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Antiferromagnetic Dirac Semimetals in Two Dimensions

The search for symmetry-protected 2D Dirac semimetals analogous to graphene is important both for fundamental and practical interest. The 2D Dirac cones are protected by crystalline symmetries and magnetic ordering may destroy their robustness. Here we propose a general framework to classify stable 2D Dirac semimetals in spin-orbit coupled systems having the combined time-reversal and inversion symmetries, and show the existence of the stable Dirac points in 2D antiferromagnetic semimetals. Compared to 3D Dirac semimetals which fall into two distinct classes, Dirac semimetals in 2D with combined time-reversal and inversion symmetries belongs to single class which is closely related to the nonsymmorphic space group symmetries. We further provide a concrete model in antiferromagnetic semimetals which supports symmetry-protected 2D Dirac points. The symmetry breaking in such systems leads to 2D chiral topological states such as quantum anomalous Hall insulator and chiral topological superconductor phases.


I. INTRODUCTION
The discovery of the time-reversal invariant topological insulators 1,2 greatly inspired the study of symmetryprotected Dirac semimetals (DSMs) 3- 14 . In a DSM, the low-energy physics is well described by pseudorelativistic Dirac fermions with linear energy dispersions along all momentum directions around Dirac points (DPs) in the Brillouin zone (BZ). To guarantee the robustness of three-dimensional (3D) DPs, the crystalline symmetry protection is a necessitate 7 , similar to the 2D DPs in graphene 15 . The 3D DSM materials 3-11 have both the time-reversal symmetry T and inversion symmetry P. The T or P breaking in general leads to Weyl semimetals [16][17][18][19][20][21][22][23][24][25][26][27] , where the DPs split into Weyl points. However, under certain conditions, such 3D DPs still remain stable with T -broken 28 . This motivates us to ask whether stable DPs could exist in 2D in the absence of T and P. The positive answer to this question will lead to many interesting physical phenomena and novel topological states, as expected from the great success in the field of graphene.
Recently, the concept of 2D DSMs in the presence of spin-orbit coupling (SOC) has been introduced by Young and Kane 29 , where the nonsymmorphic space group symmetries play a key role [29][30][31][32][33] . In this paper we study the 2D Dirac fermions in antiferromagnetic (AFM) semimetals. A general framework is proposed to classify stable 2D DSMs in SOC systems with PT symmetry. In sharp contrast to T -invariant 3D DSMs which fall into two distinct classes 7 , DSMs in 2D with PT symmetry only have one class. The DPs of 2D DSMs reside at the BZ boundary which are protected by the nonsymmorphic space group symmetries. We further provide a tight-binding model in AFM semimetals which supports symmetry-protected 2D DPs and symmetry breaking in such systems leads to exotic chiral topological states in 2D. We conclude with a brief discussion on the possible material venues for such phases.
The 2D DP with fourfold degeneracy is generated when two doubly-degenerate energy bands avoidably cross. The most common symmetry for crystalline solids to have double degeneracy of bands is PT symmetry satisfying (PT ) 2 = −1, which includes two cases with both T , P conserved and T , P broken. Under this condition, the low-energy physics can be described by the minimal fourband effective Hamiltonian, where Γ a are 4 × 4 Dirac Γ matrices satisfying {Γ a , Γ b } = 2δ ab , and the specific representation of Γ a depends on the crystalline symmetry. The particle-hole asymmetry term 0 (k) is neglected for simplicity, d a (k) (a = 1, . . . , 5) are real functions of k, and k = (k x , k y ). The energy spectrum is The DPs can be generated only when all d a (k) = 0 for certain k in the BZ. A simple case for this condition exists at the quantum critical point of the phase transition between a quantum spin Hall (QSH) and a normal insulator (NI) in the presence of T and P 16,34 . Such DPs are accidental degeneracies and not robust against perturbations. Therefore, crystalline symmetry is needed to protect the 2D DPs. The organization of this paper is as follows. After this introductory section, Sec. II describes the general framework to classify stable 2D DSM with PT symmetry. Section III presents a magnetic tight-binding model which supports symmetry-protected 2D DPs, and the symmetry-breaking phases. Section IV presents discussion on the possible material venues for 2D AFM DSMs, and concludes this paper. Some auxiliary materials are relegated to Appendixes.

II. GENERAL THEORY
The basic mechanism for the avoidable crossing of energy bands is to let the bands have different symmetry representations 7,29,35 . We start with classifying stable DSMs of 2D SOC systems in the presence of PT and crystalline symmetry such as rotational and mirror symmetries. The generic Hamiltonian has the form are Pauli matrices acting on the spin and orbital, respectively. σ 0 and τ 0 are 2 × 2 identity matrices. d ij (k) are real functions of k. The invariance of the system will set many terms to be zero or nonindependent, which reduces to the effective Hamiltonian in Eq. (1).
C 2x symmetry. The invariance of the system under π rotation along the x axis requires C 2x H(k x , k y )C −1 2x = H(k x , −k y ). k y = 0 along the k x axis, [C 2x , H(k x , 0)] = 0. Therefore we can choose a basis to make both C 2x and H(k x ) diagonal. In such a basis, the explicit forms of C 2x is obtained by the constraint [T , C 2x ] = 0. There- With the explicit representations of C 2x and P opera- iσ2K Cubic  tors, we can get the symmetry allowed forms of d a (k). Take P = ±τ 1 for example, when C 2x = iσ 3 ⊗ τ 3 , to the leading order in k, the symmetry constraints leads to d 1,2,3,4, where v i are coefficients. This is nothing but the 2D DSM with anisotropic linear dispersions. Considering the full periodic structure of the BZ, the time-reversal-invariant momentum (TRIM) (π, 0), (0, π) and (π, π) are possible locations of the 2D DPs. The (0, 0) point is simply excluded because the symmetry-protected DPs are not compatible with threefold rotation, while the space groups admit the 4D representations at (0, 0) contain where v i , u i and m are constants. This is not DSM Hamiltonian. However, it is worth to mention that critical DPs indeed exist at TRIM at the transition between QSH and NI 16 as shown in Fig. 4(b). At TRIM point, k and −k are equivalent, and all odd functions in H(k) vanishes. By tuning m, d 5 (k, m) = 0 is satisfied.
The classification of 2D DSMs for case A is shown in Table I. The details on 2D DPs protected by Mẑ and C nẑ are discussed in Appendix A. The stable DPs exist at TRIM of the BZ boundary and are protected by the lattice symmetry, including symmorphic and nonsymmorphic symmetries 29 . In nonsymmorphic symmetric systems with two sublattices, P = ±τ 1 is the inversion operator with sublattices interchanged 7 , which is realized by the inversion with respect to a lattice site followed by a fraction translation.
The crystalline symmetries fall into two classes: sym-morphic and nonsymmorphic space group symmetries, the operations of which is denoted as {g|t} and can be constructed by the point group operations g with translation t that are a full and a fraction of a Bravais lattice vector, respectively. In 2D, the representative symmorphic group symmetries are {C 2x |00}, {C 2ŷ |00}, {Mẑ|00} and C nẑ . The typical 2D nonsymmorphic group operations include screw rotation {C 2x |t}, {C 2ŷ |t}, glide mirror lines {Mx|t}, {Mŷ|t}, and glide mirror plane {Mẑ|t}, where t is a half-translation which satisfying gt = t and e iG·t = −1 for odd reciprocal lattice vector G. Explicitly, in units of Bravais lattice constant, t = ( 1 2 , 0), (0, 1 2 ), ( 1 2 , 1 2 ). In the following, we will show that stable 2D DPs can only exist in the presence of nonsymmorphic symmetries.
(i). For the symmorphic group, we take {Mẑ|00} for example, k is invariant and (σ 1 , σ 2 , σ 3 ) → (−σ 1 , −σ 2 , σ 3 ). The symmetry constraints on d a (k) is d 3,4 (k) = 0. To the lowest order in k, the general form is d a (k) = m a +v x a k x + v y a k y for a = 1, 2, 5, and m a , v x a , v y a are coefficients. The DPs are not guaranteed and can only be generated by fune tuning. Even with the extra {C 2x |00} or {C 2ŷ |00} symmetry, it only constrains d 2 (k) = v y 2 k y , still the DPs are not guaranteed.

Symmetry
Half-translation t Dispersion (v 1 k y , v 2 k y , v 3 k y , v 4 k y , v 5 k x ), where v i are real. There is an alternative way to see why topological band crossing can only exist along k y = π line. The two Bloch states |u + k , |u − k and their PT partners PT |u + k , PT |u − k are eigenstates of {C 2x | 1 2 1 2 }, which is ±ie −ikx/2 . |u ± k has the same energy as its PT partner. To have a stable band crossing, (|u + k , PT |u + k ) and (|u − k , PT |u − k ) should carry different representations of {C 2x | 1 2 1 2 } as shown in Fig. 1(b). This is only possible as {C 2x | 1 2 1 2 } = ie −ikx/2 τ 3 along k y = π.
As shown in Table II, the stable DPs can indeed exist in certain AFM semimetals respecting PT symmetry. The DPs in both T -invariant and T -broken semimetals are generated by the avoidable crossing of energy bands with distinct representations, which are protected by nonsymmorphic symmetries. However, different from the Tinvariant semimetals where the DPs reside at TRIMs of the BZ boundary, here the DPs in T -broken systems can only appear at the BZ boundary but not at TRIMs in general. Moreover, the glide mirror plane symmetry Mẑ can not protect the DPs with T -broken as listed in Appendix B, contrary to its protection of DPs with T . More importantly, the DPs in the T -broken AFM DSMs are locally permitted by crystalline symmetries, while the DPs in T -invariant DSMs are essential and cannot be gapped without lowering the specific space group symmetries.

A. Tight-binding model
Now we study a tight-binding model for a 2D AFM DSM to illustrate the nonsymmorphic symmetries protected DPs listed in Table II. For direct comparison with T invariant case, we adopted the lattice similar to Ref. 29 as shown in Fig. 2(a), the system has a layered AFM structure with two atoms in one unit cell, denoted as A and B. The lattice vectors are denoted as a 1 = (1, 0, 0), a 2 = (0, 1, 0). In each unit cell, the A and B atoms are shifted along the z axis with the position r A = (− 1 4 , − 1 4 , − c 2 ) and r B = ( 1 4 , 1 4 , c 2 ). Without AFM ordering, this structure is in space group No. 129 (P 4/nmm). Each lattice site contains a d z 2 orbital, where the square pyramidal crystal field splits d z 2 from other d orbitals. The Hamiltonian is where ij and ij denote the nearest and next-nearest neighbor sites, respectively. λ so is SOC which involves spin-dependent next-nearest neighbor hopping, whered 1 andd 2 are unit vectors along two nearest neighbor bonds hopping from site j to i 37 . s describes the electron spins.  The third term is the staggered Zeeman term (ξ i = ±1) which describes the AFM ordering (alongn direction).

B. Symmetry breaking
Lowering the symmetry by external perturbations provides a toolbox to explore a wealth of topological phases. We start with the system as shown in Fig. 2(d). The {Mx| 1 2 0} breaking by distortion in Fig. 2(c) leads to a NI with gapped DPs. Displacing the B sites along [11] or [11] always lead to NI phases. Quite different from T -, P-invariant 2D DSM where it lies at the boundary between QSH and NI 29 , the 2D AFM DSM lies in the NI phases deeply. This is simply because the edge of a physical system always breaks PT and leads to gapped edge states, even though a nontrivial bulk Z 2 index may be defined. Similarly, the structure inversion asymmetry (SIA) by applying an electric field along z axis naturally breaks PT and adds a term V τ 3 , results in nondegenerate bands where DPs split into Weyl points as shown in Fig. 3(a). Such Weyl points are located along the k x = π line and are not protected by symmetry. More interestingly, if both PT and {Mx| 1 2 0} are broken, a quantum anomalous Hall (QAH) state can be realized 38 and electrically controllable as shown in Fig. 3(b). Furthermore, with the proximity effect to an s-wave superconductor, the realization of a chiral topological superconductor with external tunability is expected [39][40][41][42] .

IV. DISCUSSION AND CONCLUSION
The AFM long range order could exist in 2D. It should be interesting to study how the magnetic fluctuation and Coulomb interaction affect the stability of DPs, which is left to future work. In terms of realistic materials, the actual existence of such phases in known materials remains an open question. However, similar to TI materials, the 2D magnetic DSMs may exist in materials with both strong SOC and specific space group symmetries. Furthermore, we comment on the possible candidate in the AMnBi 2 family of compounds (A = Ca, Sr, Eu, Yb) [43][44][45][46][47][48] . The transport experiments have already shown the 2D Dirac fermion behaviors in the bulk materials 47,48 . Take SrMnBi 2 for example, its bulk structure is in space group No. 139 (I4/mmm). For a monolayer of this material, the AFM order on Mn atoms is along the [001] direction which breaks both T and P whereas PT still holds. The Bi atoms form two layers of square lattice which is similar to the structure in Fig. 2(a), which determine most of the electronic structure [43][44][45]  In summary, we show that the 2D Dirac fermions could exist in AFM semimetals with PT symmetry, where the nonsymmorphic space group symmetries play an essential role. The realistic AFM DSM materials remain unknown. However, considering the ongoing rapid progress in the field of 2D materials 49 , together with the nonsymmorphic symmtries listed in Table I as a guidance, we are thus optimistic about the material search of the 2D AFM DSMs.
Note added: Recently, we became aware of an independent work on a similar problem 50 . However, their approach to 2D magnetic DSMs is different from our results.

ACKNOWLEDGMENTS
The author is grateful to Haijun Zhang, Biao Lian and Shoucheng Zhang for insightful discussions. This work is supported by the National Thousand-Young-Talents Program; the National Key Research Program of China This is different from C 2x and Mẑ cases, where invariant lines or plane exist in the BZ. We find that stable DPs can appear and are protected by C 2ẑ , C 4ẑ , C 6ẑ together with P = ±τ 1 . The classification of 2D DSM protected by C nẑ symmetry in case A is listed in Table. I. From Table I, we can see that the C 4ẑ case is consistent with the results obtained in Ref. 29. The C 2ẑ case is not independent from Mẑ case in the main text, for in the presence of P, C 2ẑ = MẑP.
Below we show the basic steps to see when the stable DPs can appear. As we discussed in the main text, the basic mechanism for the avoidable crossing of energy bands is to let the bands have different symmetry representations. The four Bloch states can be chosen as |u ↑ A , |u ↑ B , |u ↓ A , and |u ↓ B . Here |u ↑ A and |u ↑ B have the C nẑ eigenvalues u ↑ A and u ↑ B , respectively. There are two distinct cases, (i). When P = ±τ 0 , ±τ 3 , the inversion operation will not flip the orbitals, thus the energy E A,↑ = E A,↓ , and E B,↑ = E B,↓ at k TRIM . Therefore, the four Bloch states are grouped into (u ↑ A , u ↓ A ) and (u ↑ B , u ↓ B ). In order to have a stable band crossing, one necessary condition is that these two groups must have different eigenvalues of C nẑ . Namely, (u ↑ A , u ↓ A ) are different from (u ↑ B , u ↓ B ). Due to [C nẑ , H(k TRIM )] = 0, both C nẑ and H(k TRIM ) can be diagonal. The only possible form is H(k TRIM ) = d 5 τ 3 . However, d 5 is even under P, thus d 5 = m, where m is the external parameters. In general, m = 0, and Eq. (A1) can not constrain m to be zero, therefore, the stable Dirac points are not possible in this case.
(ii). Similarly, when P = ±τ 1 , inversion operation will switch the orbitals, therefore E A,↑ = E B,↓ , and E B,↑ = E A,↓ . Thus, (u ↑ A , u ↓ B ) must be different from (u ↑ B , u ↓ A ). This necessary condition rules out C 3ẑ . Moreover, we consider the only possible form of H at k TRIM is H(k TRIM ) = d 1 σ 3 ⊗ τ 3 . However, d 1 (k) is an odd function, thus d 1 vanishes. Furthermore, using Eq. (A1) one can get the possible forms of d a (k). Here we take C 4ẑ for example, from the above symmetry analysis, the only possible form is C 4ẑ = e ±i π 4 σ3 ⊗τ 3 . At the TRIM, all odd functions d 1,2,3,4 vanish, while d 5 (k TRIM ) is an even function. But C 4ẑ also force d 5 (k TRIM ) to be odd. Therefore d 5 vanishes at the TRIM, and DPs can appear the such TRIMs. At the TRIM, the Hamiltonian is expanded as d 1,2,3,4,5 (k) = (0, 0, vk x , vk y , 0). More interestingly, C 6ẑ symmetry could support the cubic Dirac fermions.
Here we point out, in case A with P symmetry, one can always construct the mirror symmetry Mn ⊥ (n ⊥ ⊥ẑ) from C 2n ⊥ and P as Mn ⊥ = C 2n ⊥ P. Heren ⊥ can chosen to bex orŷ. Therefore, Mn ⊥ is not independent from C 2n ⊥ , and is not listed in Table I. However, in case B without P, Mn ⊥ is different from C 2n ⊥ , as we can see from Table I.