Abstract
Analogues of the elementary particles have been extensively searched for in condensedmatter systems for both scientific interest and technological applications^{1,2,3}. Recently, massless Dirac fermions were found to emerge as lowenergy excitations in materials now known as Dirac semimetals^{4,5,6}. All of the currently known Dirac semimetals are nonmagnetic with both timereversal symmetry and inversion symmetry ^{7,8,9}. Here we show that Dirac fermions can exist in one type of antiferromagnetic system, where both and are broken but their combination is respected. We propose orthorhombic antiferromagnet CuMnAs as a candidate, analyse the robustness of the Dirac points under symmetry protections and demonstrate its distinctive bulk dispersions, as well as the corresponding surface states, by ab initio calculations. Our results provide a possible platform to study the interplay of Dirac fermion physics and magnetism.
Main
The great success in the field of topological insulators^{1,2} since last decade inspired the study of topological features of metals. Topological metals have nontrivial surface states and their bulk Fermi surfaces can be topologically characterized^{3}. Among them, Dirac semimetals^{4,5,6} have received special attention, because they host relativistic particles, the massless Dirac fermions, in a nonrelativistic setup. In such Dirac materials, two doubly degenerate bands contact at discrete momentum points called Dirac points, and disperse linearly along all directions around these points. The fourfold degenerate Dirac points are unstable by themselves; hence, symmetry protection is necessary^{7}. Following this guideline, several threedimensional Dirac semimetals have been theoretically proposed, and some of them have been experimentally verified recently^{8,9}. All of these materials have timereversal symmetry , inversion symmetry , and certain crystalline rotation symmetry.
If some of the symmetries are broken, massless Dirac fermions can in general be destroyed. For instance, when either or is broken, each doubly degenerate band is lifted, so that the Dirac cones can split into multiple Weyl cones^{10}. This gives birth to Weyl semimetals^{11,12,13,14,15,16,17}, and the chiralanomalyrelated transport phenomena can be observed as a signature^{18,19}. However, the result of both and breaking remains obscure until now. It is thus natural to ask whether Dirac fermions can still exist in the absence of both and .
In this letter, we answer the question in the affirmative, and provide a concrete example of such a Dirac semimetallic phase. We consider threedimensional systems with the antiferromagnetic (AFM) order that breaks both and but respects their combination . The lowenergy physics can be explicitly captured by the following fourband effective model
where d_{i}(k), i = 0,1, …, 5 are real functions of momentum k, and τ_{x, y, z} (σ_{x, y, z}) are Pauli matrices for orbital (spinrelated AFM) basis (see Supplementary Section 3 for details). The antiunitary symmetry satisfying is given as , where K is complex conjugation. Due to this symmetry, the last five terms in H(k) anticommute with one another; therefore, every band must be doubly degenerate (this degeneracy holds generally for all invariant systems, see Supplementary Section 1) with energy spectrum
If one has d_{i}(q) = 0 for i = 1, …, 5 at a certain isolated momentum point k = q, then the two doubly degenerate bands must cross each other there. The resulting fourfold degenerate point k = q can be further made Diraclike when additional constraints are enforced by crystalline symmetry.
To realize stable fourfold degenerate crossing points, the generic way is to let the pair of doubly degenerate bands carry different representations of certain symmetries in the system^{20,21}. For our AFM model, however, there is a simpler starting point for investigation. Let us assume that the local magnetic moments are along the z axis without loss of generality. When the spin–orbit coupling (SOC) effect is ignored, d_{3}(k) and d_{4}(k) vanish because they correspond to spinflip processes. Thus, the crossing points must be present in general, because with three momentum components one can tune d_{1}(k), d_{2}(k) and d_{5}(k) to zero simultaneously. These points can be Dirac points as long as linear dispersion is required by certain crystalline symmetry. When SOC is included, the presence of the crossing points can still be guaranteed due to the protection of the crystalline symmetry.
In accordance with our analysis, we discover that the AFM semimetals orthorhombic CuMnAs and CuMnP^{22,23} can host the Dirac fermions around the Fermi level. Their crystal structure has the nonsymmorphic space group D_{2h} (Pnma) with four formula units in the primitive unit cell (see Fig. 1a, b for the structure and the first Brillouin zone). The space group consists of eight symmetry operations that can be generated by three of them: the inversion , the gliding mirror reflection of the y plane R_{y} = {m_{y}  (0, (1/2), 0)}, and the twofold screw rotation along the z axis S_{2z} = {C_{2z}  ((1/2), 0, (1/2))}, where the two nonsymmorphic symmetries R_{y} and S_{2z} are important in our symmetry analysis (see Supplementary Section 2 for details).
CuMnAs and CuMnP have been experimentally identified as roomtemperature antiferromagnets previously^{23,24}, where nonzero magnetic moments on Mn atoms with 3d electrons order antiferromagnetically (see Supplementary Section 4). The magnetic configuration breaks some symmetries from the original space group. For the most energyfavoured AFM configuration in the orthorhombic phase (see Fig. 1a), the magnetic moments on the inversionrelated Mn atoms are aligned along opposite directions; therefore, both and are broken whereas still holds. If SOC is absent, spin internal space is decoupled from real space, so the spatial symmetries R_{y} and S_{2z} are kept. When SOC is included, however, residual symmetries depend on the orientation of magnetic moments, for example, only S_{2z} will survive if magnetic moments are along the z axis.
With the crystal structure and symmetry operations in mind, we begin to present our results of band structure calculations as well as effective model analysis (see Supplementary Section 9 for details of parameter choices). Figure 1d shows the electronic structures from firstprinciples calculations for a case where SOC is turned off in the AFM system (see Supplementary Section 5 for details). These results are consistent with the previous report^{23}, in which band crossings are visible along highsymmetry lines. Beyond these crossings, we found an entire elliptic Dirac nodal line (DNL) on the k_{y} = 0 plane around the Fermi level, with its centre at the X point (see Fig. 2a). We examined the band dispersions under various perturbations (see Supplementary Section 7), and found no gap opening along the nodal structure as long as R_{y} is present. Nevertheless, because R_{y} and commute on the k_{y} = 0 plane, no rigorous symmetry protection should hold for the band crossing here in the general sense (see Supplementary Section 2). By checking the orbital composition of the bands, we finally confirmed that the existence of such a DNL in the absence of SOC is associated with the behaviours of the underlying atomic orbitals under R_{y} (see Supplementary Section 7). Corresponding to the DNL in the bulk, a nontrivial surface state appears inside the projection of the DNL on the (010) surface (see Fig. 2f–h). This dispersive drumheadlike surface state can be measured as a clear signature of the DNL semimetal^{25,26}.
When we still exclude SOC but break R_{y} (see Supplementary Section 8), a bandgap opens along the entire DNL except at four discrete points. One pair of the fourfold degenerate points is located on the highsymmetry line X–U, and the other pair is in the interior of the Brillouin zone. We verified the first pair as Dirac points with linear dispersions shown in Fig. 2b. The Dirac points are guaranteed by the screw rotation symmetry S_{2z}. Unlike R_{y}, S_{2z} anticommutes with along the X–U line, so the doubly degenerate states at each k point on this line have the same S_{2z} eigenvalues. As a result, the crossing of one pair of the doubly degenerate bands must be stable, as long as they carry different S_{2z} eigenvalues. On the basis of ab initio results, we calculated S_{2z} eigenvalues of the bands near the Fermi level, and the results match the symmetry argument exactly (see Supplementary Sections 2 and 6).
To check the nature of the Dirac points, we derive the lowenergy effective model (see Supplementary Section 3). As we mentioned above, our AFM system without SOC is described by (we ignore the overall shift term in the following)
On the highsymmetry line X–U, the screw rotation symmetry S_{2z} is represented by S_{2z} = i{e}^{i({k}_{z}/2)}{\tau}_{z}. Expanding the Hamiltonian around one Dirac point and enforcing the symmetry constraints, we can obtain the exact Diractype Hamiltonian
where v_{ij}(i, j = 1,2,3) are velocity coefficients for different directions. These parameters are obtained from our calculations, and the resulting Dirac cones are anisotropic (see Fig. 2c, d). Splitting into two blocks that correspond to σ_{z} = ±1, we can decouple each Dirac cone into two Weyl cones with opposite chiralities (see Fig. 2e)
Since SOC is absent, the AFM basis σ_{z} = ±1 is almost equivalent to the physical spin basis (see Supplementary Section 1). We thus calculated the surface states on the (010) surface for each spin component, as shown in Fig. 2i–k. It is clear that Fermi arcs emerge on the surface, and they connect pairs of Weyl points with opposite chiralities. For either spin component, the chiralities of the Weyl points on the X–U line are found to be the same; therefore, it is reasonable that the other two Weyl points that carry opposite chiralities exist in the Brillouin zone such that the total chirality vanishes^{27}.
When SOC is turned on, the presence of the crossing points sensitively depends on the orientation of the local magnetic moments on Mn atoms, as crystalline symmetries can be broken by the magnetism. If the magnetic moments are aligned along the z axis, only S_{2z} symmetry from the space group survives. In this case, the symmetry argument for the robust crossing points along the X–U line still holds, so the fourfold degenerate points on this line are intact under the protection of S_{2z}, while the other pair of crossing points are fully gapped (see Figs 1d and 3a). The effective model near each fourfold degenerate point is derived in the same way,
where the small perturbation terms δ_{i}, (i = 1,2,3,4) originate from SOC, which can be treated as weak coupling between the two Weyl fermions at each Dirac cone. The calculated electronic structures of AFM CuMnAs are shown in Fig. 3b–d. It is clear that no gap opens at the crossing point along the X–U line, and that nontrivial surface states appear on the (010) surface that connect two gapless points. When the magnetic moments are along other directions, S_{2z} is broken generally, and the Dirac fermions will gain a small mass, which is proportional to the strength of SOC (see Supplementary Section 8). For orthorhombic CuMnAs and CuMnP, the typical energy dependence on the orientation of magnetic moments is relatively weak; therefore, to realize massless Dirac fermions here, several feasible methods, such as via proximity coupling^{28}, can be taken to pin the moments along the z axis even at finite temperatures (see Supplementary Section 10 for details).
Finally we discuss experimental detections and new physics of the Dirac fermions in AFM systems. Similar to normal Dirac and Weyl semimetals, the nontrivial surface state and the orbital texture of Dirac cones could be measured as direct evidence for the Dirac fermions by angleresolved photoemission spectroscopy^{29,30}, since there is no net magnetization in CuMnAs and CuMnP. Furthermore, large spin Hall effects could appear due to the presence of Dirac fermions, and these relativistic particles might contribute to electric control of local magnetization in invariant antiferromagnets^{24}. In addition, AFM fluctuations are inevitably present in CuMnAs and CuMnP, although our current treatment assumes that the magnetic configuration is frozen. If the Dirac fermions are massive, the fluctuations resemble the dynamical axion field, giving rise to exotic modulation of the electromagnetic field^{31}. In the case where local moments are along the z axis, the fluctuations not only directly couple to the massless Dirac fermions, but also produce fermion masses through breaking crystalline symmetries. The exact description of interplay between Dirac fermions, the AFM fluctuations, and the symmetry breaking at the moment remains an open question.
Methods
Ab initio calculations.
The firstprinciples calculations were carried out by the density functional theory method with the projector augmentedwave method^{32}, as implemented in the Vienna ab initio simulation package^{33}. The Perdew–Burke–Ernzerhof exchange–correlation functional and the planewave basis with energy cutoff of 300 eV were employed. The lattice parameters (see Fig. 1a) were chosen from experimental values^{22}, which are a = 6.577 Å, b = 3.854 Å and c = 7.310 Å for orthorhombic CuMnAs, and a = 6.318 Å, b = 3.723 Å and c = 7.088 Å for orthorhombic CuMnP respectively; and the inner atomic positions were allowed to be fully relaxed until the residual forces are less than 1 × 10^{−3} eV Å^{−1}. The Monkhorst–Pack k points were 9 × 15 × 9, and SOC was included in selfconsistent electronic structure calculations. The maximally localized Wannier functions^{34} were constructed to obtain the tightbinding Hamiltonian, which is used to calculate the bulk Fermi surface, surface electronic spectrum and surface states.
Data availability.
The data that support the plots within this paper and other findings of this study are available from the corresponding author on request.
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Acknowledgements
We acknowledge the support from the Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, under contract DEAC0276SF00515, NSF under Grant No. DMR1305677 and FAME, one of six centres of STARnet, a Semiconductor Research Corporation programme sponsored by MARCO and DARPA.
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P.T., Q.Z., G.X. and S.C.Z. conceived and designed the project. P.T. performed the firstprinciples calculations; Q.Z. performed theoretical analysis; P.T. and Q.Z. analysed the data and wrote the manuscript. All authors commented on the manuscript.
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Tang, P., Zhou, Q., Xu, G. et al. Dirac fermions in an antiferromagnetic semimetal. Nature Phys 12, 1100–1104 (2016). https://doi.org/10.1038/nphys3839
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DOI: https://doi.org/10.1038/nphys3839
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