Introduction

Rare-earth monopnictides (RPn) in the rock-salt crystal structure have been studied as promising materials for spintronics1. These compounds have attracted renewed interests because of emerging topological properties. The non-magnetic LaPn were first predicted as topologically non-trivial due to the band inversion between La d and Pn p orbitals2. The crossover and trend of such d-p band inversion in the RPn series have been studied by experiments3,4,5,6,7,8 and calculations9,10, which have largely focused on the paramagnetic state. The gradual filling of the 4 f orbitals and change of spin-orbit coupling (SOC) strength across the RPn series can potentially bring high tunability in the multitude of topological states when coupled with magnetic properties11,12,13,14,15,16. Although some RPn compounds, such as CeSb17, GdBi18, and CeBi19, have been proposed to host magnetic Weyl points (WPs) in the ferromagnetic (FM) state when fully magnetized, most RPn prefer anti-ferromagnetic (AFM) configurations. Angle-resolved photo-electron spectroscopy (ARPES) study of the RPn in the AFM state is rare except for CeBi20,21.

Very recently ARPES22 has discovered unconventional surface states (SS) in NdBi below the Neel temperature (TN = 24 K) that display a combination of hole Fermi arc-like features and SS electron pockets, as well as a peculiar temperature-dependent splitting. Unlike the Fermi arcs in cuperates23 induced by a pseudo-gap opening and those in Weyl semimetals24 by the opposite chirality of WPs, the origin of the newly discovered hole Fermi arc-like features in AFM NdBi is unclear. Figure 1a, b show the ARPES data on NdBi (001) above and below TN, respectively. Above TN and similar to other paramagnetic RBi, a large 4-fold star-shaped bulk hole pocket and another small one at the \(\bar{\varGamma }\) point overlap with the electron pocket projected from the X point in Fig. 1a. These results match reasonably well with the 2D surface Fermi surface (SFS) calculated with density functional theory25,26 (DFT) on the non-magnetic (NM) NdBi (001) in Fig. 1c. Surprisingly, when cooling below TN for NdBi (Fig. 1b), ARPES observed 4-fold unconventional hole Fermi arc-like features and SS electron pockets emerging out of the bulk band projections. The hole Fermi arc-like features connect to the bulk hole pockets and the elliptical SS electron pockets have a sharp cusp. However, the DFT-calculated SFS on NdBi (001) with the 4-fold rotational symmetry in Fig. 1e with the reported layer-stacking AFM structure27 (referred as 1q) lacks these 4-fold unconventional SS features. NdBi has a high-symmetry crystal structure without Nd-4f-level hybridization (e.g., Kondo effect), which makes the ARPES discovery more intriguing.

Fig. 1: 2D surface Fermi surface (SFS) of NdBi.
figure 1

Angle-resolved photo-electron spectroscopy (ARPES) at (a) T > TN and (b) T < TN with the unconventional surface state (SS) of hole Fermi arcs and SS electron pockets. The calculated SFS on (001) for (c) non-magnetic (NM) and anti-ferromagnetic (AFM) with (d–f) 1q, (g–i) 2q and (j, k) 3q wave vectors on different surfaces as labeled. The multi-q magnetic structures are shown in (d), (g), and (j). The orange (blue) spheres stand for Nd (Bi) atoms and the red arrows show the directions of Nd magnetic moments. The surface Brillouin zone for NM and 1q (001) is twice as large and rotated by 45 degree with respect to that for 2q and 3q (see Fig. 2b) for labels of high symmetry points). The 1q tetragonal (110) vertical diagonal surface in (f) is equivalent to the cubic (010) side surface. Blue, white and red colors stand for low, medium, and high density of states.

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The related monopnictide compounds with actinides (5 f) in the same rock-salt structure, such as UP, UAs, USb28, and NpBi29, have been reported before in type-I AFM multi-q structures, which are difficult to distinguish from the simple 1q using standard neutron diffraction techniques. The multi-q structures focused here are the commensurate magnetic structures on the Nd face-centered cubic (FCC) sublattice with superpositions of propagating vectors along the equivalent {001} directions, i.e., the type-I AFM multi-q structures30,31,32. In contrast, the type-II and III AFM on an FCC lattice have propagating vectors along the {111} or {½ 01} directions, respectively, giving different multi-q structures30,31,32. The magnetic moments in these multi-q structures can be noncollinear and noncoplanar. In general, multi-q has also been used to define the spin texture as superpositions of multiple spin helices or spin density waves, mostly incommensurate, such as skyrmions33,34,35,36,37,38,39,40,41,42 in 2D thin films or layered structures, and hedgehog43,44 in 3D structures, with a real space non-trivial topology and also a periodicity much larger than the underlying crystal lattice. Although magnetic space groups (MSG) give all the symmetry operations for Wyckoff positions with breaking time-reversal symmetry due to magnetism, multi-q as a superposition of multiple propagating vectors can include the orientation of the magnetic moments to describe both commensurate and incommensurate magnetic structures with noncollinear and noncoplanar components.

Here, we explore the effects of the type-I AFM multi-q magnetic structures on the band structure and topology of NdBi using DFT calculation. We find that NdBi AFM multi-q structures with two (2q) and three (3q) wave vectors give rise to the 4-fold unconventional surface state pairs (SSP) on (001) surface as shown in Fig. 1g–h for 2q and Fig. 1j–k for 3q, respectively; while 2q (010) and 1q (110) (equivalent to cubic (010)) have 2-fold SSP as shown in Fig. 1f, i, respectively. Considering the possibility of multiple domains of {001} directions in experimental samples, these unconventional SSP match well with the ARPES data below TN. The hybridization gap from AFM band-folding along the {001} directions is the common theme in these multi-q structures of NdBi. The difference is the number of {001} directions that the AFM band-folding occurs. It occurs along only one {001} direction in 1q, but two and three equivalent {001} directions in 2q and 3q, respectively, giving the unconventional SSP of only 2-fold in 1q, 2-fold or 4-fold in 2q, and 4-fold in 3q. Additionally, we find that NdBi 1q and 2q are Dirac semimetals, and 3q is a Weyl semimetal. Our result shows that a Fermi arc-like signature on a surface does not always imply the presence of Weyl fermions in the bulk, as in the case of 1q and 2q, making an unambiguous discovery of magnetic Weyl semimetals more elusive.

Results and Discussion

Hybridization gap opening in multi-q bulk band structures

The three different type-I AFM multi-q structures of NdBi are shown in Fig. 1d, g, j with the corresponding bulk band structures presented in Fig. 2a, c, d, respectively. The high symmetry k-points and Brillouin zones (BZ) are in Fig. 2b). The NM NdBi in space group (SG) of Fm-3m (or 225) is known to have a band inversion between the Bi p and Nd d orbitals with anti-crossing along the \(\varGamma\)-X direction to host a strong topological insulator (TI) state2 (see Supplementary Note 1 and Fig. S1). The 1q is in MSG of PI4/mnc (or 128.410) and its band structure along the \(\varGamma\)-M’ direction (Fig. 2a) retains the same band inversion in NM (neck-narrowing region), while along the AFM stacking [001] direction, the band folding opens up a hybridization gap along the \(\varGamma\)-Z direction with a Dirac point (DP) at about 0.5 eV above the Fermi energy (EF). The hybridization gap opening from the AFM band-folding of the band-inversion region is best seen in Fig. 2c for 2q in MSG of PC42/nnm (or 134.481). The band-folding in 2q happens at the X point (half-way along the \(\varGamma\)-M’ or [010] direction highlighted by the green arrows), where a bulk band gap of about 0.2 eV opens near EF due to the hybridization between the folded bands having the same Bi p orbital character. The original band inversion now becomes the upper valence bands and is pushed to higher energy. Interestingly, with the magnetic moments changing from [001] in 1q to in-plane directions in 2q, the hybridization gap along the \(\varGamma\)-Z direction is reduced and two new band crossings emerge to form two DPs. For the 3q in MSG of Pn-3m’ (or 224.113), comparing its band structure in Fig. 2d to 2q in Fig. 2c, the \(\varGamma\)-X and \(\varGamma\)-Z directions become equivalent and so are the hybridization gaps. With lifting of the band double degeneracy along the other directions in 3q, new band crossings emerge around the \(\varGamma\) point, giving WPs along the \(\varGamma\)-R direction, which will be discussed later together with the DPs in 2q because they are near or below the EF.

Fig. 2: Calculated bulk band structures of NdBi.
figure 2

a 1q, (c) 2q and (d) 3q are plotted with respect to the Fermi energy (EF) at 0.0 eV. The top valence band is in blue and the green shade stands for the projection of Bi p orbitals. b 3D bulk Brillouin zone (BZ) of 1q (dashed lines for the tetragonal unit cell) and that of 2q and 3q (solid lines for the cubic unit cell), and the 2D surface Brillouin zone (SBZ) of (001) and (010) with the high-symmetry points labeled. The green vertical arrow in (a) points to band-folding at the X point, half-way along the \(\varGamma\)-M’ direction for 1q. The green horizontal arrow in (c) points to hybridization gap opening upon band-folding along the \(\varGamma\)-X direction for 2q.

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With the inversion symmetry intact for all three magnetic structures, parity eigenvalues at time-reversal invariant momentum (TRIM) can be used to calculate \({Z}_{4}=\left(\frac{1}{2}\right)\mathop{\sum}\nolimits_{k\in {TRIM}}\left({n}_{k}^{+}-{n}_{k}^{-}\right)\,{{{{\mathrm{mod}}}}}\,4\), where \({n}_{k}^{+}\) and \({n}_{k}^{-}\) are the number of filled states with even and odd parity, respectively. Interestingly, Z4 = 2 for all the three gapless phases in the different multi-q structures, because the sum of the difference in the number of opposite parity eigenvalues over TRIM does not change. NdBi 1q and 2q are Dirac semimetals, while 3q is a Weyl semimetal. Counting both –kz and +kz directions, the one pair of DPs in 1q are about 0.5 eV above the EF, while the two pairs of DPs in 2q are near the EF. The more detailed symmetry description of the multi-q band structures and the corresponding propagating vectors can be found in Supplementary Note 2.

Unconventional surface state pairs

Next, we discuss the surface states generated by the different type-I AFM multi-q structures. The surface spectral function of 2q (001) along the \(\bar{\varGamma }\)-\(\bar{X}\)-\(\bar{\varGamma }\) direction is presented in Fig. 3b and compared side-by-side to the equivalent NM (001) \(\bar{\varGamma }\)-\(\bar{X}\)-\(\bar{{M}^{{\prime} }}\) direction in Fig. 3a. From the bulk band projection on NM (001), the band inversion is near the \(\bar{X}\) point with a dashed line, the middle point between the \(\bar{\varGamma }\) and \(\bar{{M}^{{\prime} }}\) points. The two surface Dirac cones for this strong TI are overlapped at the \(\bar{{M}^{{\prime} }}\) point with two SS stemming from band inversion region toward the \(\bar{{M}^{{\prime} }}\) point. In echo with the folding of the bulk bands in Fig. 2c for 2q, these SS of the overlapped surface Dirac cones are folded from the \(\bar{{M}^{{\prime} }}\) to \(\bar{\varGamma }\) point in Fig. 3b. Simultaneously, a bulk hybridization gap is opened in 2q with two new unconventional SSP appearing in the hybridization gap along the \(\bar{\varGamma }\)-\(\bar{X}\) direction from EF–0.150 eV to EF. The original band inversion region is pushed up into higher valence bands as marked by the green dots.

Fig. 3: Surface spectral functions of NdBi.
figure 3

a nonmagnetic (NM) (001) along \(\bar{\varGamma }\)-\(\bar{X}\)-\(\bar{{M}^{{\prime} }}\) and (b) 2q (001) along the equivalent direction of \(\bar{\varGamma }\)-\(\bar{X}\)-\(\bar{\varGamma }\). The green dots indicate the original band inversion region along \(\bar{\varGamma }\)-\(\bar{X}\), just above the bulk hybridization gap and the associated unconventional surface state pairs. Surface spectral functions on (c) 3q (001), (d) 2q (001), (e) 2q (010) and (f) 1q (110) along other directions. 2D surface Fermi surfaces at two different energies as labeled in eV with respect to Fermi energy (EF) for four surfaces: (g, k) for 3q (001), (h, l) for 2q (001), (i, m) for 2q (010), and (j, n) for 1q (110), respectively. (o, p) are the averaged cubic {001} domains for 2q and 1q, respectively. Blue, white and red colors stand for low, medium, and high density of states.

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The surface spectral functions on 3q (001) are plotted in Fig. 3c, 2q (001) in (d), 2q (010) in (e), and 1q (110) in (f), respectively. For all the four multi-q surfaces in Fig. 3 with the AFM band-folding hybridization gap along the different {001} directions, a pair of unconventional SSP emerges inside the bulk hybridization gap with similar dispersion along the \(\bar{\varGamma }\)-\(\bar{X}\) direction (or \(\bar{\varGamma }\)-\(\bar{Z}\) for 1q (110)). The SSP starts from the overlapped bulk band projection near the \(\bar{\varGamma }\) point and the splitting increases toward the surface Brillouin zone (SBZ) boundary. Then along the \(\bar{X}\)-\(\bar{M}\) (3q and 2q (001)), \(\bar{X}\)-\(\bar{R}\) (2q (010)) or \(\bar{Z}\)-\(\bar{R}\) (1q (110)) directions, the upper SS quickly merges with the bulk electron pocket, while the lower SS merges with the bulk hole pocket. Thus, the upper SS is of electron character, while the lower SS is of hole character, respectively. In contrast, on 1q (001) and (010) (or cubic (110)) surfaces, there are no such unconventional SSP (see Supplementary Note 3 and Fig. S2).

The SFS are plotted at EF–0.100 eV for 3q (001) (Fig. 3k) and EF–0.050 eV for 2q (001) (Fig. 3l). The hole SS merges with the bulk hole pocket when going off the \(\bar{\varGamma }\)-\(\bar{X}\) direction, showing a Fermi arc-like feature. The electron SS forms an elliptical pocket by itself with a cusp toward the \(\bar{X}\) point. Both of these SSP features are in good agreement with ARPES22 near the \(\bar{\varGamma }\) point. The electron SS pocket comes from the upward dispersion of the upper SS near the \(\bar{X}\) point (Fig. 3c, d). At EF, the SFS in Fig. 3g, h clearly show the upper SS envelops the bulk electron pocket when going off the \(\bar{\varGamma }\)-\(\bar{X}\) direction. For 2q bulk band structure, the hybridization gap only appears along the \(\varGamma\)-X, not the \(\varGamma\)-Z direction as seen in Fig. 2c. As a result, in the SFS on 2q (010) (Fig. 3i, m), the SSP hole Fermi arc-like feature and electron pocket also only appear along the \(\bar{\varGamma }\)-\(\bar{X}\) direction, while along the \(\bar{\varGamma }\)-\(\bar{Z}\) direction is mainly bulk band projection, giving a clearly 2-fold symmetry. Similarly, on 1q (110) surface (Fig. 3j, n), the AFM band-folding hybridization gap only appears along the \(\varGamma\)-Z direction and the SSP is also 2-fold. Because it is possible to have multiple {001} domains in experimental samples for 1q and 2q, and if the domain size is much smaller than the ARPES laser beam spot, the overlapped 2-fold SSP and bulk FS from the mixed {001} domains become effectively 4-fold SSP with a bulk FS background (see Fig. 3o, p for a direct average), which is in a good agreement with ARPES data showing that the hole Fermi arc-like feature and surface electron pocket have an underneath bulk FS background. Thus, this observation can be used to exclude 3q, because 3q only has one (001) domain (Fig. 3k) without the bulk FS background underneath the 4-fold SSP. The existence of 1q or 2q different {001} domains with the 2-fold SSP has been observed in the sibling compound NdSb45.

The detailed comparison of SSP dispersion to ARPES data will next be made with 2q (001) in Fig. 4. As highlighted by the orange box in Fig. 4a, the spin texture of the SSP hole arc and electron pockets are plotted in Fig. 4b. The orange arrows show the in-plane components of the expectation value of electron spin weighted by the SFS at the specific k-point and energy (see Eqn. (32) in Ref. 46). The opposite spin texture between the SSP hole arc and electron pocket is persistent when moving to more negative energies with the SS electron pocket shrinking in size, which is in very good agreement with the spin-polarized ARPES data22.

Fig. 4: Comparison between NdBi 2q (001) surface states and angle-resolved photo-electron spectroscopy (ARPES).
figure 4

a 2D surface Fermi surface (SFS) on 2q (001) labeled with two sets of cuts in black and white dashed lines, and an orange box focusing on the surface state pairs. b SFS zoomed in the orange box of (a) for the surface hole Fermi arc and electron pocket with the spin texture (in-plane components of the expectation value of spin weighted by the SFS) indicated by the orange arrows. The different energies are labeled in eV. c, e Comparison between ARPES and the calculated spectral functions at the cuts (black dashed lines in (a)) of 0.03, 0.08, 0.12, 0.14, 0.18, and 0.20 in the unit of the nonmagnetic surface Brillouin zone. d, f Comparison at cuts in the other direction (white dashed lines in (a)) at 0.03, 0.09, 0.17, 0.22, 0.24, and 0.27. All energies are downward shifted by 0.010 eV from the Fermi energy (EF).

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To compare with ARPES data22 side-by-side and analyze the SSP away from the high symmetry directions, the same two sets of six cuts each as for ARPES (see Fig. 3 of Ref. 22) are made with black and white dashed lines in Fig. 4a to calculate the surface spectral function on 2q (001) in detail. With the black vertical dashed cuts to focus on half of the hole Fermi arc-like feature, the spectral function in Fig. 4e shows the hole arc (the lower SS) moves to more negative energy and also gradually merges with the bulk hole band when shifted away from kx = 0 along the \(\bar{\varGamma }\)-\(\bar{X}\) direction in Cut No.1–3. These behaviors of the hole Fermi arc-like feature are in good agreement with those observed in Fig. 4c. When shifted further, both calculation and ARPES also agree on the arc tip still appearing at Cut No.4, while disappearing at Cut No.5-6 with only 3D bulk bands left behind. For the cuts with the white dashed lines rotated 45 degree and moved away from the \(\bar{\varGamma }\)-\(\bar{M}\) direction in Fig. 4a to survey both the hole Fermi arc-like feature and SS electron pocket, the emerging arc tip at Cut No.2 in Fig. 4f, the gradual emergence and shift of the asymmetric splitting between the upper and lower SS at Cut No.3-6 also agree very well with the ARPES data in Fig. 4d. The enveloping of the lower SS to bulk hole band and that of upper SS to bulk electron band are also evident. The similarly good agreement to ARPES data for the 2-fold SSP on 2q (010) and 1q (110) with the same sets of cuts can be found in Supplementary Note 4 and Fig. S3. These good agreement between the calculated unconventional SSP and the ARPES data indicates a general rule that the unconventional SSP emerges in NdBi along the {001} directions having the AFM band-folding bulk hybridization gap for the different type-I AFM multi-q structures. This common theme is the main result of the paper.

Nodal features in multi-q structures

As mentioned above, in addition to the unconventional SSP, type-I AFM multi-q NdBi also have nodal features. We will show that the unconventional SSP does not arise from the WPs in 3q or the DPs in 2q near the EF. For 3q with the all-in-all-out magnetic configuration (first predicted Weyl semimetal in iridates47), we found 16 WPs in two sets along the \(\varGamma\)-R direction (see Fig. 5a) with the momentum-energy of (± 0.0875, ± 0.0875, ± 0.0875 1/Å, EF + 12.6 meV) and (± 0.1133, ± 0.1133, ± 0.1133 1/Å, EF + 14.9 meV), respectively. In contrast to the axion insulator in NpBi with the same 3q structure as analyzed in the Xu et al.13, the even number of WPs in half of the BZ is also compatible with Z4 = 2 for NdBi 3q here. The Berry curvatures near a pair of these WPs are plotted in the vertical and horizontal planes in Fig. 5b, c to confirm the opposite chirality of ± 1. When projected on (001) surface, the WPs of opposite chirality overlap and usually Fermi arcs are not expected. But as shown by the SFS around the WP binding energy of EF + 0.014 eV in Fig. 5d, e, there are small protrusions along the \(\bar{\varGamma }\)-\(\bar{M}\) direction connecting the hole SS on both the top and bottom surfaces. After zooming in Fig. 5f, g with the surface-only contribution, it turns out that the hole SS connects the projections of outer WPs (green circles), but not the inner WPs (black squares). The usual explanation for the disappearance of open Fermi arcs when two WPs of opposite chirality overlap has the assumption that the open Fermi arcs connect in the direction of the shortest path between the WP projections. When the WP projections of opposite chirality move closer and overlap, such as the black squares, Fermi arcs disappear. However, as shown in the earlier study48, the open Fermi arc connectivity of the WP projections are not restricted to only one direction and can actually change to the opposite direction depending on the surrounding SS. In such a scenario, the overlap of WP projections of opposite chirality can give a full circle for the Fermi arcs, which is seen here for the case of the outer set of WPs in NdBi 3q. With projection to the 3q (111) surface as shown in Fig. S4 of Supplementary Note 5, there are Fermi arcs connecting the projections of the WPs without overlap of the opposite chirality.

Fig. 5: Weyl points (WPs) in NdBi 3q.
figure 5

a Locations of the two sets of 16 WPs with the momentum-energy of (± 0.0875, ± 0.0875, ± 0.0875 1/Å, EF + 12.6 meV) and (± 0.1133, ± 0.1133, ± 0.1133 1/Å, EF + 14.9 meV), respectively, where EF is the Fermi energy. Berry curvature near the WP pair on the (b) vertical kxy-kz plane and (c) horizontal kx-ky plane at kz = 0.10. Surface Fermi surface near the WP energy at EF + 0.014 eV on the (d) top and (e) bottom (001) surfaces. The corresponding surface-only contributions zoomed on the (f) top and (g) bottom (001) surface with the locations of the WP projections labeled by the black square and green circle.

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For the DPs in 2q, the bulk band structure is plotted in Fig. S5a of Supplementary Note 6 along the \(\varGamma\)-Z direction, where the top valence and bottom conduction bands cross twice with different orbital content. The two pairs of DPs have the momentum-energy of (0, 0, ± 0.1501 1/Å, EF + 73.0 meV) and (0, 0, ± 0.4330 1/Å, EF–177.5 meV), respectively. The upper DP has a switch of orbital content between Bi pz and px, while the lower DP has the switch between Nd dxy and Bi px orbitals. These DPs are protected by the 4-fold screw axis. To search for any topological SS signature of the DPs, the side (010) surface spectral function along the \(\bar{\varGamma }\)-\(\bar{Z}\) direction is presented in Fig. S5b. But the SS around the binding energy of the upper DP at EF + 0.073 eV does not originate from the DP projection. For the lower DP at EF–0.178 eV, two SS follow the narrow band gap from EF to EF–0.1 eV moving toward the DP, but they merge into the bulk band projection before reaching the DP. The lack of the closed Fermi arcs connecting two DP projections in 2q is also seen in the SFS in Fig. S5c, d at the binding energies of these DPs. Although the closed Fermi arcs connecting two DP projections are often found in some other Dirac semimetals, they are not guaranteed with topological protection and are fragile as discussed before in a theoretical model49 and also found for PtBi250. Our results for NdBi 2q provide another example regarding the fragility of the Fermi arcs for DPs.

From the perspective of using symmetry-based indicators and topological invariants to diagnose topological states for magnetic space groups, a previous theory work13 has noticed that the parity-based index Z4 = 2 can correspond to different topological states including both gapped and gapless phases. Although NpBi with 3q has been predicted13 as an axion insulator with gapped surface Dirac cones and chiral hinge states, our current study on multi-q NdBi realizes several other compatible scenarios with Z4 = 2, including a Weyl semimetal with an even number of WPs in half of the bulk BZ for 3q and a Dirac semimetal for 1q and 2q. The unconventional SSP spanning the AFM band-folding hybridization gap along the {001} directions remains very robust for all these changes in multi-q structures to give the key features of unconventional surface hole Fermi arc-like features and elliptical electron pocket as observed in ARPES. Thus, besides explaining of the unconventional SSP in NdBi below TN, our study also expands the variety of topological features that are compatible with the parity-based index Z4 = 2. However for the SSP, finding the exact topological invariants to associate with is still an open question.

We have also changed the U and J values in the DFT calculations. The DFT total energy differences with the fully relaxed unit cells in PBEsol+U + SOC are plotted in Fig. S6a of Supplementary Note 7 as a function of U. The 2q becomes the most stable phase when the U value is larger than 9.6 eV. We have calculated the 2q surface band structures with a U value of 9.8 eV on (001) and (010) in Fig. S6 and found the existence and connectivity of the unconventional SSP in the AFM band-folding hybridization gap remain the same. But the splitting between the SSP is reduced as the Nd 4 f bands are moved further away from the EF with higher U value. The simultaneous description of the right size of SSP splitting and total energy difference is challenging for the DFT + U method. The temperature dependence of the SSP splitting can be more complicated and needs further theory efforts.

Conclusion

We find that the type-I anti-ferromagnetic multi-q structures, 1q, 2q, and 3q of NdBi, can host unconventional surface state pairs (SSP) that are consistent with the ARPES data around the \(\bar{\varGamma }\) point below TN. These unconventional SS appear in pairs and reside in the bulk hybridization gap region formed by the AFM band-folding of the band inversion region along the {001} directions. The lower SS forms the Fermi arc-like features connected with the bulk hole pocket and the upper SS forms an elliptical electron pocket with a cusp toward the \(\bar{X}\) point, agreeing very well with the experiment. We also find NdBi 2q is a Dirac semimetal with two pairs of Dirac points (DPs) along the \(\varGamma\)-Z direction, but without the closed Fermi arcs connecting the DP projections. In contrast, NdBi 3q is a Weyl semimetal with two sets of 16 Weyl points (WPs) along the \(\varGamma\)-R direction. The projections of the outer set are connected by the hole SS arcs in a full circle, despite that the WP projections are overlapped with opposite chirality. Our results also expand a variety of topological nodal features that are compatible with the parity-based Z4 = 2.

Methods

All density functional theory25,26 (DFT) calculations with spin-orbit coupling (SOC) were performed with the PBE51 exchange-correlation functional using a plane-wave basis set and projector augmented wave method52, as implemented in the Vienna Ab-initio Simulation Package (VASP)53,54. Using maximally localized Wannier functions55,56, tight-binding models were constructed to reproduce closely the band structure including SOC within EF ± 1 eV with Nd s-d-f and Bi p orbitals. The surface spectral function and 2D Fermi surface (FS) were calculated with the surface Green’s function methods57,58 as implemented in WannierTools46. In the DFT calculations, we used a kinetic energy cutoff of 253 eV, \(\varGamma\)-centered Monkhorst-Pack59 (8 × 8 × 8) k-point mesh, and a Gaussian smearing of 0.05 eV. For band structure calculations of NdBi, we have used the experimental structural parameter60 of 6.41 Å. To account for the strongly localized Nd 4 f orbitals, an onsite Hubbard-like61 U = 6.3 eV and J = 0.7 eV have been used. Our DFT + U + SOC calculation gives a spin moment of 2.7 \({\mu }_{B}\) and an orbital moment of 5.8 \({\mu }_{B}\) in the opposite direction, resulting in a total magnetic moment of 3.1 \({\mu }_{B}\) on Nd, agreeing very well with the experimental data27 of 3.1 ± 0.2 \({\mu }_{B}\).