Abstract
Topological crystalline insulators (TCIs) can host anomalous surface states which inherits the characteristics of crystalline symmetry that protects the bulk topology. Especially, the diversity of magnetic crystalline symmetries indicates the potential for novel magnetic TCIs with distinct surface characteristics. Here, we propose a topological magnetic Dirac insulator (TMDI), whose twodimensional surface hosts fourfolddegenerate Dirac fermions protected by either the \({p}_{c}^{{\prime} }4mm\) or \(p{4}^{{\prime} }{g}^{{\prime} }m\) magnetic wallpaper group. The bulk topology of TMDIs is protected by diagonal mirror symmetries, which give chiral dispersion of surface Dirac fermions and mirrorprotected hinge modes. We propose candidate materials for TMDIs including Nd_{4}Te_{8}Cl_{4}O_{20} and DyB_{4} based on firstprinciples calculations, and construct a general scheme for searching TMDIs using the space group of paramagnetic parent states. Our theoretical discovery of TMDIs will facilitate future research on magnetic TCIs and illustrate a distinct way to achieve anomalous surface states in magnetic crystals.
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Introduction
The surface states of topological insulators (TIs) have anomalous characteristics that are unachievable in ordinary periodic systems^{1}. A representative example is the twofolddegenerate gapless fermion on the surface of threedimensional (3D) TIs protected by timereversal symmetry (TRS)^{2,3,4,5}. Contrary to the case of ordinary twodimensional (2D) crystals with TRS in which gapless fermions appear in pairs, a single gapless fermion can exist on the surface of TIs through its coupling to the bulk bands. Such a violation of fermion number doubling^{6,7,8} is a representative way in which the anomalous characteristics of surface states are manifested at the boundary of TIs.
In topological crystalline insulators (TCIs)^{9,10}, crystalline symmetries enrich the ways in which anomalous surface states are realized. For example, in systems with rotation symmetry and TRS, variants of the fermion doubling theorem enabled by symmetries can be anomalously violated on the surface of TCIs^{11}. Additionally, in the case of mirrorprotected TCIs^{12}, although the number of surface gapless fermions can be even, the surface band structure exhibits a chiral dispersion along mirrorinvariant lines such that anomalous chiral fermions appear in the onedimensional (1D) mirrorresolved subspace of the 2D surface Brillouin zone (BZ). More recently, studies showed that in crystals with glide mirrors, the anomalous surface states can have an hourglasstype band connection^{13}. Moreover, when the surface preserves two orthogonal glide mirrors, a single fourfolddegenerate Dirac fermion^{14} was shown to be achievable as an anomalous surface state^{15}.
In magnetic crystals, there is great potential to achieve a new type of magnetic TCI with distinct anomalous surface states^{16,17,18,19,20,21,22} because there are abundant magnetic crystalline symmetries described by 63 magnetic wallpaper groups (MWGs) and 1421 magnetic space groups (MSGs)^{23,24}, which are overwhelmingly larger than the 17 wallpaper groups and 230 space groups of nonmagnetic crystals^{25,26,27,28,29}. Very recently, exhaustive studies of magnetic topological phases and their classification have been performed^{16,17,18}, and various novel magnetic topological phases have been systematically categorized. However, as far as we can tell, all the surface states of magnetic TCIs reported up to now appear in the form of twofolddegenerate gapless fermions, whose detailed band connection depends on the surface symmetry.
Here, we propose a magnetic TCI with fourfolddegenerate gapless fermions on the surface, coined the topological magnetic Dirac insulator (TMDI). A fourfolddegenerate gapless fermion, a Dirac fermion for short hereafter, can appear on the surface of a magnetic insulator when the MWG of the surface is one of the three MWGs \(p{4}^{{\prime} }{g}^{{\prime} }m\), \({p}_{c}^{{\prime} }mm\), and \({p}_{c}^{{\prime} }4mm\), among 63 possible MWGs. Contrary to the surface Dirac fermion in nonmagnetic crystals protected by two orthogonal glides, our surface Dirac fermion is protected by symmorphic symmetries combined with either an antiunitary translation symmetry or an antiunitary glide mirror.
In particular, in magnetic crystals whose (001)surface MWG is either \(p{4}^{{\prime} }{g}^{{\prime} }m\) or \({p}_{c}^{{\prime} }4mm\), the bulk topology is characterized by the mirror Chern number (MCN) \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}\) about the diagonal mirror planes normal to either the [110] or \([1\bar{1}0]\) direction. Because of this, in TMDIs, the way in which the surface anomaly is realized is different from the case of the nonmagnetic Dirac insulator^{15} and more similar to the case of mirrorprotected nonmagnetic TCIs^{12}. Namely, along the mirrorinvariant line on the surface BZ, the Dirac fermion develops a chiral dispersion relevant to the MCN. Moreover, the MCN of TMDIs also induces hinge modes at open boundaries along the x and y directions, which respect diagonal mirrors.
Using firstprinciples calculations, we propose candidate materials for TMDIs, including Nd_{4}Te_{8}Cl_{4}O_{20} and DyB_{4}. Since the database for magnetic materials only has a limited number of materials, we construct a systematic way to find candidate magnetic materials for TMDIs using the space group of paramagnetic parent compounds.
Results
Dirac fermions and magnetic wallpaper groups
In 2D magnetic crystals, Dirac fermions with fourfolddegeneracy can be symmetryprotected at the BZ corner, M = (π, π), by three MWGs, i.e., TypeIII MWG \(p{4}^{{\prime} }{g}^{{\prime} }m\) and TypeIV MWGs \({p}_{c}^{{\prime} }mm\) and \({p}_{c}^{{\prime} }4mm\) [see Fig. 1a–c]. Here, we use the notation of Belov and Tarkhova (BT)^{23} for denoting MWGs and the notation of Belov, Neronova, and Smirnova (BNS)^{30} for denoting MSGs. Note that TypeIII MWGs have antiunitary spatial symmetries combining TRS T with spatial symmetries, while TypeIV MWGs have antiunitary translation symmetries combining T and fractional lattice translations. All three MWGs have mirrorinvariant lines, whose normal directions are \(\hat{x}\), \(\hat{y}\), or \(\hat{x}\pm \hat{y}\).
In 2D systems belonging to the TypeIII MWG \(p{4}^{{\prime} }{g}^{{\prime} }m\) described in Fig. 1c, a Dirac fermion is protected at M by twofold rotation about the zaxis C_{2z}, antiunitary glide mirror \(T{G}_{y}\equiv T\{{m}_{y} \frac{1}{2},\frac{1}{2}\}\), and offcentered diagonal mirror \({\widetilde{M}}_{x\overline{y}}=\{{m}_{x\overline{y}} \frac{1}{2},{{{\rm{}}}}\frac{1}{2}\}\). Here, the notation {g∣t} denotes the point group symmetry g followed by a partial lattice translation t. \({m}_{x\overline{y},y}\) are mirror symmetries that act on realspace coordinates as \({m}_{x\overline{y}}:(x,y,z)\to (y,x,z)\) and m_{y}: (x, y, z) → (x, − y, z). [See the conventions in Supplementary Note (SN) 1.] As detailed in SN 3, the fourfold degeneracy is formed by four states ψ_{±}, TG_{y}ψ_{±}, \({\widetilde{M}}_{x\overline{y}}{\psi }_{\pm }\), and \(T{G}_{y}{\widetilde{M}}_{x\overline{y}}{\psi }_{\pm }\), where ψ_{±} is an energy eigenstate with C_{2z} eigenvalue ±i.
In contrast, 2D systems belonging to the TypeIV MWGs \({p}_{c}^{{\prime} }mm\) and \({p}_{c}^{{\prime} }4mm\), described in Fig. 1a, b, respectively, have common symmetry elements, i.e., antiunitary translation \({T}_{G}=\{T \frac{1}{2},\frac{1}{2}\}\) and two mirrors M_{x} = {m_{x}∣0} and M_{y} = {m_{y}∣0}, where m_{x}: (x, y, z) → ( − x, y, z). At M, these symmetry elements anticommute with each other, and \({T}_{G}^{2}=1\). These relations protect the fourfold degeneracy formed by ψ_{±}, T_{G}ψ_{±}, M_{y}ψ_{±}, and T_{G}M_{y}ψ_{±}, where ψ_{±} has M_{x} eigenvalue ± i (see SN3). Note that the same symmetry representation was also studied in ref. ^{31}.
A typical band structure supported by MWG \(p{4}^{{\prime} }{g}^{{\prime} }m\) is shown in Fig. 1d. Since MWG \(p{4}^{{\prime} }{g}^{{\prime} }m\) has diagonal mirror \({\widetilde{M}}_{x\overline{y}}\), the energy bands can be divided into two different mirror eigensectors along the \(\Gamma M{\Gamma }^{{\prime} }\) direction. Here, Γ = (0, 0), and \({\Gamma }^{{\prime} }=(2\pi ,2\pi )\). Focusing on the band structures in each mirror sector, we find that the numbers of upward (chiral) and downward (antichiral) bands crossing the Fermi level [E = 0 in Fig. 1d] at M are the same. Otherwise, the mirrorresolved band structure cannot be periodic along \(\Gamma M{\Gamma }^{{\prime} }\). Hence, a Dirac fermion in 2D crystals belonging to MWG \(p{4}^{{\prime} }{g}^{{\prime} }m\) is nonchiral in each mirror sector. Similar phenomena also occur in MWGs \({p}_{c}^{{\prime} }mm\) and \({p}_{c}^{{\prime} }4mm\). In general, the mirrorresolved dispersion of Dirac fermion in 2D crystals protected by MWGs is nonchiral. Although the local dispersion near the Dirac point may exhibit either chiral or nonchiral dispersion, the full band dispersion along the mirror invariant line is always nonchiral in 2D crystals.
Chiral surface Dirac fermions
A 2D Dirac fermion, which is nonchiral in 2D systems, can be chiral on the surface of 3D magnetic TCIs, as illustrated in Fig. 1e. Here, we systematically search for 3D magnetic insulators that can host a Dirac fermion on the (001) surface. As a 2D fourfolddegenerate Dirac fermion can be protected by one of the three MWGs \(p{4}^{{\prime} }{g}^{{\prime} }m\), \({p}_{c}^{{\prime} }mm\), and \({p}_{c}^{{\prime} }4mm\), we focus on the MSGs whose (001) surface has one of these three MWGs. By studying the MSG symbols and the detailed surface symmetries, we find that there are at least 31 MSGs that can be generated from such MWGs and additional generators compatible with the MWGs. (See Supplementary Table 1.)
All 31 MSGs have mirror planes whose normal vectors are orthogonal to the (001) direction. Thus, the corresponding MCNs can give chiral dispersions along the mirrorinvariant lines on the (001) surface. First, MWG \(p{4}^{{\prime} }{g}^{{\prime} }m\) has offcentered diagonal mirrors^{32}\({\widetilde{M}}_{x\overline{y}}=\{{m}_{x\overline{y}} \frac{1}{2},{{{\rm{}}}}\frac{1}{2}\}\) and \({\widetilde{M}}_{xy}=\{{m}_{xy} \frac{1}{2},\frac{1}{2}\}\) [see Fig. 1c]. For the 11 MSGs relevant to MWG \(p{4}^{{\prime} }{g}^{{\prime} }m\), we define four MCNs \({{{{\mathcal{C}}}}}_{\pm }^{{k}_{x} = {k}_{y}}\) and \({{{{\mathcal{C}}}}}_{\pm }^{{k}_{x} = {k}_{y}}\), which are defined in the k_{x} = k_{y} and k_{x} = − k_{y} planes, respectively. Here, the ± sign denotes the mirror eigenvalues of occupied bands. All the MCNs are equivalent up to sign because of the symmetry relations among \({\widetilde{M}}_{x\overline{y}}\), \({\widetilde{M}}_{xy}\), and TC_{4z}. Hence, the bulk topology can be classified by \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}\equiv {{{{\mathcal{C}}}}}_{+}^{{k}_{x} = {k}_{y}}={{{{\mathcal{C}}}}}_{}^{{k}_{x} = {k}_{y}}\). Similarly, we can define MCNs for the 5 MSGs related to MWG \({p}_{c}^{{\prime} }4mm\), which have four mirrors, M_{x}, M_{y}, \({M}_{x\overline{y}}\), and M_{xy}. Among them, only two MCNs, \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}\) and \({{{{\mathcal{C}}}}}_{m}^{y}\equiv {{{{\mathcal{C}}}}}_{+}^{{k}_{y} = 0}={{{{\mathcal{C}}}}}_{}^{{k}_{y} = 0}\), are independent, and serve as bulk topological invariants. Finally, for the 15 MSGs related to MWG \({p}_{c}^{{\prime} }mm\), the relevant MCNs are \({{{{\mathcal{C}}}}}_{m}^{x}\equiv {{{{\mathcal{C}}}}}_{+}^{{k}_{x} = 0}={{{{\mathcal{C}}}}}_{}^{{k}_{x} = 0}\) and \({{{{\mathcal{C}}}}}_{m}^{y}\equiv {{{{\mathcal{C}}}}}_{+}^{{k}_{y} = 0}={{{{\mathcal{C}}}}}_{}^{{k}_{y} = 0}\). For more detailed discussions on the MCNs, see SN4.
Now, we classify the Wilson loop spectra^{33,34,35,36,37} according to the MCNs. We consider the k_{z}directed Wilson loop \({{{{\mathcal{W}}}}}_{z}({{{{\bf{k}}}}}_{\perp })\),
where \({P}_{{{{\rm{occ}}}}}({{{\bf{k}}}})\equiv \mathop{\sum }\nolimits_{n = 1}^{{n}_{{{{\rm{occ}}}}}}\left\vert {u}_{n}({{{\bf{k}}}})\right\rangle \left\langle {u}_{n}({{{\bf{k}}}})\right\vert\) is a projection operator for occupied bands \(\left\vert {u}_{n}({{{\bf{k}}}})\right\rangle\) and k_{⊥} = (k_{x}, k_{y}). Since the Wilson loop is unitary, its eigenvalue can be collectively denoted as \(\{{e}^{i\theta ({{{{\bf{k}}}}}_{\perp })}\}=\{{e}^{i{\theta }_{a}({{{{\bf{k}}}}}_{\perp })} {\theta }_{a}({{{{\bf{k}}}}}_{\perp })\in (\pi ,\pi ],a=1,\ldots ,{n}_{{{{\rm{occ}}}}}\}\). Then, {θ(k_{⊥})} defines the Wilson loop spectrum, or equivalently, the Wilson bands. Wilson loop spectra and surface band structures have the same spectral features^{35,38}. Thus, the band structure on the (001) surface can be systematically classified based on Wilson loop analysis. (The details on tightbinding notation and Wilson loop is provided in SN2 and SN5.)
First, let us consider the MSGs related to MWG \(p{4}^{{\prime} }{g}^{{\prime} }m\) [see Fig. 2a–e]. At \(\overline{M}=(\pi ,\pi )\), four Wilson bands form a fourfold degeneracy, which can be identified as a Dirac fermion on the (001) surface. The connectivity of Wilson bands is classified by the MCN \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}\), which is encoded in the slope of Wilson bands in each mirrorsector crossing a horizontal reference line [a green dashed line in Fig. 2b] along the \(\overline{\Gamma }\)\(\overline{M}\) direction. In Fig. 2d, for example, as two Wilson bands with mirror eigenvalue − i intersect the reference line with a negative slope, we obtain \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}=2\). See SN5 for the details on the counting rules for \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}\).
Now, we compare the Wilson loop spectra of topological phases with nonzero \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}\) and the trivial phase with zero \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}\) by focusing on the region near the fourfold degeneracy at \(\overline{M}\). Along the \(\overline{\Gamma }\)\(\overline{M}\)\(\overline{{\Gamma }^{{\prime} }}\) line, the four bands are divided into two different mirror sectors. When \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}=0\), as in Fig. 2b, the dispersion in each mirror sector is nonchiral, similar to that of Dirac fermions in 2D crystals in Fig. 1d. In Fig. 2b, chiral and antichiral modes in the same mirror sector (e.g., the +i sector) cross the green dashed reference line with opposite signs of the group velocity. In contrast, their numbers are not equal in Fig. 2d, where \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}=2\).
The Dirac fermions in Fig. 2c, e, which correspond to \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}=1\) and 4, respectively, appear with additional surface states (black arrows). When \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}=1\), the dispersion is chiral along the entire \(\overline{\Gamma }\)\(\overline{M}\)\(\overline{{\Gamma }^{{\prime} }}\) line but locally looks nonchiral near the Dirac point. However, if the dispersion along \(\overline{\Gamma }\)\(\overline{M}\)\(\overline{{\Gamma }^{{\prime} }}\) is deformed such that the additional surface states near \(\overline{\Gamma }\) are pushed away from the Fermi level (which corresponds to θ = 0 in Wilson loop spectra), the Dirac fermion in Fig. 2c becomes chiral, as in Fig. 2h. In contrast, such a deformation is impossible in Fig. 2e, where \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}=4\). In general, one can show that a nonzero MCN \( {{{{\mathcal{C}}}}}_{m}^{x\overline{y}} \le 2\) manifests as a chiral dispersion of the surface Dirac fermion along \(\overline{\Gamma }\)\(\overline{M}\)\(\overline{{\Gamma }^{{\prime} }}\), provided that there is no additional surface state other than the Dirac fermion at the Fermi level. In contrast, when \( {{{{\mathcal{C}}}}}_{m}^{x\overline{y}} \,>\, 2\), additional surface states always appear along \(\overline{\Gamma }\)\(\overline{M}\)\(\overline{{\Gamma }^{{\prime} }}\). Hence, the chiral dispersion of the Dirac fermion when \( {{{{\mathcal{C}}}}}_{m}^{x\overline{y}} \le 2\) and the coexistence of additional surface states when \( {{{{\mathcal{C}}}}}_{m}^{x\overline{y}} \,>\, 2\) are signatures of the nontrivial bulk topology of 3D TMDIs with nonzero \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}\). An exact formulation of the relation among the chiral dispersion of the Dirac fermion, MCN \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}\), and number of additional surface states is given in SN6.
Similarly, one can analyze the Wilson loop spectra of the MSGs related to MWG \({p}_{c}^{{\prime} }4mm\) [see Fig. 2f–j]. The MCN \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}\) (\({{{{\mathcal{C}}}}}_{m}^{y}\)) can be determined by examining the \({M}_{x\overline{y}}\) (M_{y}) eigenvalues and the slopes of Wilson bands crossing a reference line along \(\overline{\Gamma }\)\(\overline{M}\) (\(\overline{\Gamma }\)\(\overline{X}\)). The relation between \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}\) and the chiral dispersion of the Dirac fermion is identical to the case of the MSGs with MWG \(p{4}^{{\prime} }{g}^{{\prime} }m\). The only additional feature is that \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}\) and \({{{{\mathcal{C}}}}}_{m}^{y}\) must be equivalent up to modulo 2, i.e., \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}={{{{\mathcal{C}}}}}_{m}^{y}\) (mod 2), in insulating phases.
Finally, in the 15 MSGs related to MWG \({p}_{c}^{{\prime} }mm\), the Wilson loop spectra can be classified by the MCNs on the k_{x,y} = 0 planes, \({{{{\mathcal{C}}}}}_{m}^{x}(0)\) and \({{{{\mathcal{C}}}}}_{m}^{y}(0)\), related to M_{x,y} mirrors. The MCNs on the k_{x,y} = π planes, \({{{{\mathcal{C}}}}}_{m}^{x}(\pi )\) and \({{{{\mathcal{C}}}}}_{m}^{y}(\pi )\), are always trivial, while surface Dirac fermions can be chiral only for nonzero \({{{{\mathcal{C}}}}}_{m}^{x,y}(\pi )\). Thus, the surface Dirac fermion in the MSGs with MWG \({p}_{c}^{{\prime} }mm\) is nonchiral and trivial.
Topological magnetic Dirac insulators
According to the Wilson loop analysis, the bulk band topology of the 11 MSGs with TypeIII MWG \(p{4}^{{\prime} }{g}^{{\prime} }m\) and the 5 MSGs with TypeIV MWG \({p}_{c}^{{\prime} }4mm\) can be characterized by \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}\) and \(({{{{\mathcal{C}}}}}_{m}^{x\overline{y}},{{{{\mathcal{C}}}}}_{m}^{y})\), respectively. In these 16 MSGs, when \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}\,\ne\, 0\), a Dirac fermion whose mirrorresolved dispersion is chiral can appear on the (001) surface. Based on this, we define TMDIs as 3D magnetic TCIs with nonzero \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}\) hosting a 2D chiral Dirac fermion on the (001) surface. Additionally, according to the (001)surface MWG, TMDIs can be divided into TypeIII and TypeIV TMDIs such that TypeIII (TypeIV) TMDIs have (001)surface MWG \(p{4}^{{\prime} }{g}^{{\prime} }m\) (\({p}_{c}^{{\prime} }4mm\)).
Interestingly, TMDIs also exhibit higherorder topology^{11,29,39,40,41,42,43,44,45,46} with hinge modes at open boundaries along the x and y directions when the entire finitesize systems respect the diagonal mirror symmetries. The MCN for the diagonal mirror \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}\) follows the higherorder bulkboundary correspondence^{41,42,45}. The dispersion of hinge modes can be both chiral and helical depending on the details of the systems. Note that the number of chiral and antichiral hinge modes at each hinge can be changed by a mirrorsymmetric and bulkgappreserving perturbation because such a perturbation can close and reopen a surface gap^{42,44,47}. However, the MCN \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}\) protects at least \( {{{{\mathcal{C}}}}}_{m}^{x\overline{y}}\) hinge modes at each mirrorinvariant hinge. (See SN7 for more details.) We provide a tightbinding model for a TypeIII TMDI in SN10, which confirms the bulkboundary correspondence described above.
Candidate materials
Using firstprinciples calculations, we propose DyB_{4} and Nd_{4}Te_{8}Cl_{4}O_{20} as candidate materials for a TypeIII TMDI and a TypeIV TMDI, respectively, whose electronic and topological properties are summarized in Figs. 3 and 4, respectively. Although their band structure are metallic, as the systems have nonzero direct gap at all momenta, their mirror Chern numbers are well defined. We note that the two candidate materials, DyB_{4} and Nd_{4}Te_{8}Cl_{4}O_{20}, are not available in the existing material databases^{20,48,49,50,51,52,53,54,55}. In general, searching for candidate magnetic materials is more challenging than searching for nonmagnetic materials because the number of available materials in magnetic material databases^{20,54} is limited to approximately 1600^{54}, which is much smaller than the number of nonmagnetic materials. To overcome this limitation, we will also propose a general scheme to systematically search for candidate materials for TMDIs, whose magnetic structures and MSGs are derived from their parent paramagnetic states.
Let us first consider DyB_{4}, a member of the rareearth tetraborides family, whose crystal structure is shown in Fig. 3a. The paramagnetic parent phase of DyB_{4} has space group 127 P4/mbm. In experiments^{56}, this material was reported to have two competing spin configurations that correspond to the Γ_{2} (with MSG 127.395 \(P4/{m}^{{\prime} }{b}^{{\prime} }{m}^{{\prime} }\)) and Γ_{4} (with MSG 127.392 \(P{4}^{{\prime} }/{m}^{{\prime} }{b}^{{\prime} }m\)) magnetic states, as shown in Fig. 3b, c. Between them, the Γ_{2} state was reported to be more favored^{56}. Note that the Γ_{4} phase can support a TypeIII TMDI with (001)surface MWG \(P{4}^{{\prime} }{g}^{{\prime} }m\). Here, we investigate the conditions in which the Γ_{4} state becomes the magnetic ground state in DyB_{4} based on DFT+U(where U is the onsite Coulomb interaction) calculations. We examine the total energies of the Γ_{2} and Γ_{4} states as a function of U with fixed Hund’s coupling J = 1 eV. As shown in Fig. 3d, DyB_{4} undergoes a magnetic phase transition from the Γ_{4} to Γ_{2} state when U > 6.5 eV.
An indirect gap near the Fermi level E_{F}=0.0 eV exists in the DFT+ U result (U = 6.5 eV, J=1.0 eV), as shown in Fig. 3e. Because the gap along the ΓZ line is tiny, as shown in Fig. 3f, a topological phase transition between phases with different MCNs \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}\) can be easily induced by a small perturbation. (See SN9 for the band structures between U=6.0 eV and 6.5 eV and SN12 for a detailed analysis of this topological phase transition.) Fig. 3g, h show the k_{z}directed Wilson loop spectra for U = 6.5 eV and 6.0 eV, respectively. Because of gap closing and reopening at Z, a topological phase transition from the \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}=1\) state to the \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}=2\) state occurs as U increases from 6.0 eV to 6.5 eV. Hence, we identify DyB_{4} at U = 6.5 eV (6.0 eV) as a TypeIII TMDI with \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}=2\) ( − 1).
More specifically, let us consider DyB_{4} at U = 6.5 eV. Figure 3i–j show the surface Green’s function calculations for the top (Bterminated) and bottom (Dyterminated) surfaces. A surface Dirac fermion with fourfold degeneracy is clearly identified near E = − 0.35 eV on the bottom surface [Fig. 3j, l]. In contrast, the top surface state is buried in the bulk conduction band and thus not visible in the gap. However, the hidden surface state can be brought into the gap by applying a surface potential E_{surf} = − 0.2 eV, as shown in Fig. 3k.
The second material Nd_{4}Te_{8}Cl_{4}O_{20} is a candidate for a TypeIV TMDI with \(({{{{\mathcal{C}}}}}_{m}^{x\overline{y}},{{{{\mathcal{C}}}}}_{m}^{y})=(1,1)\). Since its magnetic properties are unknown, this material is found by using our general scheme to search for candidate materials, as briefly explained below. MSG 129.421 P_{C}4/nmm (123.16.1014 \({P}_{P}4/{m}^{{\prime} }mm\) in the OG setting) is one of the MSGs that can host TypeIV TMDIs, which is derived from nonmagnetic space group (SG) 123 P4/mmm. The crystal structure of NdTe_{2}ClO_{5}^{57} in the paramagnetic phase has SG 123 P4/mmm, as shown in Fig. 4a. Introducing a 2 × 2 × 1 supercell, Nd atoms can form a spin configuration compatible with MSG 129.421 P_{C}4/nmm, as shown in Fig. 4b, c. Assuming the magnetic ordering corresponding to MSG 129.421 P_{C}4/nmm, we study the electronic and topological properties of Nd_{4}Te_{8}Cl_{4}O_{20}. The bulk band structure in Fig. 4d shows a semimetallic state with an indirect gap between − 0.5 eV and 0.0 eV. For the bands below this gap, we compute the k_{z}directed Wilson loop spectrum, which exhibits a nonzero MCN, \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}=1\), as shown in Fig. 4g. Hence, we identify Nd_{4}Te_{8}Cl_{4}O_{20} as a candidate for a TypeIV TMDI with (001)surface MWG \({p}_{c}^{{\prime} }4mm\).
The surface Green’s function calculations for the top (Oterminated) and bottom (Ndterminated) surfaces are shown in Fig. 4e, f. Within a gap with a magnitude of 1.5 eV at \(\overline{M}\), a surface Dirac fermion with fourfold degeneracy is clearly identified near E = 1 eV on the top surface. As shown in Fig. 4h, the four states of the surface Dirac fermion are split into two nondegenerate bands and accidentally nearly degenerate bands, in a similar way as in Fig. 2h. Although bottom surface states buried in bulk states are not visible in the gap [Fig. 4f], the hidden surface state can be pushed into the gap by applying a surface potential of − 0.544 eV, as shown in Fig. 4i.
Our material search scheme used to find Nd_{4}Te_{8}Cl_{4}O_{20} can generally be applied to all 16 MSGs (11 TypeIII and 5 TypeIV MSGs) that can host TMDIs, as follows. For each MSG, (i) determine the SG of paramagnetic phases, (ii) obtain the transformation matrix of basis vectors between the magnetic unit cell and crystal unit cell, and (iii) build up the magnetic unit cell from the paramagnetic unit cell according to the transformation matrix as well as restrict the spin configuration of magnetic atoms. (See SN8 for details.) Additionally, using this material search scheme, we studied other candidate materials including two TMDI candidates with nonzero \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}\) (Ce_{2}Pd_{2}Pb, Ce_{2}Ge_{2}Mg) and four materials with \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}=0\) (HoB_{4}, FeSe, FeTe, AlGeMn, N_{3}TaTh). (See SN9 for details.)
Discussion
Let us discuss the symmetrybreaking effect on the 2D Dirac fermion, either in 2D bulk crystals or on the surface of 3D TMDIs. When a perturbation that preserves diagonal mirrors in MWGs \(p{4}^{{\prime} }{g}^{{\prime} }m\) and \({p}_{c}^{{\prime} }4mm\) is applied, the Dirac fermion in 2D crystals becomes gapped while that on the surface of 3D TMDIs remains gapless because of the nonzero MCN. In contrast, a mirrorbreaking perturbation can induce various phase transitions in both cases because a 2D Dirac fermion corresponds to a multicritical point in magnetic systems^{31}. For example, MWG \(p{4}^{{\prime} }{g}^{{\prime} }m\) is reduced to \(p2{g}^{{\prime} }{g}^{{\prime} }\) when \({\widetilde{M}}_{x\overline{y}}\) and TC_{4z} are broken by tensile strain (see SN13). In this case, a mass term is allowed; thus, the Dirac fermion can be gapped. The Chern number of the resulting gapped phase changes by 2 when the sign of the mass is reversed. This, in turn, indicates the appearance of chiral edge channels at domain walls between two gapped domains with opposite signs of the mass.
Finally, we discuss the bulk topological responses of TMDIs. Since mirror symmetry reverses the spatial orientation, it quantizes the axion angle θ to 0 or π, which is related to the MCN by \(\theta /\pi ={{{{\mathcal{C}}}}}_{m}^{x\overline{y}}\) (mod 2)^{18,58}. Hence, TMDIs with odd \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}\) correspond to an axion insulator exhibiting quantized magnetoelectric effects^{59}. In contrast, TMDIs with an even \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}\) have a vanishing axion angle. However, according to a recent theoretical proposal^{60}, even helical higherorder topological insulators (HOTIs) with θ = 0 can exhibit spinresolved magnetoelectric effects. In the case of 3D TMDIs, spinresolved bands are illdefined because of noncollinear magnetic orderings inherent in TypeIII and TypeIV MSGs. However, by combining the spin and sublattice degrees of freedom, the pseudospinresolved magnetoelectric effect can be defined in 3D TMDIs with an even \({{{{\mathcal{C}}}}}_{m}^{x\overline{y}}\), which is demonstrated using a tightbinding model in SN11. A more systematic study of bulk topological responses in magnetic TCIs with noncollinear magnetic ordering is an important subject for future research.
Methods
For the ab initio calculation, the Vienna Ab initio Simulation Package (VASP) is employed with the projector augmentedwave method (PAW)^{61}. We employ the generalized gradient approximation (PBEGGA) for exchangecorrelation potential^{62}. The default VASP potentials, an energy cutoff with 400 eV, and a 8 × 8 × 14 Monkhorstpack kpoint mesh are used. The spin–orbit coupling is considered because of the presence of the heavy rareearth atoms in the unit cell. Onsite Coulomb interaction is taken into account with U = 6.5 eV and J (Hund’s coupling) = 1 eV in DyB_{4} and U = 6 eV and J = 0 eV in NdTe_{2}ClO_{5}. Wannier Hamiltonians were constructed by WANNIER90^{63} and symmetrized by WannSymm code^{64}. The WannierTools package^{65} was used to produce the slab band structure and Wilson loop spectra based on the symmetrized Hamiltonian. The experimental crystal structure of DyB_{4}^{56} are used without structural relaxation. Details on the abinitio calculation can be found in SN S9.
Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
Code availability
The numerical codes used in this paper are available from the corresponding authors upon reasonable request.
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Acknowledgements
We thank Benjamin Wieder and Aris Alexandradinata for fruitful discussions. This work was supported by the Institute for Basic Science in Korea (Grant No. IBSR009D1), Samsung Science and Technology Foundation under Project Number SSTFBA200206, and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2021R1A2C4002773 and No. NRF2021R1A5A1032996).
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Y.H., J.K., and J.L. performed the model and symmetry analyses. Y.Q., D.R., and H.C.C. performed the ab initio calculations. B.J.Y. supervised the project. Y.H., Y.Q., H.C.C., and B.J.Y. analyzed the results and wrote the manuscript. All authors contributed to the discussion and manuscript revision.
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Hwang, Y., Qian, Y., Kang, J. et al. Magnetic wallpaper Dirac fermions and topological magnetic Dirac insulators. npj Comput Mater 9, 65 (2023). https://doi.org/10.1038/s41524023010183
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DOI: https://doi.org/10.1038/s41524023010183
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