Dirac semimetal in type IV magnetic space group
Guiyuan Hua,1, ∗ Simin Nie,2, ∗ Zhida Song,3, † Rui Yu,4, ‡ Gang Xu,1, § and Kailun Yao1
1Wuhan National High Magnetic Field Center and School of Physics,
Huazhong University of Science and Technology, Wuhan 430074, China
2Department of Materials Science and Engineering,
Stanford University, Stanford, California 94305, USA
3Beijing National Research Center for Condensed Matter Physics,
and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
4School of Physics and Technology,
Wuhan University, Wuhan 430072, China
(Dated: January 10, 2018)
Abstract
Analogues of the elementary particles, Dirac fermions in condensed matter have received ex-
tensive attention for both scientific interest and device applications. In this work, we generalize
the concept of Dirac semimetal (DSM) to the magnetic space groups (MSGs), and define a new
category of DSM in type IV MSGs, which is protected by the antiunitarity of the product of the
inversion symmetry and the nonsymmophic time-reversal symmetry. Moreover, we propose the in-
terlayer antiferromagnetic (AFM) EuCd2As2 as a promising candidate with only one pair of Dirac
points at the Fermi level. Many exotic topological states, such as the triple point semimetal and
the AFM topological insulator holding of the half-quantum Hall effect, can be derived from such
AFM DSMs by breaking certain symmetry, providing an ideal platform to study topological phase
transitions. Our results extend the range of DSM, and open a new way to study the interplay
between the DSM and other exotic topological states.
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Massless Dirac fermions (DFs) are one kind of the long-pursued elementary particles[1–4].
While their existence remains elusive in particle physics, the realization of DFs in the DSM[5–
12] has received extensive attention for both scientific interest and device applications[13–16].
In the three-dimensional (3D) materials with both time reversal symmetry (TRS) T and
inversion symmetry (IS) P , each band energy is double degenerate[17, 18]. If two double
degenerate bands cross each other at the discrete momentum point, such linearly dispersive
four-fold degenerate point is called Dirac point (DP)[19], whose low-energy excitation can be
described by the massless relativistic Dirac equation[20]. Following such guideline, several
3D DSMs have been proposed and confirmed experimentally in the nonmagnetic system[6,
8, 9, 21–28].
Generally, TRS and IS both are necessary to protect such four-fold degenerate DPs
in the nonmagnetic materials. Otherwise, the system will evolve into other exotic quan-
tum states such as Weyl semimetals[29–39] or topological insulators (TIs)[40–42]. However,
an exception is proposed in a specific antiferromagnetic (AFM) configuration of CuMnAs
recently[43], in which both P and T are broken but their combination PT is preserved. So
that Kramer’s degeneracy is reserved for every momentum k, and DPs can exist in such kind
of AFM system. Thence, a natural question is whether there exists other kind of DSMs in
solids.
In the present paper, we consider all the MSGs, and give an affirmative answer to this
question. Our studies demonstrate that the concept of DSM can be generalized to type IV
MSGs that breaks TRS T but preserves IS P and nonsymmophic time-reversal symmetry
(NTRS) T ′ = T τ , where τ is a translation operator that connect the black and white
Bravais lattices. Guiding by this idea, a concrete example of such DSM, the interlayer AFM
EuCd2As2, is predicted by our density functional theory (DFT) calculations. Many exotic
topological states can be derived from such AFM DSMs. When the three-fold rotation
symmetry C3z is broken, DSM phase can evolve into the AFM TI phase holding of the
half-quantum Hall effect as discussed by Joel Moore et al.[44]. In the case of breaking IS P ,
triple point semimetal (TPSM), rather than Weyl semimetal, can be stabilized. Our results
provide a methodology to search for DSMs in MSGs and an ideal candidate to study the
DSM and the other exotic topological states.
2
RESULTS
New type of Dirac semimetal in type IV MSGs.
In order to search for new type of DSMs, we have checked all types of MGs, among which
only the gray MSGs (also known as type II) and the black-white MSGs (including type III
and type IV) are suitable to construct the crystalline time-reversal symmetry (CTRS) [45],
which is important to realize the Kramer’s degeneracy and the DSMs. The general CTRS
operation can be defined as T ′ = T {R|τ}, where {R|τ} means a space group operation. In
the gray MSGs, {R|τ} equals to the identical operation e, and T ′ returns to the ordinary
TRS T , which describe the nonmagnetic and paramagnetic lattices. In type IV MSGs based
on the black-white Bravais lattice, there is no rotational operation {R} in CTRS, but just
a translation operation τ connecting the black and white sublattices. So that the CTRS
becomes a nonsymmophic form T ′ = T τ , and we name it as the NTRS[46] to distinguish the
other CTRS in this paper. Lastly, the rest MSGs holding an ordinary T {R|τ} are classified
as type III, which are based on the ordinary Bravais lattice.
The presence of the combined operation PT ′ whose square equals to −1 is essential
for the realization of DPs in solids, because it guarantees the Kramer’s degeneracy at the
generic momentum k. There are three categories of MSGs satisfying such requirement: the
centro-symmetric gray MSGs where both IS P and TRS T present (Category1), the type III
MSGs where {R|τ} is assumed as P (Category2), i.e. the CTRS T ′ = PT , and the centro-
symmetric type IV MSGs satisfying both P and T ′ = T τ (Category3). The classification
of DSMs in the centro-symmetric gray MSGs has been well discussed by Nagaosa et al.[11],
and the AFM DSM CuMnAs discussed by Tang et al. [43] belongs to Category2. In this
work, we focus on the centro-symmetric type IV MSGs, and complete the classification of
the DSMs in Category3.
Similar to the classification in the gray MSGs, there are two classes of DPs in the centro-
symmetric type IV MSGs, DPs created on the rotation axis of the momentum (Class-I)
and DPs created at the time reversal invariant momentum (TRIM) points (Class-II). In
the presence of an additional rotation symmetry Cn, the classification of Class-I DSMs in
the centro-symmetric type IV MSGs is homologous to the discussion given by Nagaosa et
al.[11], because the little group along the rotation momentum-axis is isomorphic with that
3
in the gray MGs, both satisfying Cnn = −1, (PT ′)2 = −1 and [Cn,PT ′] = 0. As an
illustrative example of the Class-I DSM in type IV MSGs, the interlayer AFM EuCd2As2
will be addressed in this paper, and its derived exotic states, such as AFM TI and TPSM
will be discussed too.
For DPs at TRIM points (Class-II), two cases depending on the value of k · τ need
to be distinguished. Case1, DPs at the TRIM points with k · τ = 0 are identical to the
discussion in the Ref. [10], since the little group at such kind of TRIM points in type IV
MSGs is isomorphic with that in the gray MSGs, i.e., P2 = 1, T ′2 = −1 and [P , T ′] = 0.
However, for the TRIM points with k · τ = π/2 (Case2), the algebraic relations between
P and T ′ are novel and particular, P2 = 1, T ′2 = 1, {P , T ′} = 0 and (PT ′)2 = −1,
where the minus sign of (PT ′)2 comes from the anti-commutative relation {P , T ′} = 0.
To our best knowledge, such kind of algebraic relations have never been addressed in the
fermion system. So the conclusion that DPs can exist at Case2 TRIM points in Ref.[10]
is not applicable. In this paper, we just provide an example to illustrate that DPs are not
inevitable at Case2 TRIM points of type IV MSGs, but leave the general classification for
the (P , T ′)-anticommuting protected topological node states in the future work. In this
example, we assume the system holding IS P , NTRS T ′ = T {(0, 0, 1/2)} and an additional
screw symmetry C ′2x = {C2x|(1/2, 0, 0)}. It is easy to certify that four-fold degenerate
node line is realized at TRIM points (π, 0, π) and (π, π, π), while there is no any node
existing around TRIM points (0, 0, π) and (0, π, π). Taking (π, 0, π) as example, the little
group satisfies P2 = 1, C ′22x = 1, T ′2 = 1, {P,C ′2x} = 0, {P , T ′} = 0 and {C ′2x, T ′} = 0.
Then it is direct to prove that the irreducible representation of such little group must be
four-dimensional. Concretely, we give an explicit irreducible representation as: P = sy,
C ′2x = sxσy, T ′ = K, where K is the complex conjugation operator, and sx,y,z, σx,y,z are
the Pauli matrices describing the orbital and spin degrees of freedom, respectively. Such
symmetry restriction would result in a four-fold degenerate node line in the kx = π plane.
AFM Dirac semimetal EuCd2As2.
Guided by the DSM classification of the MSGs, we discover that one pair of Class-I
DPs (belonging to type IV MSGs) can be hosted at the Fermi level in the interlayer AFM
EuCd2As2. As shown in Fig. 1a, EuCd2As2 crystallizes into the CaAl2Si2-type structure
4
TABLE I. Total energies of four different magnetic structures for EuCd2As2 calculated by
GGA+SOC+U. The converged magnetic moments of each Eu atom are given too.
Config. Eu1(µB) Eu2 (µB) Eu3 (µB) Energy (eV/f.u.)
NM (0, 0, 0) (0, 0, 0) (0, 0, 0) -18.021
FM (0, 0, 6.88) (0, 0, 6.88) (0, 0, 6.88) -24.368
AF(L) (0, 0, 6.88) (0, 0,-6.88) -24.370
AF(F) (3.44,5.95, 0) (-6.87, 0, 0) (3.44, -5.95,0) -24.365
with space group D33d (P 3m1) [47–49], in which Cd2As2 layers are separated by the trigonal
Eu layers. Considering that Eu2+ has a half-filled 4f -shell, we have calculated three different
magnetic configurations for EuCd2As2 by the generalized gradient approximation (GGA) +
Hubbard U (GGA+U) method with U = 5 eV, including the ferromagnetic (FM), frustrated
AFM (AF(F) in Table 1) and the interlayer AFM (AF(L) in Table 1) configurations as
shown in Fig. 1a, 1b and 1c, respectively. The calculated total energies and moments are
summarized in Table 1, which demonstrates that all magnetic states are lower than the
non-magnetic (NM) state about 6.3 eV/ f.u., and the interlayer AFM is the most stable one,
further lowering the total energy about 2 meV/f.u. than the FM states, which are consistent
with recent experiment measurement very well[50].
The projected band structures of the interlayer AFM EuCd2As2 are shown in Fig. 1d.
Our DFT calculations indicate that the low-energy bands near the Fermi level are mainly
contributed from the p orbitals of As atoms and the s orbitals of the Cd atoms. In particular,
the double degenerate s-s bonding states of Cd atoms (even parity) invert with the p-p anti-
bonding states of As atoms (odd parity) at the Fermi level around the Γ point. For the first
glance, it is a little surprising that the crossing along Γ − A line has no gap but a stable
four-fold degenerate DP, because TRS T is broken. However, after a detailed symmetry
analysis, one can find a NTRS T ′ = T ⊕ c, connecting the up spin momentum layer at z = 0
and the down spin momentum layer at z = c, exists in this interlayer AFM system. The
MSGs of the interlayer AFM EuCd2As2 can be expressed as D43d⊕T ′D4
3d, whose generators
include T ′, IS P , three-fold rotation C3z and two-fold screw C ′2x = C2x ⊕ c. Combining
the NTRS T ′ = T ⊕ c with the IS P together, the antiunitarity of PT ′ would prohibit the
hopping terms between the nonsymmophic time-reversal pair of states, such as |3/2,±3/2〉
5
or |1/2,±1/2〉. So that every energy state is double degenerate in such interlayer AFM
system. As we discussed above, it is possible to host the DPs with the help of the proper
rotation symmetry, which is exactly what happened on the Γ−A line in the interlayer AFM
EuCd2As2.
The little group on the Γ − A line can be described as C3v ⊕ PT ′C3v. When spin-orbit
coupling (SOC) is included, the topology of the system and the band inversion are dominated
by the four states |3/2,±3/2〉− from the p − p antibonding state of As and |1/2,±1/2〉+
from the s − s bonding state of Cd. Under the symmetry restrictions, an effective 4 × 4
k · p model (in the order of |1/2, 1/2〉+, |3/2, 3/2〉−, |1/2,−1/2〉+, |3/2,−3/2〉−) around the
Γ point can be written in the following form,
H = ε0(k) +
M(k) Ak+ 0 Bk+
Ak− −M(k) Bk+ 0
0 Bk− M(k) −Ak−Bk− 0 −Ak+ −M(k)
where ε0(k) = C0 +C1k
2z +C2(k
2x +k2y), k± = kx± iky and M(k) = M0−M1k
2z −M2(k
2x +k2y)
with parameters M0,M1,M2 < 0 to guarantee the band inversion. We note that our effective
model is very similar to the Hamiltonian in Na3Bi, except that the off-diagonal terms take
the leading order Bk± rather than high-order Bkzk2± as in Na3Bi. By diagonalizing the above
Hamiltonian, we can get two double degenerate egenvalues E+ = ε0 + ∆ and E− = ε0 −∆
with ∆ =√
(A2 +B2)(k2x + k2y) +M(k)2. Evaluating the eigenvalues, it is clear that two
linear DPs at kc = (0, 0,±√M0/M1) exist on the Γ − A line. This fact is confirmed by
the calculated surface states (Fig. 1e,1f) and Fermi arcs (Fig. 1g) based on the Green’s
functions[51, 52] of the semi-infinite system. For this purpose, the maximally localized
Wannier function (MLWF) method[53] from the DFT calculations have been constructed.
The (001) surface states plotted in Fig. 1e exhibit a clear band touching at the Γ point and
Fermi level, where one pair of DPs are projected to the same point on the (001) face. More
evidence supporting the DSM phase comes from the Fermi arcs on (010) face as plotted
in Fig 1f and Fig 1g, where one pair of arc states connect the DPs from the bulk state
unambiguously. As shown in the inset of Fig. 1g, even though the Fermi arcs look like
closed, their Fermi velocities are discontinuous at the DPs. Such kind of closed Fermi arcs
in DSMs have been discussed by Zhijun Wang et al. [8] and Peizhe Tang et al.[43].
6
TI phase and triple point semimetal in AFM system.
In addition to the generic features discussed above, our AFM DSM has its own uniqueness.
Such uniqueness can be reflected by its derivatives, which makes our AFM DSM different
from Na3Bi and CuMnAs.
We would like to discuss the derivative TI phase from the AFM DSM EuCd2As2 first.
As pointed out by Zhijun Wang et al. [8] and Nagaosa et al.[11], three-fold rotation C3z
is important for the stability of the DPs in the interlayer AFM EuCd2As2. When this
symmetry is broken, jz is no longer the good quantum number along Γ − A line. So that
the hopping terms between |jz = ±1/2〉 and |jz = ±3/2〉 can be introduced, and the system
evolves into the strong TI phase due to the inverted band structure. In Fig. 2a, we have
plotted the band structures by enlarging the a-axis 1%, where an insulating gap of 9 meV
opens up clearly.
The TI phase realized in Fig. 2a has unique topological properties. It is not the con-
ventional 3D TI as Bi2Se3, but the AFM TI (AFTI) protected by T ′ as discussed by Joel
Moore et al.[44]. To illustrate the difference, one can see the (001) surface states plotted
in Fig. 2c, where the surface states are intrinsically gapped at the Γ point, because the T ′
symmetry is broken when open boundary is applied along the c-axis. However, when open
boundary is applied along the other directions, such as a-axis, where T ′ is preserved, the
surface states remain gapless as that in the conventional TI (see Fig. 2d). These characters
conform with the discussion of the AFTI [44] exactly, which can be taken as a product state
of the 2D Chern insulators stacked along the c-axis, and each pair of the nearest neighbors
are connected by the NTRS T ′ = T ⊕ c (Fig. 2e). Therefore, a nontrivial AFM Z2 invari-
ant related to T ′ can be defined, and the half-quantum Hall effect can be realized on the
intrinsically gapped (001) face of such AFTI (Fig. 2c).
The other big difference between our AFM DMS and the other DSMs (Na3Bi and CuM-
nAs) can be reflected by breaking IS P . As we all know, the DPs usually split into two pairs of
Weyl points with opposite chirality in the conventional DSM when IS P is broken. However,
two pairs of triple points are obtained in our AFM DMS when IS P is broken. Such results
are plotted in Fig. 2b, in which |jz = ±3/2〉 states are split, while the |jz = ±1/2〉 states
remain double degenerate. The origin of the TPSM realized in the P breaking EuCd2As2
can be understand as following. In absence of PT ′ symmetry, the little group along Γ− A
7
reduces to the magnetic point group C3v, which has a 2D irreducible representation E1/2
(| ± 1/2〉) and two one-dimensional irreducible representations E3/2 ( 1√2|3/2〉 ± i√
2| − 3/2〉).
Therefore the degeneracy between | ± 3/2〉 states, which is originally protected by PT ′, is
broken; while the degeneracy between | ± 1/2〉 remains, leading to two pairs of triple points
on the Γ − A line naturally. We calculate the Fermi surfaces of the TPSM phase and plot
them in Fig. 2f, where two pairs of tangent Fermi pockets exist, and each pocket encloses
one triple point. Similar to the nonmagnetic TPSM[54–56], two touching Fermi pockets hold
opposite spin winding number. Finally, it’s worthy to note that this is the first reported
TPSM in the magnetic material. which provides an ideal platform to study the interplay
between triple point and magnetic order.
CONCLUSION
In summary, we have studied all the MSGs, and generalized the concept of DSM to the
centro-symmetric type IV MSGs, where the antiunitarity of the product between IS and
NTRS, i.e. (PT ′)2 = −1 is essential for the Kramer’s degeneracy and the AFM DPs. Based
on our DFT calculations, we propose that the interlayer AFM EuCd2As2 is a candidate
of such AFM DSM. Many exotic topological states can be derived from the AFM DSMs.
For example, when the three-fold rotation symmetry is broken, it can evolve into the AFTI
discussed by Joel Moore et al.[44], where the half-quantum Hall effect can be realized on
the intrinsically gapped (001) face. If the IS P is broken, it can result in the TSM phase,
rather than the Weyl semimetal. Our results provide a new direction to study the DSM and
the other exotic topological states.
Note added. The evidence of a Dirac-cone type dispersion in EuCd2As2 has been observed
recently in the angle-resolved photoemission spectroscopy (ARPES) measurements by Ma
et al.[61], which greatly support our theoretical prediction.
Methods
The DFT calculations are carried out using the projector augmented wave method im-
plemented in Vienna ab initio simulation package (VASP)[57, 58]. Perdew-Burke-Ernzerhof
type[59] of GGA + Hubbard U (GGA + U) approach[60] with U = 5 eV on Eu’s 4f orbitals
is used to treat with the exchange and correlation potential. SOC is taken into account
8
self-consistently. The cut-off energy of 500 eV and the 10 × 10 × 5 sampling of Brillouin
Zone are used and carefully checked to ensure the convergence. MLWFs for the p orbitals
of As and s orbitals of Cd have been generated and used to calculate the surface states
iteratively[51, 52].
∗ These authors contributed equally to this work.
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Acknowledgements We thank Tian Qian and Yulin Chen for useful discussions.
G. X. and R. Y. is supported by the National Thousand-Young-Talents Program and the
the National Natural Science Foundation of China. S. N. are supported by Stanford Energy
3.0.
Author contributions G. X., Z.-D. S. and R. Y. conceived and designed the
project. G.-Y. H. and S.-M. N. performed all the DFT calculations, G. X. and Z.-D. S. did
the theoretical analysis. All authors contributed to the manuscript writing.
Competing financial interests The authors declare no competing financial in-
terests.
Correspondence Correspondence and requests for materials should be addressed to
Z.-D Song (email: [email protected]), R. Yu. (email: [email protected]), G. Xu (email:
14
ΓZ Z'
Ene
rgy
(meV
)
0
10
-20
-10
ΓM K
Ene
rgy
(eV
)
0
0.2
-0.2
0
0.02
-0.02
-0.004 0.0040k
y (1/Å)
k z (1/
Å)
0
1
2
-2
-1
a
b
c d
e
f
g
a
b
a b
c
Eu
Cd
As
c
√3a
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c
c0.1
−0.1
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Ene
rgy
(eV
)
0.1
−0.1
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|1/2,±1/2>+
Figure 1. | Crystal structure and DPs in the interlayer AFM EuCd2As2. (a) The side
view of the FM EuCd2As2. The blue, pink and light green balls represent Eu, Cd and As atoms,
respectively. (b) The top view of the frustrated AFM EuCd2As2 with a√
3a×√
3a reconstruction.
(c) The side view of the interlayer AFM EuCd2As2. The red arrows in 1a, 1b and 1c mean the
directions of the magnetic momentum. (d) The GGA+U+SOC calculated band structures of the
interlayer AFM EuCd2As2. The insets are the zoomin of the band structures around the Γ point
to show the DP obviously, where the red and the light blue dots represent the projections of the As
p and Cd s orbitals, respectively. (e), (f) are the calculated surface states of the interlayer AFM
EuCd2As2 on the (001) and (100) faces, respectively. (g) The Fermi arcs and their spin textures
on the (100) face of the interlayer AFM EuCd2As2, where the inset reveals the discontinuity at the
DPs between two Fermi arcs.
15
Ener
gy (
meV
)0
30
60
-30
-60
ΓY Z
Ener
gy (
meV
)
ΓM K
0
20
-20
A
Г M K ГA AL H
Ener
gy (
eV)
0
1
2
-2
-1
a
b
c
d
e
f
Г M K ГA L H
Ener
gy (
eV)
0
1
2
-2
-1
0.03
−0.03
0
−
+
c
|3/2,±3/2>
|1/2,±1/2>
0.03
−0.03
0−
+
K
H
M
A L
Г
Figure 2. | The derived exotic topological states from the AFM DSM EuCd2As2.
(a), (b) The GGA+U+SOC calculated band structures of the interlayer AFM EuCd2As2 in the
case of C3z breaking (AFTI) and IS P breaking (TPSM), respectively. The insets are the zoomin
of band structures along the Γ − A direction to exhibit the insulating gap in 2a and the triple
points in 2b distinctly. (c), (d) The calculated surface states of the AFTI on the (001) and (100)
faces, respectively. The inset of 2c shows the intrinsic gapped surface states clearly. (e) Schematic
of the AFTI by stacking the 2D Chern insulators along the c-axis, where the red and blue balls
represent the up spin layers (Chern number C = 1) and down spin layers (C = −1), respectively.
(f) The calculated Fermi surfaces of TPSM derived from the interlayer AFM EuCd2As2 with IS
P breaking, where the Fermi surfaces are magnified 60 times to exhibit them and their tangency
visibly.
16