Antiferromagnetic topological insulator with selectively gapped Dirac cones

Antiferromagnetic (AF) topological materials offer a fertile ground to explore a variety of quantum phenomena such as axion magnetoelectric dynamics and chiral Majorana fermions. To realize such intriguing states, it is essential to establish a direct link between electronic states and topology in the AF phase, whereas this has been challenging because of the lack of a suitable materials platform. Here we report the experimental realization of the AF topological-insulator phase in NdBi. By using micro-focused angle-resolved photoemission spectroscopy, we discovered contrasting surface electronic states for two types of AF domains; the surface having the out-of-plane component in the AF-ordering vector displays Dirac-cone states with a gigantic energy gap, whereas the surface parallel to the AF-ordering vector hosts gapless Dirac states despite the time-reversal-symmetry breaking. The present results establish an essential role of combined symmetry to protect massless Dirac fermions under the presence of AF order and widen opportunities to realize exotic phenomena utilizing AF topological materials.


Introduction
To realize exotic quantum states in the topological insulator (TI), it is often necessary to break the time-reversal symmetry by introducing ferromagnetism into the crystal [1][2][3][4] , as highlighted by the observation of quantum anomalous Hall effect [5][6][7] .As a natural extension, antiferromagnetic (AF) TIs are recently attracting particular attention because it is expected to show exotic properties 1,[8][9][10][11][12][13][14][15][16][17] , such as quantized magneto-electric effect accompanied by surface Hall conductivity 1,8,18,19 , gigantic magneto-optical responses by AF fluctuations 9,20 , and dynamic axion field useful for detecting dark-matter axions 21 .Also, AF TI is a useful platform applicable to spintronic devices owing to the ultrafast spin response and zero stray magnetic field 22,23 .To provide a pathway toward realizing exotic properties associated with antiferromagnetism and topology, it is essential to establish a new AF topological material, in particular, AF TI.
Prediction of AF TI was first made by Mong et al. in 2010 in their tight-binding model for the NaCl lattice with type-I anfiferromagnetism 2 .While the time-reversal symmetry (Θ) is broken in the AF phase, the combined symmetry (S = ΘTD), where TD represents the translation by the D vector that inverts the spin direction (Fig. 1a), is preserved 8,12,13 .Materials having this S symmetry are characterized by the Z2 topological invariant as in the case of time-reversal-invariant TIs 8,12,13 and have been predicted to show a weak-TIlike behavior wherein the Dirac-cone SS is protected for the crystal plane parallel to the D vector, otherwise gapped.Thus, the AF TI has a unique experimental advantage distinct from strong TIs and ferromagnetic TIs; namely, the controllability of Dirac-cone SS through the manipulation of the D vector.This advantage manifests as intriguing characteristics of surface Dirac fermions; namely, the Dirac mass is strongly anisotropic and depends on the configuration of the AF structure, distinct from so-far established strong 3D TIs and topological semimetals.The protection by the S symmetry in AF TI is a key ingredient to realize exotic properties [8][9][10][11]15,19,24 . It is ths essential to spectroscopically establish AF TI by distinguishing surfaces protecting/breaking S symmetry.
As a candidate of AF TIs protected by S symmetry, some materials such as MnBi2nTe3n+1 (MBT), EuIn2As2, and EuCd2As2 have been theoretically predicted 14,15,25- 27 .The transport property that supports the topological nature of AF TIs such as the quantum anomalous Hall effect in MBT (n = 1) has been intensively investigated 8,14,28 , and various attempts to examine the associated magnetic gap of the Dirac-cone state in magnetic TI candidates have been made 14,[29][30][31][32][33] .However, these materials have a layered structure and spectroscopies can access only a single surface parallel to the layer, making it difficult to obtain deep insights into the relationship between the AF order and topological SS.Also, in the case of MBT, there exists a fierce controversy on the magnitude and gapless/gapful nature of the Dirac-cone SS, which is not settled at the moment [34][35][36][37][38][39][40] .In this regard, the rare-earth monopnictide NdBi that exhibits type-I AF order (Fig. 1a) below Néel temperature of TN = 24 K is an excellent platform from the spectroscopic viewpoint, because (i) the cubic crystal structure is suited to access surfaces with different AF domains 41 and (ii) the large magnetic moment (3μB) and large chemical ratio (50 %) of Nd ions may lead to a large magnetic Dirac gap.This provides us a precious opportunity to investigate the topological property unique to the AF TI, although the bulk semimetallic nature of monopnictides may hinder the quantum transport expected for AF TIs.A recent angle-resolved photoemission spectroscopy (ARPES) study on NdBi reported an unusual SS in the AF phase 42 , making this material more interesting to study the entanglement between the AF order and electronic states (note that such SS was recently proposed to be a topologically trivial state and unlikely to be directly related to the AF TI properties 43 ).Here, by utilizing the AF-domain-selective micro-focused ARPES, we established the electronic states of both S-preserving and Sbroken surfaces with different Dirac-electron characteristics in NdBi.

Results and Discussion
Z2 topology in the paramagnetic phase.First, we present the overall band structure of NdBi in the paramagnetic (PM) phase.Rare-earth monopnictide (RXp) is a semimetal characterized by two hole pockets at the Γ point and electron pockets at the X point in the face-centered cubic (fcc) Brillouin zone (BZ) (Fig. 1b; refs. 44,45).Bulksensitive ARPES with soft-X-ray photons together with first-principles band-structure calculations signifies bulk bands forming these pockets (Fig. 1c, d) together with the signature of bulk band inversion (for details, see Supplementary Note 1 and Supplementary Note 2).The Fermi-surface (FS) mapping in Fig. 1e obtained with vacuum ultraviolet (VUV) photons reveals corresponding features in the surface BZ.At the Γ __ point, we found inner square-like (h1) and outer diamond-like (h2) pockets which originate from the Bi-6p hole bands (Fig. 1d).In Fig. 1e, one can also recognize vertically (e2) and horizontally (e1) elongated pockets at the M __ point originating from Nd-5d bands at X points of different kz's (kz = 0 and 2π/a, respectively; Fig. 1b) due to the strong kz broadening of VUV photons [45][46][47] .The utilization of surface-sensitive VUV photons enables us to visualize the topological SS.One can identify in Fig. 1f a Dirac-cone band (called D1) [47][48][49][50] around the Γ __ point.49]51 ), called here D2 and D3, respectively.All these D1-D3 bands are confirmed to be of surface origin from their hv-independent energy position (for details, see Supplementary Note 3).The difference in the number of Dirac cones at two different time-reversal-invariant momenta (TRIM) at the surface corresponds to the difference in the number of band inversion in the bulk, namely, a single bulk X point is projected onto the Γ __ point whereas two inequivalent bulk X points onto the M __ point (see Fig. 1b).Odd numbers of Dirac cones in total suggest that NdBi in the PM phase is a Z2 TI with a negative band gap (Fig. 1i), consistent with the parity analysis of the bulk-band structure which suggests the strong TI nature with (v0; v1, v2, v3) = (1; 0, 0, 0); for details, see Supplementary Table S1.Our surface-projection calculations for the PM phase also reproduce the D1-D3 Dirac-cone states (Fig. 1h), consistent with the band picture obtained from the experiment (Fig. 1i; for details, see Supplementary Note 2).
AF-induced reconstruction of the Dirac-cone SS.Next, we show how the AF order influences the topological nature of NdBi.As summarized in Fig. 2a, the FS topology in the PM phase projected onto the surface BZ is characterized by the h1 and h2 hole pockets at Γ __ , the D1 SS, the elongated e1 and e2 pockets at M __ , and the D2 SS.As shown by the FS mapping around the Γ __ point at T = 5 K in Fig. 2b, all the pockets observed in the PM phase (h2, h1, and D1) are also resolved in the AF phase.A comparison of the band dispersion across the Γ __ point between the PM (Fig. 2c; T = 30 K) and AF phases (Fig. 2d; T = 5 K) shows that the D1 band is kept observed at both temperatures, indicating that the bulk-band inversion is preserved in the AF phase; however, a careful look at the D1 band reveals that the V-shaped upper branch (D1U) moves upward in the AF phase relative to that in the PM phase, while the Λ-shaped lower branch (D1L) moves downward, resulting in the opening of a Dirac gap associated with time-reversal-symmetry breaking due to the AF order.This gap is confirmed in more careful analyses of the energy distribution curve (EDC) at the Γ __ point (Fig. 2e) which signifies a double peak at T = 5 K as opposed to a single peak at T = 30 K, and its AF origin is also supported by the detailed temperature-dependent ARPES-intensity variation at the Γ __ point (Fig. 2f; for details, see Supplementary Note 4 and Supplementary Fig. 5).The magnitude of the Dirac gap at Γ __ estimated from the EDC is 125 ± 5 meV.This value is unexpectedly large despite the zero net magnetization and low TN (24 K) in NdBi and may be related to the effectively large local exchange field although the exact mechanism is unclear at the moment (for more detailed discussion, see Supplementary Note 4).Such a sizable Dirac gap that is directly linked to the AF order is in sharp contrast to that of an intrinsic magnetic TI candidate MBT where the Dirac gap is elusive 14,[29][30][31][32][33][34][35][36][37][38][39][40] .Now we turn our attention to the AF-induced reconstruction of the other Dirac-cone bands located at the M __ point.Although the spectral features are complicated by the presence of two Dirac-cone bands D2 and D3 in the PM phase (Fig. 2h), these bands still survive in the AF phase (Fig 2i), as also seen in the FS mapping in the AF phase in Fig. 2g, where a small circular pocket originating from the upper branch of the D2 band (D2U) is observed inside the elongated bulk electron pockets, e1 and e2.The gapless X-shaped dispersion of the D2 band in the PM phase (Fig. 2h) turns into the upper (D2U) and lower (D2L) bands separated by a Dirac gap in the AF phase (Fig. 2i).This is also evident from the EDCs at the M __ point in Fig. 2j where a peak located at EB = 0.19 eV at T = 40 K splits into two peaks at 0.12 and 0.28 eV at T = 5 K, exhibiting a Dirac gap of 160 ± 5 meV.
The Dirac gap opening for the D2 and D3 bands in the AF phase is also supported by our slab calculations for the AF phase (for details, see Supplementary Fig. 2).
To validate the AF origin of the observed spectral change, we have performed temperature-dependent ARPES measurements across TN.The ARPES intensity at the M __ point plotted against temperature in Fig. 2k signifies no discernible change in the intensity profile above TN, as evident from the T-invariant EB position of the D2 band.On the other hand, the D2 band splits into two bands as soon as the sample is cooled down below TN.On lowering the temperature, the splitting is gradually enhanced and almost saturated below T = 15 K.This unambiguously demonstrates the AF origin of the Dirac gap.We note that the splitting of the D3 band was not clearly observed because of its weak intensity; the gap is likely much smaller than that of the D2 band (< 40 meV).This trend is also recognized in the slab calculation shown in Supplementary Fig. 2. Domain-selective electronic states in the AF phase.Since NdBi crystal is expected to inherently contain multiple AF domains at the surface without magnetic field owing to the cubic structure (Fig. 1a), we surveyed the band structure in the AF phase by scanning a micro-focused beam on the surface and found that there exists another domain (called domain B) that exhibits a spectral feature markedly different from that discussed above (called domain A).As shown in Fig. 3a, on domain A, the bulk h1 and h2 bands outside the D1 band smoothly disperse toward EF without anomalies, resembling the band dispersion of the PM phase (Fig. 1f).On the other hand, the energy bands of domain B look significantly reconstructed (Fig. 3b).While the h1 band shows a similar dispersion for two domains, the broad feature originating from the h2 band seen in domain A (Fig. 3a) turns into a couple of sharp features (S1 and S2; white arrows) crossing EF with a shallower dispersion.The S1 and S2 bands were assigned to the magnetically split Fermiarc SS 42 .
The definitive domain-dependent nature also shows up in the Dirac-cone SS, D1.Although domain A hosts a Dirac gap of 125 meV (Figs.3a and 2d), an X-shaped band with no signature for a Dirac gap is observed in domain B (Fig. 3b).A critical difference is also visualized by a comparison of the EDC at T = 5 K in Fig. 3c which signifies a double peak for domain A in contrast to a single peak for domain B. This suggests symmetry protection of the Dirac-cone SS (D1) in domain B despite the time-reversalsymmetry breaking.In contrast to the D1 band, the D2 band at the M __ point gapped out by 160 meV for domain A (Fig. 3d) still shows the band separation (D2U and D2L) by 80 meV for domain B (Fig. 3e).Persistence of the gap of the D2 band is also visualized by the EDCs (Fig. 3f) which signify the double-peaked structure (D2U and D2L) for both domains (note that the D3 peak at EB ~ 0.4 eV is sharper for domain B than domain A, indicative of a change in the gap magnitude, although its quantitative estimation is difficult).According to the Z2 classification, when the number of Dirac-cone SS is odd on a given surface, the Dirac cone is topologically protected, whereas it is not protected for the even-number case 8,12,13 .In the present case for domain B, the single Dirac-cone SS (D1) is protected, whereas the other two Dirac-cone SSs (D2 and D3) are not.Thus, the residual gap opening for the D2 band may not violate the Z2 topological protection, but indicates its topologically more fragile nature.This point needs to be further examined, as detailed in Supplementary Note 2.
We systematically scanned micro-focused VUV photons on the cleaved surface (Fig. 3g), and probed the local band structure as a function of real-space position (x, y) in the AF phase, as highlighted by the mapping of photoelectron intensity around the Γ __ point (Fig. 3h).As a result, the ARPES spectra in the AF phase were categorized into either domain A, domain B, or other indistinguishable region, as indicated by different colorings in Fig. 3i.The distinction of domains was made by looking at the local band dispersion together with specifying the aforementioned anomalies in the h2 and D1 bands, as exemplified in Fig. 3j.We found that the ARPES spectrum in the PM phase shows no meaningful (x, y) dependence, indicative of the disappearance of such domains, confirming the AF origin of domains A and B. Such two types of domains have not been resolved in the previous ARPES study of NdBi 42 , while some micro-ARPES studies have been already applied to other rare-earth pnictides such as CeSb 52,53 .The existence of multiple AF domains in NdBi was also confirmed by our polarizing microscope measurements, as detailed in Supplementary Note 5 and Supplementary Fig. 6.

S-symmetry protection of the Dirac-cone SS.
To obtain further insights into the origin of intriguing domain-dependent electronic structure, we directly compare the FS topology between domains A and B as shown in Fig. 4a-d.By a side-by-side comparison of the FS around the M __ point (Fig. 4b, d), one can recognize that there exist small pockets at both corners of the horizontally elongated electron pocket, e1, only for domain B (white arrows).In fact, the ARPES intensity along a k cut crossing this pocket (red lines in Fig. 4b, d) signifies an additional shallow electron band that likely arises from the SS (S4) 42,43 besides the e1 band (inset to Fig. 4d) whereas it is completely absent for domain A (inset to Fig. 4b).Around the Γ __ point, while domain A shows a normal FS image (Fig. 4a) similar to the case of PM phase (Fig. 1e), domain B obviously has an anomaly at both corners of the h2 pocket (Fig. 4c); a small pocket (S2) that resembles the pocket of the M __ -centered FS (Fig. 4d) seems to appear.
The observed critical differences in the FS topology between the two domains are explained by taking into account the existence of two types of AF domains at the same surface and the relevant band reconstruction, as highlighted in Fig. 4e-h.Our data for domain A are consistent with the magnetic structure in which the AF stacking occurs along the out-of-plane direction and the magnetic moment of Nd ions at the topmost surface layer aligns ferromagnetically (Fig. 4g).Since the observed bulk-band structure is significantly broadened along kz already in the PM phase, it does not show an effective change even when entering into the AF phase because the band folding occurs along the kz direction (Fig. 4e).The C4 symmetric pattern in Fig. 4a also supports the AF-ordering along kz.On the other hand, our data for domain B are consistent with the magnetic structure in which the AF stacking occurs along the in-plane direction (e.g.along the xaxis), corresponding to the antiparallel configuration of the magnetic moment at the topmost layer (Fig. 4h).In this case, the band folding occurs with respect to the magnetic BZ boundary at halfway between the Γ __ and M __ points (kx = π/a; Fig. 4f).Consequently, the e1 pocket around M __ is folded to Γ __ , and the h1 and h2 pockets around Γ __ to M __ (note that the intensity of the folded bulk bands is weak).Because the magnetic BZ has the C2 symmetry along the kx axis, small pockets associated with the AF-induced SS (S1-S4) appear only at the horizontal side of the original bulk pocket, reflecting the magnetic BZ boundary (see Supplementary Note 6 for details).We have confirmed that the C2 symmetric electronic structure is not associated with the matrix-element effect of photoelectron intensity, as detailed in Supplementary Note 7. Our argument is further corroborated by the domain-selective ARPES measurements on a cousin material NdSb 54,55 which signified the existence of all three types of AF domains (out-of-plane, in-plane horizontal/vertical) together with the non-topological (non Z2) nature of the S1-S4 pockets.This conclusion is different from that of the recent study which suggests the C4 symmetric FS 42 , probably because of the better spatial resolution of the present study.
The configuration of the AF order plays a crucial role in the massive vs massless characteristics of the Dirac-cone SS.When the translation vector (D) lies on the surface (domain B; Fig. 4h), the combined symmetry S = ΘTD is expected to be preserved and the Dirac cone is protected 8,12,13 .As we demonstrated in Fig. 3a-f, although the timereversal symmetry (Θ) is broken in the AF phase, the D1 band for domain B is still protected by the S symmetry and maintains the massless character (Fig. 4j), in contrast to the massive character for domain A whose surface breaks the S symmetry (Fig. 4i).Such distinction, which owes to the high spatial resolution of micro-focused ARPES, together with our first-principles band-structure calculations that signify the domain-dependent Dirac gap (for details, see Supplementary Note 2), firmly verifies the S-symmetry protection of the Dirac cone, and thus validates the long-awaited AF TI phase proposed by theory 8,9 .
Topological protection by the S symmetry suggests a high tunability of the Diraccone SS via controlling the AF domain and the surface index 8,15,19 .Such tunability would be useful for realizing exotic topological matters such as 3D axion insulators and highorder TI.For example, when we cut a NdBi crystal so as to make all facets have a domain-A-type configuration where the D vector contains a finite out-of-plane component, an axion insulator with a negative band gap may be realized 8,15,19 .Further, on a certain crystal hinge between these facets, a 1D chiral hinge state may emerge within the surface Dirac gap -a signature of higher-order TI 11,15,19 .At the step edge of the ferromagnetic layer, 1D chiral edge mode is proposed to appear 8,20,25 .Thus, besides the significance of the discovery of AF TI protected by the S symmetry, NdBi also serves as a useful platform to realize exotic quantum states.

Sample fabrication.
Single crystals of NdBi were grown by the flux method using indium flux.The raw materials were mixed in a molar ratio of Nd: Bi: In = 1: 1: 20 and placed in an alumina crucible.The crucible was sealed in an evacuated Quartz tube filled with Ar gas of 50 mbar.The ampoule was heated to 1100 °C, kept for 10 hours, and then cooled to 700 °C in 160 h.The excessive indium was removed in a centrifuge.Obtained crystals were characterized by X-ray diffraction measurements.ARPES measurements.SX-ARPES measurements were performed with an Omicron-Scienta SES2002 electron analyzer with energy-tunable synchrotron light at BL2 in Photon Factory (PF), KEK.We used linearly polarized light (horizontal polarization) of 515-601 eV.VUV-ARPES measurements were performed with micro-focused VUV synchrotron light at BL28 in PF 56 , BL5U in UVSOR, and I05 in Diamond Light Source.We used linearly or circularly polarized light of 60-75 eV.The energy resolution for the SX-and VUV-ARPES measurements was set to be 150 and 10-20 meV, respectively.Samples were cleaved in situ along the (001) plane of the cubic crystal in an ultrahigh vacuum of 1×10 -10 Torr.Prior to the ARPES measurement, the crystal orientation was determined by the X-ray Laue backscattering measurement which signifies clear fourfold symmetric diffraction spots consistent with the (001) cleaved plane.The Fermi level (EF) of samples was referenced to that of a gold film electrically in contact with the sample holder.
Calculations.First-principles band-structure calculations were carried out by using a projector augmented wave method implemented in Vienna Abinitio Simulation Package (VASP) code 57 .To calculate the band structure for the PM phase, the modified Becke-Johnson (mBJ) potential 58 which is known to properly reproduce the band gap in RXp 59 , was used for the exchange-correlation functional.The total energy was calculated selfconsistently with the tetrahedron sampling of 8×8×1 k-point mesh taking into account SOC.The surface states were obtained with the surface Green's function method implemented in WannierTools code 60 after the maximally localized Wannier functions for Bi-s, Bi-p, and Nd-d orbital states were obtained by using Wannier90 code 61 .For the AF phase, we have carried out slab calculations with 12 atomic-layer slabs by taking into account the actual type-I AF structure (Supplementary Fig. 2 and 3).To properly take into account the magnetic moment of Nd ions, we included the strong correlation effect of Nd 4f electrons by using GGA+U calculation potential 62 instead of mBJ potential which has a nonconvergence problem in the slab calculations as in the previous study 57 .We have carried out the unfolding of bands for the superstructure in the AF phase for domain B by using a method proposed in the previous literature 63 .
Polarizing microscopy.Polarizing microscopy measurements were performed by using a home-built UHV microscope system at Tohoku University.We have used a 100 W halogen lamp (U-LH100L-3, Olympus) to obtain bright reflectance images.Polarizing images were obtained in the crossed Nicols configuration with the optical principal axes along [110] axis.Samples were cleaved in situ along the (001) plane of cubic crystal in a UHV of 1×10 -10 Torr.The sample was cooled by liquid helium cryostat and the temperature was controlled in the range of 5-30 K.  ) ) ) Here, topological invariants v1, v2, and v3 were calculated for the cleaved planes of (100), (010), and (001), respectively, to obtain direct correspondence with the ARPES experiments.The result shows that NdBi is a strong TI in the PM phase with (v0; v1, v2, v3) = (1; 0, 0, 0), in good agreement with the band inverted character shown in Fig. 1c.
Regarding (iii), we have carried out tight-binding calculations for 100 atomic layers in the PM phase of NdBi, and calculated the surface projected spectral weight along high symmetry lines of the surface BZ as shown in Fig. 1h of the text (note that the Nd-4f orbitals were assumed to be the core states as in the case of the bulk-band calculations because this assumption was necessary for constructing Wannier functions by fitting the calculated DFT bands to correctly simulate the surface-projected spectral weight).The result signifies the Dirac-cone SS, D1, at the Γ !point, together with the double Dirac-cone SSs, D2 and D3, at the M !point, consistent with the ARPES results for the PM phase.
Regarding (iv), we have carried out slab calculations in the AF phase with 12 atomiclayer slabs by taking into account the actual type-I AF structure (Supplementary Fig. 2a).To properly take into account the magnetic moment of Nd ions, we included the strong correlation effect of Nd-4f electrons by using GGA+U potential instead of mBJ potential which has a nonconvergence problem in the slab calculations as in the previous study 13 .In GGA+U slab calculations, the Nd-4f bands also appear in the calculation at the binding energy of 5-6 eV (out of the energy range of Supplementary Fig. 2).Since this treatment made it difficult to calculate the surface-projected spectral weight because of the difficulty in constructing the proper Wannier functions by fitting the calculated DFT bands due to the presence of 4f states, we show in Supplementary Fig. 2b

Supplementary Note 4: AF origin of the Dirac gap for the D1 band
We have carried out detailed temperature-dependent ARPES measurements for the D1 band to clarify whether the energy splitting in the AF phase is indeed associated with the AF transition.As shown in Supplementary Fig. 5, one can recognize a clear energy splitting of the upper Dirac-cone (D1U) and lower Dirac-cone (D1L) bands at T = 15 K (Supplementary Fig. 5a1).On increasing temperature, these bands gradually broaden and approach each other.The splitting appears to persist at least up to T = 21 K (Supplementary Fig. 5a7) and becomes invisible at 24 K (= TN) and 27 K.As shown by the ARPES intensity at the Γ !point plotted against temperature in Supplementary Fig. 5b, the D1 band starts to split into the D1L and D1U bands just at TN.These results strongly support the AF origin of the Dirac gap.
It is noted here that the energy position of the DP in the PM phase (0.21 eV) in Fig 2c of the main text (as represented by the ARPES data in Supplementary Fig. 5a11) estimated by the linear extrapolation of MDC peak positions slightly deviates from the peak position of the EDC at the Γ !point in Fig. 2e (0.18 eV).We think that this difference is associated with the local deviation of the Dirac-band dispersion from the linear behavior around the DP.The upper Dirac-cone band is rounded around the Dirac point and connected to a highly dispersive lower Dirac-cone band with weaker intensity at the Γ !point.Indeed, such behavior was recognized in previous ARPES and DFT-calculation studies in other rare-earth monopnictides [2][3][4][5] .Another mechanism to cause a deviation of the DP and EDC-peak energies may be a complication of the overall spectral lineshape (in particular EDCs) due to the strong surface-bulk interaction for the D1 band as supported by our first-principles band calculations.
Here we briefly discuss the reason why the observed gap (125 meV) in the AF state is so large.The gap size in NdBi is indeed much larger than those in other AF TI candidates such as MnBi2Te4 (MBT) which shows the AF-induced gap of at most 85 meV [15][16][17] (note that the reported gap size significantly varies depending on the group).While the exact reason for the observed large gap is unclear at the moment, we speculate that the Dirac gap size is linked to the density and magnetic moment of magnetic ions (Nd/Mn).Specifically, the magnetic moment for Nd 3+ ion (3.1 µB) is ~70 % of that of Mn 2+ ion (4.6 µB), while the magnetic ion density in NdBi (Nd:Bi = 1:1) is 350 % of that of MBT (Mn:Bi2Te4 = 1:6).In total, the effective exchange field for the surface Dirac electrons is expected to be much larger in NdBi than in MBT, likely producing a larger gap in NdBi.However, the exchange constants estimated from the neutron diffraction experiments are J1 = -0.008meV and J2 = 0.016 meV for the nearest neighbor and next nearest neighbor, respectively 18 , far smaller than the observed Dirac gap.Thus, although a qualitative argument on the larger Dirac gap may be possible, it is hard to quantitatively explain the magnitude of the Dirac gap in NdBi.as a white colored region (i.e.no birefringence) and domain B with the in-plane magnetic moment as a red/blue colored region.Importantly, we found that the spot size of the microbeam shown by the black circle is much smaller than the typical AF domain size.This result supports that the spatially dependent band structure observed by ARPES is related to the difference in the AF domain.

Fig. 1|
Fig. 1| Band structure of NdBi in the paramagnetic phase.a Rock-salt crystal structure of NdBi with spin orientation in the antiferromagnetic (AF) phase.Translation D vector is also indicated by the green arrow.b Schematic Fermi surface (FS) and bulk fcc Brillouin zone (BZ) of NdBi, together with the surface BZ projected onto the (001) plane (orange rectangle) and Dirac-cone surface state (SS) at the Γ __ Fig. 2| Observation of massive Dirac-cone bands in the AF phase.a Schematic FS of NdBi projected onto the surface BZ. e1 and e2 (blue curves) represent electron pockets elongated along kx and ky, respectively, and are located at different X points in the bulk BZ. h1 and h2 (red curves) represent inner and outer hole pockets, respectively.b FS mapping around the Γ __ point in the AF phase (T = 5 K) measured at hν = 60 eV.c, d ARPES intensity around the Γ __ point in the PM phase (T = 30 K) and the AF phase (T = 5 K), respectively, measured at hν = 75 eV.e EDCs at the Γ __ point at T = 5 K (orange curve) and 30 K (black curve).Dashed lines are a guide for the eyes to trace the upper and lower D1 (D1U and D1L).f Temperature dependence of ARPES intensity at the Γ __ point.Dashed curves are a guide for the eyes to highlight the gap opening across TN for D1.gi Same as b-d but around the M __ obtained at hν = 60 eV.Dashed curves in h and i are a guide for the eyes to trace the band dispersion of D2 and D3 as well as their lower and upper branches.j, l Same as e and f but measured at the M __ point.Dashed curves in k are a guide for the eyes to highlight the gap opening across TN for D2 and D3.

Fig. 3|
Fig. 3| Domain-dependent band structure of NdBi.a, b ARPES intensity around the Γ __ the original band dispersion for the 12 atomic-layer slab instead of the surface-projected weight.Supplementary Fig. 2 | Calculated band structure in the AF phase.a 12 atomic layer slab structure with type-I AF configuration adopted for the calculations to simulate the band structure for domain A. b, c Calculated band structure in the AF and PM phases along high-symmetry lines in the surface BZ.Possible D2 and D3 bands are also indicated.D3 bands irrespective of hv's, despite the strong modulation of their intensities associated with the matrix-element effect.This result strongly suggests the surface origin of the D1-D3 bands.