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.

† song.zhida@iphy.ac.cn

‡ yurui@whu.edu.cn

§ gangxu@hust.edu.cn

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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: song.zhida@iphy.ac.cn), R. Yu. (email: yurui@whu.edu.cn), G. Xu (email:

gangxu@hust.edu.cn).

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

√3a

aГ M K ГA AL H

c

c0.1

−0.1

0

|3/2,±3/2>−

|1/2,±1/2>+

Ene

rgy

(eV

)

0.1

−0.1

0|3/2,±3/2>−

|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