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# Universal Role of Combined Symmetry for the Protection of the Dirac Cone in
Antiferromagnetic Topological Insulators

Asuka Honma  Department of Physics, Graduate School of Science, Tohoku
University, Sendai 980-8578, Japan  Noriyuki Kabeya  Department of Physics,
Graduate School of Science, Tohoku University, Sendai 980-8578, Japan  Seigo
Souma  Center for Science and Innovation in Spintronics (CSIS), Tohoku
University, Sendai 980-8577, Japan  Advanced Institute for Materials Research
(WPI-AIMR), Tohoku University, Sendai 980-8577, Japan  Yongjian Wang
Institute of Physics II, University of Cologne, K$\ddot{o}$ln 50937, Germany
Kunihiko Yamauchi  Center for Spintronics Research Network (CSRN), Osaka
University, Toyonaka, Osaka 560-8531, Japan  Kosuke Nakayama  Department of
Physics, Graduate School of Science, Tohoku University, Sendai 980-8578, Japan
Daichi Takane  Department of Physics, Graduate School of Science, Tohoku
University, Sendai 980-8578, Japan  Kenichi Ozawa  Institute of Materials
Structure Science, High Energy Accelerator Research Organization (KEK),
Tsukuba, Ibaraki 305-0801, Japan  Miho Kitamura  National Institutes for
Quantum Science and Technology (QST), Sendai 980-8579, Japan  Koji Horiba
National Institutes for Quantum Science and Technology (QST), Sendai 980-8579,
Japan  Hiroshi Kumigashira  Institute of Multidisciplinary Research for
Advanced Materials (IMRAM), Tohoku University, Sendai 980-8577, Japan  Tamio
Oguchi  Center for Spintronics Research Network (CSRN), Osaka University,
Toyonaka, Osaka 560-8531, Japan  Takashi Takahashi  Department of Physics,
Graduate School of Science, Tohoku University, Sendai 980-8578, Japan  Noriaki
Kimura  Department of Physics, Graduate School of Science, Tohoku University,
Sendai 980-8578, Japan  Yoichi Ando  Institute of Physics II, University of
Cologne, K$\ddot{o}$ln 50937, Germany  Takafumi Sato  Department of Physics,
Graduate School of Science, Tohoku University, Sendai 980-8578, Japan  Center
for Science and Innovation in Spintronics (CSIS), Tohoku University, Sendai
980-8577, Japan  Advanced Institute for Materials Research (WPI-AIMR), Tohoku
University, Sendai 980-8577, Japan  International Center for Synchrotron
Radiation Innovation Smart (SRIS), Tohoku University, Sendai 980-8577, Japan
Mathematical Science Center for Co-creative Society (MathCCS), Tohoku
University, Sendai 980-8577, Japan

###### Abstract

Antiferromagnetic topological insulators (AF TIs) are predicted to exhibit
exotic physical properties such as gigantic optical and topological
magnetoelectric responses. While a key to achieving such phenomena relies on
how to break the symmetry protecting the Dirac-cone surface state (SS) and
acquire the mass of Dirac fermions, the mechanism has yet to be clarified. To
address this issue, we carried out micro-focused angle-resolved photoemission
spectroscopy for GdBi hosting the type-II AF order, and uncovered the stripe-
type 2$\times$1 reconstruction of the Fermi surface associated with the AF
band folding. Intriguingly, in contrast to NdBi with the type-I AF order
displaying the surface-selective Dirac-fermion mass, GdBi shows massless
behavior irrespective of AF domains due to the robust topological protection.
These results strongly suggest a crucial role of the $\mathit{\Theta}$TD
(time-reversal and translational) symmetry to create the Dirac-fermion mass in
AF TIs.

##  I INTRODUCTION

TIs are characterized by the boundary state showing linearly dispersive energy
bands with a band crossing at the Dirac point [1, 2]. Symmetries that
guarantee such a Dirac-band crossing play a key role in realizing various
topological quantum phenomena [3, 4, 5]. For example, in nonmagnetic TIs, the
introduction of ferromagnetic (FM) order breaks the time-reversal symmetry
(TRS; $\mathit{\Theta}$) and lifts the band degeneracy at the Dirac point,
leading to quantum anomalous Hall effect [6, 7, 8]. Unlike TRS-broken
ferromagnets, many antiferromagnets with a periodic arrangement of magnetic
moments have a symmetry analogous to the TRS [9]. This is called S symmetry
which is the combination of time-reversal operation $\mathit{\Theta}$ and
translational operation TD (S = $\mathit{\Theta}$TD) where the D vector
connects antiparallel spins so that the S symmetry preserves the structure of
antiferromagnets. The AF TIs with the S symmetry at the surface are
characterized by the Z2 topological invariant and host the Dirac-cone SS as in
the case of time-reversal-invariant TIs [9, 10]. Intriguingly, when the S
symmetry is broken at the surface with a particular mirror index, an energy
gap (Dirac gap) opens at the Dirac point and as a result the Dirac fermions
become massive. This S-symmetry-broken surface leads to the emergence of
chiral edge modes [9, 11, 12, 13] and was theoretically predicted to show
unique phenomena such as quantized magnetoelectric effect [4, 9, 14, 15, 16],
giant optical responses [14, 17, 18, 19], and axion electrodynamics [20, 21].
AF TIs also provide a unique opportunity to manipulate the topological
properties via controlling the magnetic domains and Néel vectors [22, 23, 24].

Despite these theoretical predictions, it is still experimentally unclear
whether or not the S symmetry plays an essential role in characterizing the
Dirac-cone SS in AF TIs. This is highlighted in the case of MnBi2Te4 (MBT)
which was proposed as the first material platform of AF TIs. MBT is a van-der
Waals AF insulator containing a septuple layer in a unit cell with a
ferromagnetic Mn layer placed at the center layer wherein adjacent Mn layers
are antiferromagnetically coupled [25]. At the surface parallel to the
septuple layer obtained by the cleaving of the crystal, the S symmetry is
broken, resulting in the opening of a Dirac gap of 70 meV [25]. The predicted
Dirac gap was indirectly supported by the transport measurements with thin
flakes of MBT [26, 27]. On the other hand, it is hard to access the
S-symmetry-preserved surface because the cleaving plane is limited to that
parallel to the layer. This hinders the understanding of the relationship
between the S symmetry and the Dirac gap.

Recently, rare-earth monopnictide NdBi was proposed to be an AF TI
characterized by the topological SS protected by the S symmetry [28]. NdBi has
a cubic rock-salt crystal structure and shows an inverted bulk-band structure
that supports the strong TI nature in the paramagnetic (PM) phase. Below the
Néel temperature ($T_{\mathrm{N}}=24\ \mathrm{K}$), the type-I AF order with
the propagation vectors of [100], [010], and [001] directions (equivalent to
$<001>$ in a cubic lattice) is stabilized, creating three different types of
AF domains on the cleaved (001) surface. Angle-resolved photoemission
spectroscopy (ARPES) study has identified the AF-domain-dependent Dirac-
fermion mass, supporting the protection of Dirac cones by the S symmetry [28].
On the other hand, an unusual SS, distinct from the Dirac cone, has been
observed in NdBi and other type-I AF monopnictides, requiring deeper
understanding of the role of symmetries in magnetic topological materials [29,
30, 31]. A key to resolve such a complex situation on the interplay between
the symmetry and the surface band structure lies in the investigation of
topological SS for materials with different AF structures. Here, we
deliberately chose GdBi as a target material of ARPES investigation, because
it exhibits the type-II AF order [Fig. 1(a)] with the propagation vector along
the $<111>$ direction, distinct from the type-I order of NdBi with the $<001>$
direction.

##  II EXPERIMENTS

GdBi single crystals were grown by the Bridgman method in a high-vacuum
furnace equipped with a tungsten heater. High-purity starting materials of Gd
(3N) and Bi (6N) with a ratio of 1:1 were sealed in a tungsten crucible using
an electron-beam welder. The crucible was heated above the melting point of
sealed materials (2,100 ${}^{\circ}\mathrm{C}$) and then slowly pulled down
from the heater. After that, the crucible was slowly cooled down to room
temperature. Obtained crystals were characterized by x-ray diffraction and
magnetization measurements.

Vacuum-ultraviolet (VUV)-ARPES measurements were performed at BL-28A in Photon
Factory (PF) with circularly polarized 60–200 eV photons using a micro beam
spot of 12 $\times$ 10 $\mathrm{\mu m}^{2}$ [32]. Soft-x-ray (SX) ARPES
measurements were performed at BL-2A (Musashi) in PF with linearly polarized
light (horizontal polarization) of 536 eV. The energy resolution for VUV and
SX-ARPES measurements was set to be 10–30 meV and 150 meV, respectively.
Samples were cleaved in situ in an ultrahigh vacuum of $\sim 1\times 10^{-10}$
Torr along the (001) crystal plane. The Fermi level (EF) of the samples was
referenced to that of a gold film evaporated onto the sample holder.

First-principles band-structure calculations were carried out by using the
projector augmented wave method implemented in the Vienna Abinitio Simulation
Package (VASP) code [33]. To calculate the band structure in the PM phase, the
modified Becke–Johnson (mBJ) potential [34], which is known to properly
reproduce the band gap in rare-earth monopnictides [35], was used for the
exchange-correlation functional. The total energy was calculated self-
consistently with the tetrahedron sampling of $8\times 8\times 1$ k-point mesh
taking into account spin-orbit coupling. The SS was obtained with the surface
Green’s function method implemented in WannierTools code [36] after the
maximally localized Wannier functions for Bi-s, Bi-p, and Gd-d orbital states
were obtained by using Wannier90 code [37].

##  III RESULTS AND DISCUSSION

First, we present a characterization of a GdBi single crystal grown by the
Bridgman method. Temperature dependence of magnetic susceptibility $\chi(M/H)$
in Fig. 1(b) signifies a peak at 27.5 K associated with the AF transition,
consistent with the previous report [38]. The $1/\chi$ shows a linear
temperature dependence above $T_{\mathrm{N}}$ of 27.5 K (inset), suggesting
that contributions from core diamagnetism, Pauli paramagnetism, and van Vleck
paramagnetism are small. As shown in Fig. 1(c), the core-level photoemission
spectrum in a wide binding energy ($E_{\mathrm{B}}$) region is composed only
of the Bi- and Gd-derived peaks, and the Laue diffraction pattern shows sharp
spots with four-fold symmetry of the (001) cleaving plane. These suggest the
high-quality nature of our bulk crystal.

To clarify the existence of topological Dirac-cone SS in the PM phase, we
carried out surface-sensitive ARPES measurements with VUV photons in a wide
valence-band region along the $\bar{\mathrm{\Gamma}}\bar{\mathrm{M}}$ cut of
surface Brillouin zone (BZ), as shown in Fig. 1(d). The existence of
topological SS in the PM phase is a necessary condition for GdBi to become an
AF TI below $T_{\mathrm{N}}$. Due to the spectral broadening along the wave
vector perpendicular to the surface associated with the short escape depth of
photoelectrons, the ARPES signal for the bulk states reflects the electronic
structure integrated over a large $k_{z}$ area [28, 39]. In the experiment,
one can recognize two hole bands centered at the $\bar{\mathrm{\Gamma}}$
point, h1 and h2, showing stronger and weaker intensities, respectively, both
of which are ascribed to the bulk Bi 6p bands at the $k_{z}=0$ plane. We also
find a shallow and weak electron band e1 which is attributed to the Gd 5d band
at $k_{z}=0$. DFT calculations along the $\mathrm{\Gamma}$X cut of bulk BZ
(corresponding to the $\bar{\mathrm{\Gamma}}\bar{\mathrm{M}}$ cut in the
surface BZ) shown by black curves in Fig. 1(e) reproduce well the experimental
h1, h2 and e1 bands. Importantly, according to the DFT calculations, the h2
and e1 bands show an inverted band structure at the $\bar{\mathrm{M}}$ point
producing a small spin-orbit gap at their intersection slightly away from the
$\bar{\mathrm{M}}$ point (black arrows; see Appendix A for details). In fact,
our parity analysis on the calculated band structure at eight time-reversal-
invariant momenta (TRIMs) shows topological invariants of
$(\nu_{0};\nu_{1},\nu_{2},\nu_{3})=(1;0,0,0)$, supporting that GdBi is a
strong $Z_{2}$ TI with a negative band gap in the PM phase, as in other RBi
compounds such as NdBi [28]. Associated with the bulk-band inversion at the
bulk X point, single (called D1) and double (called D2 and D3) Dirac-cone SSs
appear at the $\bar{\mathrm{\Gamma}}$ and $\bar{\mathrm{M}}$ points,
respectively, corresponding to the number of band inversions projected onto
the surface BZ. Such Dirac-cone SSs are identified in both the experiment
[Fig. 1(d)] and the theoretically-calculated surface spectral weight obtained
with the Green’s function method [Fig. 1(e)]. It is noted that the energy
separation between the Dirac points of the D2 and D3 SSs at the
$\bar{\mathrm{M}}$ point is very close to each other, and the energy
dispersion of the D3 SS is less clear due to the stronger interaction with the
bulk states.

To examine the influence of type-II AF order on the fermiology, we carried out
Fermi-surface mapping across $T_{\mathrm{N}}$. As shown in Fig. 2(a), the
ARPES intensity in the PM phase is dominated by the warped square-like outer
hole pocket at the $\bar{\mathrm{\Gamma}}$ point (h2) and the cross-shaped
electron pockets (e1 and e2) at the $\bar{\mathrm{M}}$ point, consistent with
the semimetallic bulk-band structure shown in Fig. 2(b) under the influence of
strong $k_{z}$ broadening (note that the h1 and D1 pockets are inside the h2
pocket and not clearly visible). We have examined homogeneity of the observed
electronic structure by scanning the whole area of the cleaved surface with
the micro beam spot. As a result, we found that the intensity pattern is
homogeneously distributed over the entire surface in the PM phase, whereas the
intensity in the AF phase exhibits two different types of patterns depending
on the $(x,y)$ position of the surface. We categorize these as domains A and
B, respectively, and their distribution is visualized by green and orange
colors in Fig. 2(c). As shown in Fig. 2(d), the intensity distribution for
domain A is apparently different from that of the PM phase [Fig. 2(a)].
Besides the Fermi surface seen in the PM phase, we found a replica of the
$\bar{\mathrm{\Gamma}}$-centered square pocket with a reduced intensity midway
between the first and second $\bar{\mathrm{\Gamma}}$ points; specifically, at
$(k_{x},k_{y})=(1.0,-1.0)$ and (-1.0, 1.0) in a unit of $\pi/a$. This results
in a 1D array of square pockets running along the lower-right/upper-left
direction [see also schematics in Fig. 2(e)]. Such replicas are also observed
for the $\bar{\mathrm{M}}$ centered cross-shaped pockets at
$(k_{x},k_{y})=(1.0,1.0)$ and (-1.0, -1.0), resulting in the stripe-type
reconstruction of the Fermi surface. In domain B [Figs. 2(f) and 2(g)], all
these replica pockets are resolved, but the direction of 1D array is
different; it runs along the lower-left/upper-right direction, rotated by
$90^{\circ}$ from that of domain A. All these features are well explained in
terms of the AF-induced band folding with two types of surface-projected AF
ordering vectors depicted by $\mathrm{\mathbf{G}_{AF}}$ (green and orange
arrows) in Figs. 2(d)–2(g). When we take into account the cubic symmetry of
bulk crystal and the type-II AF order, there exist four types of AF domains in
the bulk characterized by four AF-ordering vectors, $[111]$,
$[\bar{1}11]$,$[1\bar{1}1]$, and $[\bar{1}\bar{1}1]$ [orange and green arrows
in the bulk BZ in Fig. 2(b)]. When these vectors are projected onto the (001)
surface [surface BZ with purple shade in Fig. 2(b)], two of them, e.g. $[111]$
and $[\bar{1}\bar{1}1]$ exhibit the same band folding, leading to the
distinction of two types of AF domains by ARPES. Despite such strong domain-
dependent Fermi-surface modulation, we found that the Dirac-cone band (D1) at
the $\bar{\mathrm{\Gamma}}$ point seen in the PM phase remains unchanged even
in the AF phase. As shown in Figs. 2(h) and 2(i), one can commonly recognize
the gapless X-shaped D1 band for both domains A and B.

We have confirmed that the D1 band in the AF phase is indeed of surface origin
by observing that the band dispersion is robust against $h\nu$ variation
despite a change in the intensity relative to the bulk hole bands h1 and h2
[Figs. 3(a)–3(h)]. We also found that the D2 band shows a negligible change
across $T_{\mathrm{N}}$ and preserves the gapless character below
$T_{\mathrm{N}}$ [Fig. 3(d)], as in the case of the D1 band [note that it was
difficult to trace the whole dispersion of the D3 band in the experiment
likely due to the stronger interaction with the bulk states, as inferred from
the DFT calculation in Fig. 1(e)]. Such AF-transition-insensitive nature of
the D1 band is distinctly different from the behavior of isostructural NdBi
[28] with the type-I AF order showing an energy gap of $\sim 125\
\mathrm{meV}$ in the AF phase on the top surface [see also Fig. 4(c)], as
highlighted by a side-by-side comparison of the experimental D1-band
dispersion between GdBi and NdBi in Fig. 3(e). This is also evident from the
EDC at the $\bar{\mathrm{\Gamma}}$ point [Fig. 3(f)], which shows a single
peak in GdBi while a two-peaked structure is seen in NdBi. Intriguingly, a
similar trend was also identified for the D2 band; the D2 band in NdBi shows
an energy gap [blue circles in Fig. 3(g)] accompanied by the double-peaked
feature in the EDC [blue curve in Fig. 3(h)], while in GdBi such an energy gap
is not clearly seen [red markers in Fig. 3(g)] and the EDC shows a single peak
[red curve in Fig. 3(h)].

It is emphasized here that, in FM TIs, the out-of-plane component of magnetic
moment at the surface is essential to create a finite Dirac-fermion mass
associated with the TRS breaking. On the other hand, GdBi with the type-II AF
structure always shows a massless Dirac cone at the surface with a finite out-
of-plane magnetic component [Fig. 1(a)]. This strongly suggests that the TRS
alone cannot account for the Dirac-cone protection in GdBi, requiring
consideration of other symmetries unique to the AF structure.

Now we discuss a critical difference in the nature of the surface Dirac-
fermion mass between GdBi and NdBi. In the PM phase, both compounds commonly
show bulk-band inversion and gapless Dirac-cone SS associated with the $Z_{2}$
topology on the (001) cleaved surface [Fig. 4(a)], and their topological
properties are essentially the same with each other. On the other hand, in the
AF phase, GdBi and NdBi show different (type-II and type-I, respectively) AF
order [Figs. 4(b) and 4(c)]. In NdBi, the type-I AF order leads to three types
of AF domains on the (001) surface with the magnetic propagation vectors
running along the [100], [010], and [001] axes, respectively. In a rock-salt
crystal, equivalently, one can consider the single AF domain with the fixed
ordering vector and three cleaving planes (100), (010), and (001), as shown in
Fig. 4(c). Since the (100) and (010) surfaces (side surfaces) have the D
vector (the translation that inverts the spin direction) lying on the surface,
the combined symmetry S is preserved, and the Dirac cone keeps the massless
character. On the other hand, the S symmetry is broken on the (001) surface
(top surface) and the Dirac cone becomes massive because the D vector cannot
be defined parallel to the surface. This leads to the weak-TI-like behavior on
the Dirac-cone SS in NdBi. In GdBi, there exist two types of AF domains in the
(001) surface with propagation vectors of [110] and $[1\bar{1}0]$ (see also
Fig. 2). In this case, one can define the D vectors parallel to the surface
for both the top and side surfaces [Fig. 4(b)], leading to the emergence of
gapless Dirac cones and strong-TI-like characteristics. This consideration led
us to conclude that the S symmetry is crucial for determining the massive vs.
massless character of Dirac fermions in AF TIs independently of the AF
structures. This conclusion is reasonable because only the S symmetry is a
common symmetry between surface and bulk among various crystal symmetries
(e.g. I, $C_{3,[111]}$, $\\{m_{\bar{1}10}|1,1,1\\}$) in the AF phase of GdBi
(magnetic space group; $R_{\mathrm{I}}\bar{3}c$, No. 167.108 [40, 41]).

The present results confirm the theoretically proposed condition
$\mathrm{\mathbf{N}}\cdot\mathrm{\mathbf{D}}=0$ ($\mathrm{\mathbf{N}}$: a
vector normal to the surface) to keep the gapless Dirac-cone SS in AF TIs with
the S symmetry [9, 10]. Here, the $\mathrm{\mathbf{D}}$ vector hosting the
gapless Dirac-cone SS is generally written as,

$\displaystyle\mathrm{\mathbf{D}}=\sum^{3}_{i=1}(n_{i}+\frac{x_{i}}{2})\mathrm{\mathbf{a}}^{\prime}_{i}$
(1)

where $\mathrm{\mathbf{a}}^{\prime}_{i}$ [shown by open black arrows in Figs.
4(a)–4(c)] and $n_{i}$ are the unit vector in the AF phase and an arbitrary
integer, respectively, and $x_{i}$ is 0 or 1 (at least one of $x_{i}$’s must
be 1) [10]. The gapless surfaces of NdBi and GdBi in the AF phase satisfy the
above condition, because the shortest D vector for NdBi can be written with
the set of $(n_{1},n_{2},n_{3})=(0,0,0)$ and $(x_{1},x_{2},x_{3})=(1,1,1)$,
whereas in GdBi the $(n_{1},n_{2},n_{3})$ and $(x_{1},x_{2},x_{3})$ are (-1,
0, 0) and (1, 1, 1), respectively [orange arrows in Figs. 4(b) and 4(c)].
Thus, in AF TIs, the deliberate choice of a surface with a suitable mirror
index can lead to the S-symmetry-broken massive Dirac-cone SS which is a
prerequisite to realizing $Z_{2}$-originated optical and electromagnetic
responses [10, 14]. In this respect, the AF TIs with S symmetry are
advantageous to realize the exotic quantum phenomena, compared to the FM case
which often requires the complex hetero-junction of TI and ferromagnet. Such
surface engineering utilizing the S-symmetry-characterized Dirac-cone SS is
the next step to realizing exotic phenomena unique to AF TIs.

##  IV CONCLUSION

We have performed AF-domain-selective micro-ARPES measurements on type-II
antiferromagnet GdBi, and uncovered (i) stripe-type reconstruction of Fermi
surface associated with the AF band folding and (ii) robust gapless Dirac-cone
SS in the AF phase in contrast to surface-selective Dirac gap in isostructural
type-I antiferromagnet NdBi. Our result strongly suggests the universal role
of S symmetry for the protection of the Dirac cone in AF TIs, and paves a
pathway toward realizing exotic quantum phenomena in AF TIs.

###### Acknowledgements.

We thank T. Kato, T. Osumi, and Y. Kondo for their assistance in the ARPES
experiments. This work was supported by JST-CREST (No. JPMJCR18T1), Grant-in-
Aid for Scientific Research (JSPS KAKENHI Grant Numbers JP21H04435 and
JP24K00564), Grant-in-Aid for JSPS Research Fellow (No: JP23KJ0210), KEK-PF
(Proposal number: 2021S2-001, 2024S2-001, and 2022G652), and UVSOR. The work
in Cologne was funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) - Project number 277146847 - CRC 1238 (Subproject A04).
A.H. thanks GP-Spin and JSPS, and D.T. thanks JSPS and Tohoku University
Division for Interdisciplinary Advanced Research and Education.

##  Appendix A BULK BAND INVERSION IN GdBi

Bulk-sensitive SX ARPES is advantageous in visualizing the bulk band
dispersion of GdBi. Figure 5 shows the plot of the second derivative intensity
of the ARPES spectrum along the $\mathrm{\Gamma X}$ cut of bulk BZ measured
with $h\nu=536\ \mathrm{eV}$. The plot signifies a weak Gd 5d electron band e1
(dashed blue curve) which crosses $E_{\mathrm{F}}$ midway between the
$\mathrm{\Gamma}$ and X points, together with the inner (h1) and outer (h2)
Bi-6p hole bands around the $\mathrm{\Gamma}$ point. The e1 and h2 bands
appear to show a hybridization gap at the intersections (white arrows),
suggestive of the inverted band structure as in the case of other RBi
compounds [39, 35, 28]. As shown in Fig. 5(b), ARPES-derived band dispersions
are well reproduced by our band-structure calculations including the spin-
orbit coupling which signifies the bulk band inversion with a small spin-orbit
gap (see black arrows). These results, together with the observation of Dirac-
cone SS in the PM phase [Fig. 1(d)], support the TI nature of GdBi in the PM
phase.

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Figure 1: (color online). (a) Crystal structure and spin configuration of GdBi
in the type-II AF phase. FM layers are antiferromagnetically stacked along the
$<111>$ direction. (b) Temperature dependence of the $M/H$ curve obtained
under the magnetic field along the [001] direction. Blue and red circles
represent data obtained with the zero-field cooling (ZFC) and field-cooling
(FC) conditions, respectively. Black arrow corresponds to the Néel temperature
($T_{\mathrm{N}}=27.5\ \mathrm{K}$). Inset shows the $H/M$ plot for the ZFC
data. Black solid line shows a linear fit based on the Currie-Weiss’s law. (c)
Photoemission spectra in a wide $E_{\mathrm{B}}$ region measured with
$h\nu=200\ \mathrm{eV}$ at $T=7\ \mathrm{K}$ (AF phase; red curve) and 30 K
(PM phase; blue curve). Inset shows the Laue diffraction pattern. (d) Plot of
ARPES intensity as a function of $E_{\mathrm{B}}$ and wave vector along the
$\bar{\mathrm{\Gamma}}\bar{\mathrm{M}}$ cut measured in the PM phase ($T=30\
\mathrm{K}$) with $h\nu=90\ \mathrm{eV}$. Red, blue, and black dashed curves
are a guide for the eyes to trace the Bi 6p (h1 and h2), Gd 5d (e1), and
surface Dirac-cone (D1) bands, respectively. (e) Calculated surface spectral
weight along the $\bar{\mathrm{\Gamma}}\bar{\mathrm{M}}$ cut projected onto
the (001) plane, obtained with the Green’s function method for a semi-infinite
slab of GdBi in the non-magnetic phase. Black dashed curves trace the band
dispersion of the D1, D2 and D3 SSs. Black solid curves are calculated bulk
band dispersion at $k_{z}=0$ ($\mathrm{\Gamma X}$ cut in bulk BZ).

Figure 2: (color online). (a) ARPES-intensity mapping at $E_{\mathrm{F}}$ for
GdBi plotted as a function of $k_{x}$ and $k_{y}$ at $T=40\ \mathrm{K}$
obtained with $h\nu=105\ \mathrm{eV}$. Solid lines indicate the surface BZ
boundary whereas the dashed lines indicate the cross-section of both $k_{z}=0$
and $\pi$ planes of the bulk fcc BZ boundary. (b) Bulk fcc BZ together with
the bulk Fermi surface. Purple rectangle indicates the surface BZ projected
onto the (001) surface. Orange and green arrows indicate the type-II AF
ordering vectors in bulk BZ and projected surface BZ, respectively. (c) (Top)
microscope image of the cleaved surface of GdBi, and (bottom) distribution of
AF domains visualized by scanning ARPES with $26\times 22$ mesh. The pixel
size is $50\times 50\ \mathrm{\mu m^{2}}$. Regions shown with orange and green
color correspond to domains A and B where the AF ordering vector points to
lower right and upper right directions, respectively. White region represents
the area where the domain identification was difficult, while the black region
has no signal from the sample. (d) Same as a but obtained at $T=6\ \mathrm{K}$
(AF phase) for domain A. $\mathrm{\mathbf{G}_{AF}}$ (green arrow) is the AF
ordering vector projected onto the surface. White arrows indicate the replica
of Fermi surfaces and red rectangles represent the magnetic surface BZ. (e)
Schematics of band folding in the surface BZ, together with the folded Fermi
surfaces for domain A. Red rectangle represents the magnetic surface BZ. (f),
(g) Same as (d) and (e), respectively, but for domain B. (h), (i)
Near-$E_{\mathrm{F}}$ ARPES-intensity plots around the $\bar{\mathrm{\Gamma}}$
point obtained at $T=6\ \mathrm{K}$ along the $k_{x}$ cut for domains A and B,
respectively.

Figure 3: (color online). (a)–(c) Plots of ARPES intensity around the
$\bar{\mathrm{\Gamma}}$ point for GdBi along the $k_{x}$ axis obtained at
$T=7\ \mathrm{K}$ with $h\nu=105,90,\mathrm{and}\ 60\ \mathrm{eV}$,
respectively. (d) Plot of ARPES intensity around the $\bar{\mathrm{M}}$ point
for GdBi obtained at $T=7\ \mathrm{K}$ with $h\nu=60\ \mathrm{eV}$. (e)
Experimental D1-band dispersion for GdBi and NdBi in the AF phase. Open
circles and cross markers represent the band dispersion obtained by tracing
the peak position of energy distribution curves (EDCs) and momentum
distribution curves (MDCs), respectively, measured with $h\nu=90\
\mathrm{eV}$. Magenta triangles represent the EDC/MDC peak plot for the data
obtained with $h\nu=105\ \mathrm{eV}$. D1U and D1L stand for upper and lower
branches of gapped D1 band, respectively. (f) EDCs for GdBi (red curve) and
NdBi (blue curve) at the $\bar{\mathrm{\Gamma}}$ point in which the energy
zero is set to be at the Dirac point of the D1 band in the PM phase. (g), (h)
Same as (e) and (f), respectively, but for the D2 and D3 bands in which the
energy zero is aligned to the Dirac point of the D2 band in the PM phase.

Figure 4: (color online). (a) Schematic of surface Dirac cones in the PM phase
for both NdBi and GdBi, which are attributed to the strong TI phase with
surface-independent massless Dirac cones protected by TRS. (b), (c) Same as a
but for the type-II (GdBi) and type-I (NdBi) AF phases, respectively. Dirac
cones are protected by the combined symmetry S, except for the
ferromagnetically aligned surface for the type-I case. The unit lattice
vectors $\mathrm{\mathbf{a}}’_{i}$ ($i=1–3$) that take into account the
magnetic structure for the AF case are shown by open black arrows. D vectors
are shown by orange color in (b) and (c).

Figure 5: (color online). (a) Plot of second derivative intensity of ARPES
spectrum of GdBi in the PM phase ($T=40\ \mathrm{K}$) as a function of $k_{x}$
and binding energy ($E_{\mathrm{B}}$) along the $\mathrm{\Gamma X}$ cut of
bulk BZ measured with $h\nu=536\ \mathrm{eV}$. Red and blue dashed curves are
a guide for the eyes to trace the Bi-6$p$ (h1 and h2) and Gd 5$d$ (e1) bands,
respectively. (b) Calculated band structure along the $\mathrm{\Gamma X}$ cut
of bulk BZ, vertically expanded by a factor of 1.45 to obtain a better
matching with the experiment.