Magnetic Dirac Semimetals in Three Dimensions

3.3 MB·

by  Jing Wang

Magnetic Dirac Semimetals in Three Dimensions
Jing Wang1,2
1State Key Laboratory of Surface Physics and Department of Physics, Fudan University, Shanghai 200433, China
2Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China
(Dated: January 5, 2017)
We present a new type of three-dimensional essential Dirac semimetal with magnetic ordering.
TheDiracpointsareprotectedbythemagneticspacegroupsandcannotbegappedwithoutlowering
such symmetries, where the combined antiunitary symmetry of half-translation operator and time-
reversalplaysanessentialrole. Weintroducetwoexplicittight-bindingmodelsforspacegroups16
and 102, which possesses Dirac point at time-reversal-invariant momenta of surface Brillouin zone.
7
In contrast to the time-reversal-invariant essential Dirac semimetal, the magnetic space groups
1
here can be either symmorphic or non-symmorphic, and the magnetic DSM is symmetry tuned to
0
the boundary between weak topologically distinct insulating phases. Interestingly, the symmetry-
2
breaking perturbations could lead to an ideal Weyl semimetal phase with only two minimal Weyl

n pointspinnedexactlyattheFermienergyforfillingν ∈4Z+2. Byreducingthedimensionalitywe
a are able to access the Dirac and Weyl semimetal phases in two dimensions.
J

4
Theexperimentaldiscoveryofthetime-reversalinvari- of a primitive-lattice vector. We introduce two explicit

] anttopologicalinsulators(TIs)[1,2]hasinspiredintense tight-bindingmodelsforSGs16and102,whichpossesses
l
l research interest in the symmetry-protected topological DP at time-reversal-invariant momenta (TRIM) of sur-
a
h phases of matter. Recently, great attention has been face BZ with S2 = −1. Like Θ-invariant essential DSM,
- given to the topological semimetals in various spatial di- the magnetic DSM is symmetry tuned to the boundary
s
e mensions, which have nontrivial surface states (SSs) and betweentopologicallydistinctinsulatingphases. Wecon-
m may be characterized by Fermi surface topological in- clude with a brief discussion of the measurable conse-
. variants [3–6]. The three-dimensional (3D) semimetals quences and possible material venues for these phases.
t
a can be classified in terms of degeneracies of the crossing
Ina3DΘ-invariantsystem,DSMemergesatthetran-
m
points in the electronic structure, which includes Dirac
sition between a normal insulator (NI) and a TI or weak
- semimetal(DSM),Weylsemimetal(WSM),DoubleDSM TI/topological crystalline insulator (TCI) [11, 16]. The
d
and Spin-1 WSM [7–30]. Among them, DSM is particu-
n DP is further stabilized by crystallographic symmetries,
o larly interesting because it is the parent state of various where Kramers degeneracy from Θ plays a key role. In
c exotic quantum states such as TI and WSM. The DSM a 3D Θ-broken magnetic system, a new Z invariant can
[ hosts massless Dirac fermions with linear energy disper- 2
bedefinedina2DBZwhenS-symmetryispresent,sepa-
1 sions as the low-energy excitation, where the conduction ratingaNIfromAFMTI[39,40]. Therefore,amagnetic
v andvalencebandscontactonlyattheDiracpoints(DPs) DSMisexpectedtoariseatthephaseboundary. Tohave
6 intheBrillouinzone(BZ).TheDSMmaterialshaveboth
anintuitivepicture,wefirstconsiderthesufficientcondi-
9
the time-reversal (Θ) and inversion (I) symmetries [7–
tion for the existence of a fourfold degeneracy at certain
8
15], which fall into two distinct classes. The first class is
0 k in the BZ. For high symmetry k points which are left
0 thetopological DSMsuchasCd3As2 whichisinducedby invariantunderspatialoperationsG andG ,ifforexam-
1 2
1. band inversion and locally permitted by crystalline sym- ple, G1,G2 satisfy {G1,G2} = 0 and G12 = ±1, G22 = ±1,
metries,whilethesecondclassistheessential DSMsuch
0 there exists a twofold degeneracy. The Θ symmetry will
7 as BiO2 where the nodal features are filling-enforced by set constraints at TRIM. If {Θ,G }={Θ,G }=0, there
1 2
1 specific space group (SG) symmetries [31–35].
exitsafourfolddegenracy. Thisisevidentbyconsidering
:
v TheΘbreakingingeneralsplitsaDPintoWeylpoints an eigenstate of G1 satisfying G1ψ = λψ, then G2ψ, Θψ
i andΘG ψ arealsoeigenstatesofG witheigenvalues−λ,
X and a WSM is obtained. The concept of magnetic DSM 2 1
−λ∗ and λ∗. All of these eigenstates are orthogonal to
in 2D has been introduced recently [36, 37]. The goal in
r
a thispaperistoexplainhowamagneticDSMwithbroken eachotherduetodifferenteigenvaluesandantiunitaryΘ.
In fact, this is the case for DSM predicted in BiZnSiO
Θsymmetrycanneverthelessexistin3D.Quitedifferent 4
where G ={M |011} and G ={I|110} [10].
from the antiferromagnetic (AFM) DSM considered in 1 zˆ 22 2 22
Ref. [38], where DPs are created through band inversion Now in a Θ-broken but S-invariant system, S is an-
and belongs to the first class. Here the magnetic DSM is tiunitary like Θ, and the Hamiltonian is S-invariant as
essential,wherethebulkDPisprotectedby(eithersym- S H(k)S−1 =H(−k). However,thereisakeydifference:
k k
morphic or non-symmorphic) magnetic SG and cannot while Θ2 = −1 for the spin-1/2 system, S2 ≡ S S =
−k k
be gapped without lowering such symmetries. An essen- Θ2T = −T . Therefore, S2 = −1 only at the TRIM
2d 2d
tial role is played by the combined symmetry of Θ and satisfying k·d = nπ where the Kramers’ degeneracy is
translation operator, namely S = ΘT , where d is half preserved, but S2 = +1 at k·d = (n+ 1)π where the
d 2
2
bands are generically nondegenerate. Therefore, a four- (a) (b) k
z
folddegeneracyisguaranteedtoexistatthekpointwith
Ζ T
k·d=nπ, if either (+,−)
U R
(−,−) y
{G1,G2}={S,G1}={S,G2}=0, (1) (+,+)x Γ Y ky
z Χ Μ
with G12 =G22 =±1; or (−,+) kx
{S,G }=[S,G ]=0, G G G =G , (2) (c)
1 2 1 2 1 2
1.0
with G4 =−1 the fourfold rotation.
1 t
ModelA.Wenowstudyanexplicittight-bindingmodel E0.0
for a 3D magnetic DSM to illustrate the symmetry-
−1.0
protected DPs listed in Eq. (1). The lattice has an
orthorhombic primitive structure of SG 16 (Shubnikov Γ Χ Μ Y Γ Ζ U R Τ ΖY ΤU ΧΜ R
group P2(cid:48)2(cid:48)2) as shown in Fig. 1(a). The lattice vectors
are (cid:126)a1 = (100), (cid:126)a2 = (010), (cid:126)a3 = (001). The system has FIG. 1. (color online). (a) The lattice for the orthorhombic
a layered structure with four sublattices in one unit cell, primitive structure of SG 16. The four sublattices are de-
indexed by (τz,σz)=(±1,±1) associated with the basis notedassolidcirclesandlabeledby(τz,σz)=(±1,±1). The
vectors t =−τ σ (100)−σ (010)−τ (001). The sym- magneticmomentsarealong±yˆdirection. (b)BZ.Thegreen
0 z z 4 z 4 z 4 solid circles are TRIM with k·d = nπ. (c) Energy band of
metrygeneratorsandtheirrepresentationsinthesublat-
the AFM system in (a), which is described by the model in
tice space are {C |000} = τ σ and {C |000} = τ σ .
2xˆ x x 2yˆ x z Eq.(3),witht=1.0,λ =0.25,λ =0.6,λ =0.4,λ =0.2,
1 2 3 4
Each lattice site contains an s orbital, which in general
λ = 0.4, and λ = 0.15. Two DPs at M and R are pro-
5 6
leads to an eight-band model. We then introduce the tected by the magnetic SG in Eq. (1). Here only the upper
AFM ordering along ±yˆ direction, thus the system re- fourbandsofaneight-bandmodelisshown. Thebandalong
spects S symmetry associated with d = (110), which is M-R is dispersionless when λ6 =0.
22
represented by S = eik·diσ K, and K is complex con-
y
jugation. We further assume AFM interaction is much
M and R are protected by S, {C |000} and {C |000},
2xˆ 2yˆ
stronger than hopping and spin-orbit coupling (SOC)
which can be seen by examining the effective model near
terms, therefore the system is decoupled into two four-
these points. However, they need not be at the same
bandmodelsastime-reversalpartners,eachwithonespin
energy. The representations of symmetry operations at
per sublattice. The upper subsystem consists of | ↑ (cid:105),
++ M are, {C |000} = iτ σ , {C |000} = iτ and S =
2xˆ x y 2yˆ y
| ↑ (cid:105), | ↓ (cid:105) and | ↓ (cid:105), with ↑ and ↓ denoting the
−+ +− −− iτ σ K. Therefore,thegenerick·pHamiltonianatM is
z y
spin up and down states, respectively. Here, we consider
agenericbutsimplifiedmodelwhichrespectsallthesym- HM =(u τ −u τ σ +u τ σ )k
a 1 x 2 z x 3 z z x
metries and are sufficient to characterize all the essential +(v τ σ +v σ +v σ )k +w τ σ k , (4)
1 y y 2 z 3 x y 1 z y z
degeneracies of the band structure. The Hamiltonian is
which leads to the dispersion
k k k k
H =tτ cos x cos z +λ τ cos x sin z
a x 2 2 1 y 2 2 E2 (k)=|u|2k2+|v|2k2+|w|2k2±2k k |u×v|. (5)
a,M± x y z x y
k k k k
+λ2τzσxcos 2x sin 2y +λ2σxsin 2x cos 2y where u ≡ (u1,u2,u3), v ≡ (v1,v2,v3) and w ≡
(w ,0,0). When u ⊥ v, i.e., |u×v| = |u|×|v|, one of
k k k k 1
+λ3τyσycos 2y cos 2z +λ4τxσycos 2y sin 2z the branches vanishes on the line |u|kx = |v|ky, kz = 0.
Fromthetight-bindingmodel,wehaveu=(−t,λ2,λ5),
+λ5σzcoskxsinky+λ5τzσzsinkxcosky 2 2 2
+λ τ σ sinkx sinky sink . (3) svy=mm(−etλr23y,-pλ2r5o,t−ecλ2t2e)d,sDoPasaltonMg.asLtikλe2w(cid:54)=ise0,,athteRre,wthilelbreepa-
6 z y 2 2 z
resentations of symmetry operations are, {C |000} =
2xˆ
Here t describes the nearest neighbor hopping, where we iτ σ , {C |000}=iτ and S =iσ K. The k·p Hamil-
y y 2yˆ x y
fixthegaugewithinthesublatticesuchthatHa(k+G)= tonian at R is
e−iG·t0(τz,σz)Ha(k)eiG·t0(τz,σz). λiisSOCwhichinvolves
spin-dependent nearest (i=1,2,3,4), third nearest (i= HaR =(u(cid:48)1τy−u(cid:48)2τzσx+u(cid:48)3τzσz)kx
5), and fourth nearest (i=6) neighbor hopping. +(−v(cid:48)τ σ +v(cid:48)σ +v(cid:48)σ )k +w(cid:48)τ σ k . (6)
1 x y 2 z 3 x y 1 z y z
Fig.1(c)showsenergybandassociatedwithH ,which
a This leads to the dispersion E2 (k) = |u(cid:48)|2k2 +
features two inequivalent DPs at M and R with linear a,R± x
|v(cid:48)|2k2 + |w(cid:48)|2k2 ± 2k k |u(cid:48) × v(cid:48)|, where from the
dispersion along all directions. Moreover, we find that y z x y
there are two additional Weyl points along the line Z-U model (3), we have u(cid:48) ≡ (u(cid:48),u(cid:48),u(cid:48)) = (−λ1,λ2,λ5),
1 2 3 2 2 2
when2(λ1/λ2)(λ5/λ4)=(λ1/λ2)2+1. ThesetwoDPsat v ≡ (v1(cid:48),v2(cid:48),v3(cid:48)) = (λ24,λ25,−λ22) and w ≡ (w1(cid:48),0,0), so
3
RAep1s Pertur1bations PDhSaMses # of2Nodes Χ ΓΖkzΜ YT (a) Y Μ Ζ kz T (bZ) Z Μ
B1 τz WSM→NI 4→0 U R U R Μ Γ
τyσx,τxσx WSM 4 kx Χ Γ Μ Y ky Γ Χ kx Χ Γ Μ Y Χ ky Γ Χ
τ σ ,τ σ WSM 4
y z x z
1.0 1.0
B τ ,τ WSM→AFM TCI 4→2→0
2 x y
B3 τyσy,τxσy WSM→AFM TCI 4→2→0 Et0.0 Et0.0
−1.0 −1.0
Γ Χ Μ Γ Y Μ Γ Χ Μ Γ Z Μ
TABLE I. Perturbations to the DP of Model A in SG 16,
classified by their symmetry under D point group [41]. The
2
resultinginsulatingandsemimetallicphases(with#ofnodes) FIG.3. (coloronline). EnergybandsforModel Ainthethin
asafunctionoftheincreasedperturbationstrengthareindi- filmlimit,with2DBZshownabove. (a)Thebilayerstructure
cated. along[001]direction. ThesingleDPatM¯ ismarkedasagreen
dot. (b) The bilayer structure along [010] direction preserves
{C |00}and{C |00}butbreaksS,theDPssplitintoWeyl
2xˆ 2yˆ
there are no symmetry respecting terms at R that could points (marked as yellow dots).
lift the degeneracy.
We further consider the symmetry-breaking perturba-
serves {C |000}. Such reduced symmetry adds a term
tions which may lead to a wealth of topological phases. 2yˆ
Ha =γ sin(k /2)[sin(k /2)τ +cos(k /2)τ ]. Whenγ is
The general S-invariant perturbations are listed in Ta- 2 2 x z x z y 2
small, the DP at M (R) splits into a pair of Weyl points
ble I. Adding these mass terms results in either insulat-
locatedalongthelineM-X (R-U)asshowninFig.2(a).
ing or WSM phases. Take τ for example, it breaks both
z
Withincreasedγ ,theWeylpointsalongU-Rannihilate
{C |000}and{C |000}whichallowsatermHa =γ τ . 2
2xˆ 2yˆ 1 1 z pairwisely, and the system becomes a minimal and ideal
It corresponds to a staggered on-site potential, which re-
WSM with only two Weyl points pinned to the Fermi
sults in WSM when γ is small. Each DP splits into
1 energy for filling ν ∈ 4Z+2. These fillings are allowed
a pair of Weyl points located along the line M-R, and
due to absence of multiple nonsymmorphic symmetries.
strong γ will further push the Weyl points annihilating
1
With further increased γ , these two Weyl points meet
pairwise,resultinginaNIwithoutgaplessSSs,regardless 2
andannihilateatX,resultinginabulkinsulatingphase.
of surface termination [42]. More interestingly, we con-
Such insulating phase is a AFM TCI with trivial Z in-
sideralatticedistortionwhichbreaks{C |000}butpre- 2
2xˆ
dex ν =0, which has gapless SSs on certain S-invariant
0
surfaces such as (110) plane. However, it is topological
(a) in a weaker sense than strong TI because these SSs are
1.0
not generally immune to disorder.
t We close this section by studying how the 3D DP
0.0
E
evolves by reducing dimensionality. The energy band for
−1.0 a [001]-oriented thin film is shown in Fig. 3(a). The 2D
Γ Χ Μ Y Γ Ζ U R Τ ΖY ΤU ΧΜ R latticerespectsS,{C2xˆ|00}and{C2yˆ|00},whichprotects
(b) asingle2DDPatM¯ followingthealgebrainEq.(1). In
1.0
fact, the 3D lattice is viewed as a magnetic layer group
t0.0 along [001] direction [41]. Therefore, the magnetic crys-
E
talline symmetry could protect the essential DP in both
−1.0 3D and 2D [37]. While for a [010]-oriented thin film in
(c) (110) plane (d) (110) plane Fig.3(b),thelatticepreserves{C2xˆ|00}and{C2yˆ|00}but
breaksS,theDPisnolongerprotectedbutsplitsinto2D
0.5 0.5
t t Weyl points [42]. This is in sharp contrast to topological
E E
0.0 0.0 DSM in the first class, where gapped quantum spin Hall
state can be obtained by dimension reduction [9].
−0.5 −0.5
Model B.Wemayconstructasimilarmodelwhichhas
Γ Χ Μ Γ Z Γ Χ Μ Γ Z
the symmetry-protected DPs described by Eq. (2). The
lattice has a tetragonal structure of SG 102 (P4(cid:48)n(cid:48)m).
2
FIG.2. (coloronline). SymmetrybreakingphasesofModelA.
As illustrated in Fig. 4(a), there are four sublattices in
(a) Bulk band when Ha is added with γ =0.3. (b) Minimal
2 2 a unit cell indexed by (τ ,σ )=(±1,±1) with the basis
and ideal WSM with γ = 0.7. (c) & (d) Energy dispersion z z
foraslabconfiguration2withγ2 =0.9on(110)&(1¯10)plane. vectorst0 =(1−τz)(41140)+(1−σz)(0014). Similarly,each
The (110) plane has gapless Dirac SSs with DPs located at latticesitecontainsansorbital. TheAFMinteractionis
surface TRIM Γ¯ and Z¯. strong with the ordering being along ±zˆ direction. S =
4
eik·diσyK with d = (21120). The system is characterized Reps Perturbations Phases
by the symmetry generators {C4zˆ|0012} = τxeiπσz/4 and A1 1 DSM
{M |111}=τ σ . Thegenericandsimplifiedmodelfor A τ σ WSM
xˆ 222 x y 2 x z
the upper subsystem is B τ NI
2 z
τ AFM TCI
k y
H =tτ cos z +t τ sink sink (7) E (τ σ ,τ σ ) WSM
b x 2 1 z x y x x x y
(cid:18) (cid:19)
k k k k
+λ τ σ cos x sin y +σ sin x cos y
1 z x 2 2 y 2 2 TABLE II. Perturbations to the DP of Model B in SG 102,
(cid:18) k k k k (cid:19) k classifiedbytheirsymmetryunderC4v pointgroup[41]. The
+λ2τx σxsin 2x cos 2y +σycos 2x sin 2y cos 2z resulting insulating and semimetallic phases are indicated.
(cid:18) (cid:19)
k k k k k
+λ3τy σxcos 2x sin 2y +σysin 2x cos 2y sin 2z Frommodel(7), wehaveζ =(λ21,λ23,0)andζ0 =0, soa
symmetry-protected DP exists at Z. A similar analysis
Here t describes the hopping, λ is SOC. Fig. 4(c)
i i appliestoA. Moreover, thesymmetry-breakingtopolog-
shows energy band associated with H , which features
b icalphasesofModel B arestudiedinTableII,wherethe
two inequivalent DPs at Z and A with linear disper-
perturbations are classified by their symmetry under the
sion. These two DPs are protected by S, {C |001} and
{kM·xpˆ|12m12o12d}e,lwnheiacrhtchaensebepofuinrttsh.erTseheenrbeypreexsea4znˆmtiantii2nognsthoef Cstoe4rvvaeplsao{tintCitceg|rd0o0ius1tpo}.r.tTiToahnkeewrτhexdiσcuhzcfebodrresaeyxkmasmm{Mpetlexrˆ,y|21iat12lcl12oo}wrrsbeusapttopenrrdmes-
s{yMmm|1e1tr1y}op=erτatiσonsanatdZSa=re,i{τCσ4zˆ|K00. 12}Th=erτexfoeirπeσzt/h4e, H1b =κ1s4inzˆ(kz2/2)τxσz, which leads to a WSM as shown
xˆ 222 x y z y in Fig. 4(d). It is worth mentioning that the lattice con-
generic k·p Hamiltonian at Z is
sidered here is not a magnetic layer group, therefore by
HZ =(ζ σ +ζ τ σ +ζ τ σ )k reducing the dimensionality we cannot access the 2D es-
b 0 x 1 z y 2 y y x
sential DSM phase.
+(ζ σ +ζ τ σ +ζ τ σ )k +η τ k , (8)
0 y 1 z x 2 y x y 1 x z
Discussion. The two models studied above provides
which leads to the dispersion an explicit extension of the relation between filling and
essentialnodalpoints[32–35]tomagneticSGs. Thecom-
E2 (k)=(|ζ|2+ζ2)(k2+k2)+|η|2k2
b,Z± 0 x y z plete study of such relation in 1651 magnetic SGs is left
(cid:113)
±2 ζ2|η|2(k2+k2)k2+4k2k2ζ2|ζ|2.(9) to future work. Note that in both models, there are two
0 x y z x y 0
symmetry-inequivalent DPs which is symmetry tuned to
where ζ ≡(ζ1,ζ2,0) and η ≡(η1,0,0). When |ζ|=|ζ0|, theboundarybetweendistinctTCIphases. Itispossible
one of the branches vanish on the line kx = ky, kz = 0. to have an intrinsic magnetic DSM with only symmetry-
equivalentDPsandnoadditionalstatesattheFermien-
ergy, which separates NI and AFM TI with a nontrivial
(a) (c)
Z index ν = 1 [39]. However, practically, constructing
1.0 2 0
suchamodelrequiresamorecomplicatedAFMordering.
t
(−,−) E0.0 Moreover, instead of S, there also exists magnetic DSMs
(−,+) protected by Θ¯ ≡ C Θ [43], which is beyond the scope
−1.0 n
z y of this paper.
(+,+)x (+,−) Γ Χ Μ Γ ΖΧ RΜ Α R Ζ Α Finally,wediscusssomeinterestingphysicsandexper-
(b) k (d)
z imentalconsequenceoftheessentialmagneticDSM.The
1.0
Ζ R nontrivial SSs [42] and spin-orbit texture of Dirac cones
t
A E0.0 can be directly measured by angle-resolved photoemis-
Γ
X ky sion spectroscopy. The broken Θ and finite orbital mag-
Μ −1.0
netic moments of Fermi surface in magnetic DSMs may
k
x Γ Χ Μ Γ ΖΧ RΜ Α R Ζ Α lead to a novel magnetopiezoelectric effect [44]. Macro-
scopically, S-symmetry implies a topological magneto-
FIG. 4. (color online). (a) Model lattice for the common electric response ∂P/∂B =(θ/2π)(e2/h) with θ (cid:54)=0 but
tetragonal primitive structure of SG 102. The four sublat- not necessarily quantized on certain ferromagnetic sur-
tices in a unit cell are denoted as solid circles and labeled by
faces with broken S, provided the surface spectrum is
(τ ,σ ) = (±1,±1). The magnetic moments are along ±zˆ
z z gapped. Moreover, the magnetic fluctuations in this sys-
direction. (b)BZ.ThegreendotsareTRIMwithk·d=nπ.
tem may lead a dynamical axion field [45, 46]. In terms
(c) Energy band for SG 102, which is described by the tight-
binding model of Eq. (7), with t = 1.0, t = 0.2, λ = 0.6, of realistic materials, the actual existence of such phases
1 1
andλ =λ =0.4. (d)Thesymmetry-breakingperturbation in known materials remains an open question. However,
2 3
Hb leads to WSM, with κ=0.5. unlikethebandinversioninducedtopologicalDSM,here
1
5
onlythespecificmagneticSGsymmetryisneededforthe [21] S.-M. Huang, S.-Y. Xu, I. Belopolski, C.-C. Lee,
essential magnetic DSM, which seems to be compatible G. Chang, B. Wang, N. Alidoust, G. Bian, M. Neupane,
with the narrow band width from AFM d- or f-orbitals. C. Zhang, S. Jia, A. Bansil, H. Lin, and M. Z. Hasan,
Nat. Commun. 6, 7373 (2015).
We are thus conservatively optimistic about the experi-
[22] H.Weng,C.Fang,Z.Fang,B.A.Bernevig, andX.Dai,
mental prospects of 3D essential magnetic DSMs.
Phys. Rev. X 5, 011029 (2015).
We acknowledge Naoto Nagaosa and Zhong Wang
[23] S.-Y. Xu, I. Belopolski, N. Alidoust, M. Neupane,
for valuable discussions. This work is supported by G. Bian, C. Zhang, R. Sankar, G. Chang, Z. Yuan, C.-
theNationalThousand-Young-TalentsProgram; theNa- C. Lee, S.-M. Huang, H. Zheng, J. Ma, D. S. Sanchez,
tional Key Research Program of China under Grant B. Wang, A. Bansil, F. Chou, P. P. Shibayev, H. Lin,
No.2016YFA0300703;theOpenResearchFundProgram S. Jia, and M. Z. Hasan, Science 349, 613 (2015).
[24] B. Q. Lv, H. M. Weng, B. B. Fu, X. P. Wang, H. Miao,
of the State Key Laboratory of Low-Dimensional Quan-
J.Ma,P.Richard,X.C.Huang,L.X.Zhao,G.F.Chen,
tum Physics, through Contract No. KF201606; and by
Z.Fang,X.Dai,T.Qian, andH.Ding,Phys.Rev.X5,
Fudan University Initiative Scientific Research Program.
031013 (2015).
[25] A.A.Soluyanov,D.Gresch,Z.Wang,Q.Wu,M.Troyer,
X. Dai, and B. A. Bernevig, Nature 527, 495 (2015).
[26] Y. Sun, S.-C. Wu, M. N. Ali, C. Felser, and B. Yan,
Phys. Rev. B 92, 161107 (2015).
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