett. 33, 077501 (2016).
Yang, S. et al. Giant quantum correction to the anomalous Hall effect (2019).
Chang, G. et al. Topological Hopf and chain link semimetal states and their application to Co_{2}MnGa. Phys. Rev. Lett. 119, 156401 (2017).
Manna, K., Sun, Y., Muechler, L., Kübler, J. & Felser, C. Heusler, weyl and berry. Nat. Rev. Mater. 3, 244–256 (2018).
Carbonari, A. W. et al. Magnetic hyperfine field in the Heusler alloys Co_{2}YZ (Y = V, Nb, Ta, Cr; Z = Al, Ga). J. Magn. Magn. Mater. 163, 313–321 (1996).
Klaer, P. et al. Tailoring the electronic structure of halfmetallic Heusler alloys. Phys. Rev. B 80, 144405 (2009).
Hughes, T. L., Prodan, E. & Bernevig, B. A. Inversionsymmetric topological insulators. Phys. Rev. B 83, 245132 (2011).
Soluyanov, A. A. et al. TypeII Weyl semimetals. Nature 527, 495 (2015).
Chadov, S., Wu, S.C., Felser, C. & Galanakis, I. Stability of Weyl points in magnetic halfmetallic Heusler compounds. Phys. Rev. B 96, 024435 (2017).
Kübler, J. & Felser, C. Noncollinear antiferromagnets and the anomalous Hall effect. Europhys. Lett. 108, 67001 (2014).
Chen, H., Niu, Q. & MacDonald, A. H. Anomalous hall effect arising from noncollinear antiferromagnetism. Phys. Rev. Lett. 112, 017205 (2014).
Zhang, Y. et al. Strong anisotropic anomalous Hall effect and spin Hall effect in the chiral antiferromagnetic compounds Mn_{3} X (X = Ge, Sn, Ga, Ir, Rh, and Pt). Phys. Rev. B 95, 075128 (2017).
Nakatsuji, S., Kiyohara, N. & Higo, T. Large anomalous Hall effect in a noncollinear antiferromagnet at room temperature. Nature 527, 212 (2015).
Kiyohara, N., Tomita, T. & Nakatsuji, S. Giant anomalous hall effect in the chiral antiferromagnet Mn_{3}Ge. Phys. Rev. Appl. 5, 064009 (2016).
Nayak, A. K. et al. Large anomalous Hall effect driven by a nonvanishing Berry curvature in the noncolinear antiferromagnet Mn_{3}Ge. Sci. Adv. 2, e1501870 (2016).
Zhang, W. et al. Giant facetdependent spinorbit torque and spin Hall conductivity in the triangular antiferromagnet IrMn_{3}. Sci. Adv. 2, e1600759 (2016).
Ohgushi, K., Murakami, S. & Nagaosa, N. Spin anisotropy and quantum Hall effect in the kagomé lattice: chiral spin state based on a ferromagnet. Phys. Rev. B 62, R6065–R6068 (2000).
Xu, G., Lian, B. & Zhang, S.C. Intrinsic quantum anomalous hall effect in the kagome lattice Cs_{2}LiMn_{3}F_{12}. Phys. Rev. Lett. 115, 186802 (2015).
Zyuzin, A. A., Wu, S. & Burkov, A. A. Weyl semimetal with broken time reversal and inversion symmetries. Phys. Rev. B 85, 165110 (2012).
Tomiyoshi, S. & Yamaguchi, Y. Magnetic structure and weak ferromagnetism of Mn_{3}Sn studied by polarized neutron diffraction. J. Phys. Soc. Jpn 51, 2478–2486 (1982).
Tomiyoshi, S., Yamaguchi, Y. & Nagamiya, T. Triangular spin configuration and weak ferromagnetism of Mn_{3}Ge. J. Magn. Magn. Mater. 31–34, 629–630 (1983).
Sticht, J., Höck, K.H. & Kübler, J. Noncollinear itinerant magnetism: the case of Mn_{3}Sn. J. Phys.: Condens. Matter 1, 8155–8176 (1989).
Brown, P. J., Nunez, V., Tasset, F., Forsyth, J. B. & Radhakrishna, P. Determination of the magnetic structure of Mn_{3}Sn using generalized neutron polarization analysis. J. Phys.: Condens. Matter 2, 9409–9422 (1990).
Sandratskii, L. M. & Kübler, J. Role of orbital polarization in weak ferromagnetism. Phys. Rev. Lett. 76, 4963–4966 (1996).
Guo, G.Y. & Wang, T.C. Large anomalous Nernst and spin Nernst effects in the noncollinear antiferromagnets Mn_{3} X (X = Sn, Ge, Ga). Phys. Rev. B 96, 224415 (2017).
Seemann, M., Ködderitzsch, D., Wimmer, S. & Ebert, H. Symmetryimposed shape of linear response tensors. Phys. Rev. B 92, 155138 (2015).
Suzuki, M.T., Koretsune, T., Ochi, M. & Arita, R. Cluster multipole theory for anomalous Hall effect in antiferromagnets. Phys. Rev. B 95, 094406 (2017).
Suzuki, M.T. et al. Multipole expansion for magnetic structures: a generation scheme for a symmetryadapted orthonormal basis set in the crystallographic point group. Phys. Rev. B 99, 174407 (2019).
Kuroda, K. et al. Evidence for magnetic Weyl fermions in a correlated metal. Nat. Mater. 16, 1090 (2017).
Higo, T. et al. Large magnetooptical Kerr effect and imaging of magnetic octupole domains in an antiferromagnetic metal. Nat. Photonics 12, 73–78 (2018).
Liu, J. & Balents, L. Anomalous hall effect and topological defects in antiferromagnetic weyl semimetals: Mn_{3}Sn/Ge. Phys. Rev. Lett. 119, 087202 (2017).
Li, X. et al. Chiral domain walls of Mn_{3}Sn and their memory, http://arxiv.org/abs/1903.03774v2 (2019).
Yamane, Y., Gomonay, O., Sinova, J. Dynamics of noncollinear antiferromagnetic textures driven by spin current injection, http://arxiv.org/abs/1901.05684v1 (2019).
Richard, W. & Irina, A. Citation for: half antiperovskites. III. Crystallographic and electronic structure effects in Sn_{2−x}In_{x}Co_{3}S_{2}. Z. für. anorganische und Allg. Chem. 632, 1531–1537 (2006).
Vaqueiro, P. & Sobany, G. G. A powder neutron diffraction study of the metallic ferromagnet Co_{3}Sn_{2}S_{2}. Solid State Sci. 11, 513–518 (2009).
Schnelle, W. et al. Ferromagnetic ordering and halfmetallic state of Sn_{2}Co_{3}S_{2} with the shanditetype structure. Phys. Rev. B 88, 144404 (2013).
Holder, M. et al. Photoemission study of electronic structure of the halfmetallic ferromagnet Co_{3}Sn_{2}S_{2}. Phys. Rev. B 79, 205116 (2009).
Yang, H. et al. Giant anomalous Nernst effect in the magnetic Weyl semimetal Co_{3}Sn_{2}S_{2}, http://arxiv.org/abs/1811.03485v2 (2018).
Beck, H. P. & Strobel, C. Zur Hochdruckpolymorphie der Seltenerdsulfidiodide LnSI. Z. anorg. allg. Chem. 535, 229–239 (1986).
Šmejkal, L., Železný, J., Sinova, J. & Jungwirth, T. Electric Control of Dirac Quasiparticles by SpinOrbit Torque in an Antiferromagnet. Phys. Rev. Lett. 118, 106402 (2017).
Šmejkal, L., Jungwirth, T. & Sinova, J. Route towards Dirac and Weyl antiferromagnetic spintronics. Phys. Status Solidi RRL 11, 1700044 (2017).
Wadley, P. & Edmonds, K. W. Spin switching in antiferromagnets using Nelorder spinorbit torques. Chin. Phys. B 27, 107201 (2018).
Šmejkal, L., Mokrousov, Y., Yan, B. & MacDonald, A. H. Topological antiferromagnetic spintronics. Nat. Phys. 14, 242–251 (2018).
Emmanouilidou, E., Liu, J., Graf, D., Cao, H. & Ni, N. Spinflop phase transition in the orthorhombic antiferromagnetic topological semimetal Cu_{0}.95MnAs. J. Magn. Magn. Mater. 469, 570–573 (2019).
Máca, F. et al. Roomtemperature antiferromagnetism in CuMnAs 324, 1606–1612 (2012).
Wadley, P. et al. Electrical switching of an antiferromagnet. Science 351, 587–590 (2016).
Kim, Y., Wieder, B. J., Kane, C., Rappe, A. M. Dirac line nodes in inversionsymmetric crystals. Phys. Rev. Lett. 115, 036806 (2015).
Yu, R., Weng, H., Fang, Z., Dai, X., Hu, X. Topological nodeline semimetal and dirac semimetal state in antiperovskite Cu_{3}PdN. Phys. Rev. Lett. 115, 036807 (2015).
Katmis, F. et al. A hightemperature ferromagnetic topological insulating phase by proximity coupling. Nature 533, 513–516 (2016).
Xu, S.Y. et al. Observation of Fermi arc surface states in a topological metal. Science 347, 294–298 (2015).
Xu, S.Y. et al. Discovery of a Weyl Fermion semimetal and topological Fermi arcs. Science 349, 613–617 (2015).
Li, R., Wang, J., Qi, X.L. & Zhang, S.C. Dynamical axion field in topological magnetic insulators. Nat. Phys. 6, 284–288 (2010).
Artmann, A., Mewis, A., Roepke, M. & Michels, G. AM_{2}X_{2}Verbindungen mit CaAl_{2}Si_{2}Struktur. XI. Struktur und Eigenschaften der Verbindungen ACd_{2}X_{2} (A: Eu, Yb; X: P, As, Sb). Z. Anorg. Allg. Chem. 622, 679–682 (1996).
Schellenberg, I., Eul, M., Hermes, W. & Pöttgen, R. A 121Sb and 151Eu Mössbauer spectroscopic investigation of EuMn_{2}Sb_{2}, EuZn_{2}Sb_{2}, YbMn_{2}Sb_{2}, and YbZn_{2}Sb_{2}. Z. Anorg. Allg. Chem. 636, 85–93 (2009).
Sancho, M. P. L., Sancho, J. M. L. & Rubio, J. Quick iterative scheme for the calculation of transfer matrices: application to Mo (100). J. Phys. F: Met. Phys. 14, 1205 (1984).
Sancho, M. P. L., Sancho, J. M. L., Sancho, J. M. L. & Rubio, J. Highly convergent schemes for the calculation of bulk and surface Green functions. J. Phys. F: Met. Phys. 15, 851 (1985).
Mong, R. S. K., Essin, A. M. & Moore, J. E. Antiferromagnetic topological insulators. Phys. Rev. B 81, 245209 (2010).
Yang, S.Y. et al. Symmetry demanded topological nodalline materials. Adv. Phys.: X 3, 1414631 (2018).
Wang, J. Antiferromagnetic topological nodal line semimetals. Phys. Rev. B 96, 081107 (2017).
Yu, J., Yan, B. & Liu, C.X. Model Hamiltonian and time reversal breaking topological phases of antiferromagnetic halfHeusler materials. Phys. Rev. B 95, 235158 (2017).
Chen, B. et al. Magnetic properties of layered itinerant electron ferromagnet Fe_{3}GeTe_{2}. J. Phys. Soc. Jpn 82, 124711 (2013b).
May, A. F., Calder, S., Cantoni, C., Cao, H. & McGuire, M. A. Magnetic structure and phase stability of the van der Waals bonded ferromagnet Fe_{3−x}GeTe_{2}. Phys. Rev. B 93, 014411 (2016).
Bradlyn, B. et al. Topological quantum chemistry. Nature 547, 298 (2017).
Po, H. C., Vishwanath, A. & Watanabe, H. Symmetrybased indicators of band topology in the 230 space groups. Nat. Commun. 8, 50 (2017).
Song, Z., Zhang, T., Fang, Z. & Fang, C. Quantitative mappings between symmetry and topology in solids. Nat. Commun. 9, 3530 (2018).
Song, Z., Zhang, T. & Fang, C. Diagnosis for nonmagnetic topological semimetals in the absence of spinorbital coupling. Phys. Rev. X 8, 031069 (2018).
Zhang, T. et al. Catalogue of Topological Electronic Materials, http://arxiv.org/abs/1807.08756v1 (2018).
Zhang, R.X. & Liu, C.X. Topological magnetic crystalline insulators and corepresentation theory. Phys. Rev. B 91, 115317 (2015).
Okuma, N., Sato, M., Shiozaki, K. Topological classification under nonmagnetic and magnetic point group symmetry: application of realspace AtiyahHirzebruch spectral sequence to higherorder topology, http://arxiv.org/abs/1810.12601v3 (2018)
Pesin, D. & Balents, L. Mott physics and band topology in materials with strong spinorbit interaction. Nat. Phys. 6, 376 (2010).
Morimoto, T. & Nagaosa, N. Weyl mott insulator. Sci. Rep. 6, 19853 (2016).
Acknowledgements
We thank the support by the Ministry of Science and Technology of China (2018YFA0307000), and the National Natural Science Foundation of China (11874022); G.X. is supported by the National ThousandYoungTalents Program.
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J.Z. and G.X. contributed to the collection of references and outline of the review paper. Z.H. participated in the mathematical formalisms and interpretations of the basic theory. All authors contributed to writing the manuscript.
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Zou, J., He, Z. & Xu, G. The study of magnetic topological semimetals by first principles calculations. npj Comput Mater 5, 96 (2019). https://doi.org/10.1038/s4152401902375
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DOI: https://doi.org/10.1038/s4152401902375
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