Spin-orbit-induced exotic insulators in a three-orbital Hubbard model with $(t_{2g})^5$ electrons

0.59 MB·English

by  Toshihiro Sato

Spin-orbit-induced exotic insulators in a three-orbital Hubbard model
with (t )5 electrons
2g
Toshihiro Sato1, Tomonori Shirakawa1,2, and Seiji Yunoki1,2,3
1Computational Condensed Matter Physics Laboratory, RIKEN, Wako, Saitama 351-0198, Japan
2Computational Materials Science Research Team,
RIKEN Advanced Institute for Computational Science (AICS), Kobe, Hyogo 650-0047, Japan
3Computational Quantum Matter Research Team,
RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan
(Dated: February 3, 2015)
5
On the basis of the multi-orbital dynamical mean field theory, a three-orbital Hubbard model
1
0 with a relativistic spin-orbit coupling (SOC) is studied at five electrons per site. The numerical
2 calculations are performed by employing the continuous-time quantum Monte Carlo (CTQMC)
method based on the strong coupling expansion. We find that appropriately choosing bases, i.e.,
n
the maximally spin-orbit-entangled bases, drastically improve the sign problem in the CTQMC
a
calculations, which enables us to treat exactly the full Hund’s coupling and pair hopping terms.
J
This improvement is also essential to reach at low temperatures for a large SOC region where the
1
SOC most significantly affects the electronic structure. We show that a metal-insulator transition
3
is induced by the SOC for fixed Coulomb interactions. The insulating state for smaller Coulomb

interactions is antiferromagnetically ordered with the local effective total angular momentum j =
]
l 1/2, in which the j = 1/2 based band is essentially half-filled while the j = 3/2 based bands are
e completely occupied. More interestingly,for larger Coulomb interactions, we findthat an excitonic
-
r insulatingstateemerges,wherethecondensationofanelectron-holepairinthej =1/2andj =3/2
t based bandsoccurs. Theorigin of theexcitonic insulator as well astheexperimentalimplication is
s
. discussed.
t
a
m PACSnumbers: 71.27.+a,71.30.+h,75.25.Dk
-
d
Recent experiments have reported interesting obser- also by the multi-orbital nature. Therefore, a question
n
o vations for 5d transition metal Ir oxides in the layered naturallyarises: whatisthegroundstateofmulti-orbital
c perovskite structure such as Sr2IrO4 and Ba2IrO4 [1–5]. systems with the competition between the electron cor-
[ In these materials, along with moderate electron corre- relations and the SOC. This is precisely the main issue

1 lations [6, 7], there exists a strong relativistic spin-orbit of this paper and we will demonstrate the emergence of
v coupling (SOC), which splits t2g orbitals, already sepa- exotic insulators in a three-orbital Hubbard model with
8 rated from e orbitals due to a large crystal field, into the SOC.
g
0
the effective total angular momentum j = 1/2 doublet
1 Here,onthebasisofthemulti-orbitaldynamicalmean
and j = 3/2 quartet orbitals in the atomic limit [8].
0 field theory (DMFT) [29], we numerically study a three-
0 Sincetherearenominallyfive5delectronsperIrion,the
orbital Hubbard model with the SOC at five electrons
. j = 1/2 orbital is half filled while the j = 3/2 orbitals
2 are fully occupied. persite, correspondingto (t2g)5 electronicconfiguration.
0 The continuous-time quantum Monte Carlo (CTQMC)
5 As opposed to simple expectation from strongly cor-
method based on the strong coupling expansion is em-
1 related 3d and 4d transition metal oxides [9–13], the ex-
ployed as a multi-orbital impurity solver [30]. We find
v: perimentshaverevealedthatthegroundstateoftheseIr
that the sign problem is significantly improved by ap-
i oxidesisaj =1/2antiferromagnetic(AF)insulator[14–
X propriately choosing the maximally spin-orbit-entangled
17]. The theoretical understanding of the j = 1/2 AF
r bases, which allows us to treat exactly the full Hund’s
insulator has been also reported [14, 18–24]. Moreover,
a coupling and pair hopping terms. This improvement is
evenpossibleunconventionalsuperconductivityhasbeen
crucialalsoforthecalculationsinalowtemperatureand
proposedoncemobilecarriersareintroducedintothein-
a strong SOC regions where the metal-insulator transi-
sulating state [25–28]. However, the electronic structure
tion (MIT) occurs. We find that for fixed Coulomb in-
in multi-orbital systems with the competition between
teractions the MIT is induced by increasing the SOC.
theelectroncorrelationsandtheSOChasnotbeenthor-
The insulating phases include the j =1/2 antiferromag-
oughlyunderstood. WhentheSOCissignificantlylarge,
netically ordered phase as well as a multi-orbital AF in-
the j = 1/2 based band is completely separated from
sulating (AFI) phase. In addition, we find an excitonic
the j = 3/2 based bands, and thus a single-orbital de-
insulating (EXI) phase, where an electron-hole pair in
scription of the j = 1/2 based band is expected to be
the j =1/2 and j =3/2 based bands are condensed.
valid. On the other hand, when the SOC is small, this
picture breaks down and the electronic structure should The three-orbital Hubbard model studied
be largelyaffectednotonly by Coulombinteractionsbut here is described by the following Hamiltonian:
2
H = H0 + HI, where H0 = Phi,i′iPγ,σtγc†iγσci′γσ − in QMC calculations is a basis dependent problem and
µPi,γ,σnγiσ + λPi,γ,δ,σ,σ′hγ|Li|δi · hσ|Si|σ′ic†iγσciδσ′ Aitlcthanoubgehiwmepernocvoeudnbtyeratphpersoeprrioiautseslyigcnhporoosbinlegmbawsheesn[3t5h]e.
represents the non-interacting part of the model
aUn′d HI =nγUnδPi,γnγi↑nγi↓J + U′2−Jc†Pci,γ6=δc,†σncγiσnδiσ ++ iomriaglilnyalspt2ing-boarbseits-e(ncitγaσn)glaerde jusbeda,sewse(fiainjdm)thiamtptrhoevemtahxe-
2 Pi,γ6=δ,σ iσ iσ¯ − Pi,γ6=δ iγ↑ iγ↓ iδ↓ iδ↑ sign problem significantly. These j bases are the eigen-
iJn′tPerai,cγt6=ioδncs†iγ.↑Hc†ieγr↓ec,iδt↓γcisδe↑tsdtehsecrnibeeasrestth-eneilgohcablorChoouplpominbg st2tgatbeassoefs,H0 in the atomic limit and are related to the
amplitude for t2g orbitals γ = (dyz,dzx,dxy) on the
a(aBtsdeγsotuhpm=etinlaagtt/ts√tiechmZeei)ew,sllaiitmwphtheicecrobedoarentdndhiswneitaiydtDitohMonfFfonsTrutamttisehbseeerxftoahZrcrteZ[e3[13o]1.→r,bi3tWa2∞l]es. aaai32ii3221−222ss3s = √16−√√s23s −−i−i√√i23s √−022s ccciiidddxzyxzyσσσ¯¯ ,(1)
c† (c ) is an electron creation (annihilation) op- where s = 1( 1) for σ = ( ). In the j bases represen-
eirγaσtor wiγiσth spin σ(= , ) and orbital γ at site i and tation,thenu−mberofoff-d↑iag↓onalelementsinGδ,σ′(τ)is
↑ ↓ γ,σ
nγ = c† c . λ is the SOC and L (S ) is the orbital reduced and as a result the sign problem can be allevi-
iσ iγσ iγσ i i
(spin) angular momentum operator at site i. The ated.
chemical potential µ is tuned to be at five electrons per Figure1showstypicalresultsoftheaveragesign sgn
h i
site. σ¯ denotes the opposite spin of σ. HI includes the (see Ref. 35 for the definition) for U = 8 calculated for
intra (inter) orbital Coulomb interaction U (U′), the the paramagnetic and orbital disordered solutions. The
Hund’s coupling (J), and the pair hopping (J′). We set λ dependence of sgn at T = 0.08 in Fig. 1(a) clearly
h i
U = U′ +2J and J = J′ = 0.15U [33]. Employing the shows that sgn 1 for the j bases, much larger than
h i ≈
CTQMC method as an impurity solver [30] to calculate sgn for the t2g bases particularly for large λ. Simi-
h i
the imaginary-time Green’s functions at the impurity larly,the T dependence of sgn at λ=0.24in Fig. 1(b)
site, Gδ,σ′(i,τ) T c (τ)c† (0) , we can solve demonstrates the remarkabhle imi provement of sgn for
γ,σ ≡ −h τ iγσ iδσ′ i h i
numerically exactly the model. In what follows, U, low T when the j bases are used. These improvements
λ, temperature T, and frequency ω are in units of t. of sgn guarantee higher accuracy of our CTQMC cal-
h i
We also omit the site index in the Green’s function, culationswith lesscomputationalcostina wide rangeof
Gδ,σ′(τ)=Gδ,σ′(i,τ),unlessGδ,σ′(i,τ)issitedependent. λ even at very low T.
γ,σ γ,σ γ,σ
Let us first examine the numerical accuracy of the First,weshallbrieflyexaminetheMITinducedbythe
CTQMC calculations. Generally, a negative sign prob- SOC in paramagnetic and orbital disordered states. A
lem is one of the most serious issues in QMC calcula- typicalexampleofthe evolutionofsingle-particleexcita-
tions. Specially, when QMC methods are employed for tion spectrum Aj,m(ω) = −1/πImGjj,,mm(ω +i0+) in the
the multi-orbital DMFT calculations, the sign problem real frequency ω with increasing λ is shown in Fig. 2(a).
seriously prevents us from simulating at low tempera- Here, 0+ is positive infinitesimal. The maximum en-
tures, which often enforces to approximate the Hund’s tropy method is employed to calculate Aj,m(ω) from
coupling and pair hopping terms [34]. Indeed, for our Gj,m(τ) [36]. As shown in Fig. 2(a) for U = 8, the
j,m
model, we find that the sign problem becomes destruc- SOC induces the transition from the metallic to insulat-
tively serious at low temperatures specially when the ing state at λ 0.31,where the quasi-particlepeak near
∼
SOC is large (see Fig. 1). theFermilevelvanishesandthesingle-particleexcitation
gap starts to open. The insulating state for λ & 0.31 is
It is now important to recall that the sign problem
the j = 1/2 Mott insulator, where the j = 1/2 based
band is half-filled while the j = 3/2 based bands are
1 fully occupied [see Fig. 2(b)]. Figure 2(b) also indicates
j bases j bases that the j = 1/2 based band, which is degenerate with
the j = 3/2 based bands at λ = 0, gradually separates
>
n
g0.5 formthe j =3/2basedbands with increasingλ [see also
s
< t2g bases t2g bases Fig. 2(c)]. This is essential for the SOC induced Mott
(a) (b) transition because the critical U for the MIT decreases
0
0.1 0.2 0.3 0.4 0.05 0.1 0.15 as the orbital degeneracy is degraded [37].
T Now,weshallconsiderpossibleorderedstates. Tothis
end,weintroducemagneticorderparameters,M (l)=
FIG. 1: (Color online) (a) λ dependence of the average sign j,m
hsgni for U = 8 at T = 0.08. (b) T dependence of hsgni for 12Pm′=±msign(m′)ha†ljm′aljm′i, where l(= A,B) indi-
U =8 at λ=0.24. For both t2g and j bases in the CTQMC cates two sublattices [38] and aljm is defined in Eq. (1)
calculations, the paramagnetic and orbital disordered solu- with site i being on sublattice l. In addition, we in-
tions are assumed. vestigate excitonic orders formed by an electron-hole
3
(a) (b) (a) (c)
2
λ＝0.08 j,|m|=1/2,1/2 j,m=1/2,1/2
0 j , | m | ==33//22,,31//22 nj,m1.5 jjj,,,mmm===331///222,,,131///222 Mj,m 0.2 jj,,mm==33//22,,13//22 ×5 ×5
λ＝0.30 0
0.1
0λ＝0.31 10 0.1 0.2 0.3 0.4 j’,m’Δj,m-0.10 Δ31//22,,11//22 Δ31//22,,--11//22
(c) 2
ω ω
0λ＝0.36 0 j=1/2=3/2 0 jj=1/2 nj,m 1.5
4t 1
0 3/2λ jjjj=3/2 0.1 0.2 0.3 0.1 0.2 0.3
-8 -4 0 4 8
λ＝０ λ≠０ (b) j,m=1/2,1/2 (d)
j,m=1/2,-1/2
FspIeGc.tru2m: A(jC,mol(oωr) oannlidne)(b)(a)eleScitnrgolne-pdaertnisciltey enxjc,imtatio=n λ＝0.08 jjj,,,mmm===333///222,,,13-1///222 λ＝0.02 F31//22,,--11//22
Pm′=±mha†ijm′aijm′i for U = 8 at T = 0.06 with varying 0 j,m=3/2,-3/2 0
λ. Paramagneticandorbitaldisorderstatesareassumed. (c)
Schematicdensityofstatesforthenon-interactinglimit with λ＝0.32 λ＝0.32 F31//22,,11//22
λ=0 and λ6=0. Fermi level is at ω=0. The j =3/2 based
bandsare completely filled when λ≥4/3. 0 0
-8 -4 0 4 8 -8 -4 0 4 8
pair in different (j,m) bands with excitonic order pa-
r(ja′m,met′e)r.s,We∆ajj,′lm,smo′c(la)lcu=latehath†ljemeallejc′mtr′oin, dwehnesritey(njj,,mm()l)=6= Fg∆eIjrG,e′md.′3m:(cae(gnCntoeetlroi)zr,aatoinnodlnineMele)cj,t(mrao)n(atdonpedn),s(icte)yx:cniλtondiec(pbeoonrtddtoeernmcp)e.aro(abfm)steaatnegdr-
ePxcmit′=at±iomnhas†ljpme′catlrjumm′i,FAj′j,m,m′((ll,,ωω)),=an1/dπItmheGja′,nmo′m(l,aωlou+s O(dtj),h:meSrinpgalrea-mpaertteircsleuesxecditaatrieonTss=pe0c.t0r6umja,mnAdj,Um(=ω)8fo(rat,wb)oaλn’sd.
i0+) [39, 40]. Since wejh,mave found−that thesje,morder pa- 9.25 (c,d). Anomalous excitation spectrum F13//22,,±±11//22(ω) is
rameters satisfy M (A)= M (B) and ∆j′,m′(A)= also shown in (d).
j,m − j,m j,m
∆jj,′−,−mm′(B), as wellas nj,m(A)=nj,m(B), Aj,m(A,ω)= As shown in Fig. 3(c), the j = 1/2 AFI phase is found
A (B,ω) and Fj′,m′(A,ω) = Fj′,−m(B,ω), we will with no exitonic order for λ 0.25, the same phase dis-
j,−m j,m j,−m ≥
drop the sublattice index l in these quantities hereafter. cussed above for smaller U. Aj,m(ω) shown in Fig. 3(d)
clearly exhibits a finite gap only in the j = 1/2 based
Let us first explore a case with U = 8. Figure 3 (a)
showstheλdependenceofM ,∆j′,m′,andn . First band while the j = 3/2 based bands are fully occupied.
j,m j,m j,m The difference appears as compared with the case with
of all, it is apparent that there is no excitonic order for
smaller U when λ decreases. For 0.12 λ 0.25, all
fialnlitvealwueitshofnλ. Second1, faonrdλn≥ 0.25,=onlny M1/2,1/2 2is, three Mj,m’s are now finite and n1/2,1/2≤(n3/≤2,1/2 and
revealingthe1j/2=,1/12/2≈AFI state.3T/2h,1i/rd2,forλ3/2,30/.225≈,the n3/2,3/2) increases (decrease) from one (two) with de-
≤ creasingλ[seenFig.3(c)]. Wefurtherconfirmafinitegap
magnetic order disappears and n =n starts
3/2,1/2 3/2,3/2 inA (ω)forbothj basedbands. Theseresultsindicate
to decrease from two with decreasing λ, implying the j,m
that this phase is a AFI state but apparently breaks the
breakdown of the single-orbital description of the half-
single-orbital description of the half-filled j = 1/2 based
filled j =1/2 based band.
band.
Figure 3(b) shows A (ω) for two λ’s at U =8. Sim-
j,m More remarkably, for λ 0.12, we find that the exci-
ilar to the paramagnetic cases (see Fig. 2), for large tonic order parameter ∆3/≤2,±1/2 is finite with non zero
λ = 0.32 in the AFI phase, a finite gap is clearly open 1/2,±1/2
AF order [see Fig. 3(c)]. Although the magnetic order
in the j = 1/2 based band while the j = 3/2 based
3/2,±1/2
bands are fully occupied (i.e., band insulators). How- eventually disappears for λ≤0.04,∆1/2,±1/2 remains fi-
ever, notice that A (ω) [A (ω)] has larger nite. The most important feature in this phase is that
1/2,1/2 1/2,−1/2
(smaller) weight in the occupied states than in the un- ∆3/2,1/2 +∆3/2,−1/2 = 0 but ∆3/2,1/2 ∆3/2,−1/2 is fi-
1/2,1/2 1/2,−1/2 1/2,1/2 − 1/2,−1/2
occupied states for sublattice A because of the AF or- nite[41]andstaggeredbetweenthetwosublattices,indi-
der. For smaller λ = 0.08, besides broad structures cating that the excitonic order is accompanied by trans-
aroundω U/2,thesharpquasi-particlepeakappears lationalsymmetrybreaking. ItisalsonoticedinFig.3(c)
∼±
aroundtheFermilevelforbothj basedbands,indicating thatn =n inthisphasebecauseofthepres-
3/2,1/2 3/2,3/2
a metallic state with strong correlations. ence of non-z6ero excitonic order. Moreover,we find that
Next, we shall examine a case with larger U = 9.25. a single-particle excitation gap is always finite in both
4
(a) (b) state [see Fig. 4(b)] in the case where the degeneracy of
0.4
10 =0.02 t2g orbitalsisliftedduetoatetragonalcrystalfield,caus-
0.08 jjj===111///22 AAFFII ing a finite splitting δ between the dxy orbital and the
0.16 0.3
8 other two orbitals. We have considered both magnetic
and excitonic orders in the DMFT calculations [44]. Al-
/ 6 0.2
MMOAFI though the crystal field splitting reduces the exciton or-
4 der parameter as compared to the one at δ =0, we have
0.1
MMeettaall EEEEXXII++AAFF foundthattheEXIstateisrobustagainstthecrystalfield
2 EXI splitting as long as δ is smaller than the single-particle
0 | |
6 7 8 9 8 9 OOI 10 excitation gap at δ =0.
U U
In summary, we have studied the three-orbital Hub-
bardmodelwiththeSOCusingthemulti-orbitalDMFT.
FIG.4: (Coloronline)(a)U dependenceoftheeffectiveSOC
We have employed the CTQMC method based on the
λ∗ (see the text for definition) for λ=0.02 at T =0.06. For
strong coupling expansionand found that the signprob-
comparison, theresultsfor λ=0.08 and0.16 arealso shown.
lem is significantly improved by using the maximally
(b) U-λ phase diagram at T = 0.06. (MO)AFI, EXI, and
OOI stand for (multi-orbital) antiferromagnetic insulating, spin-orbit-entangled j bases. The improvement is essen-
excitonic insulating, and orbital ordered insulating phases, tial to treat exactly the full Hund’s coupling and pair
respectively. A plus mark indicates a set of U and λ where hopping terms and alsoto reachatlow enoughtempera-
theeffect of a tetragonal crystal field is examined. tures for large SOC.
We have applied this method to determine the finite
temperaturephasediagramontheU-λplaneatfiveelec-
j based bands, including the phase where only the exci-
tronper site. The U-λphase diagramatthe lowesttem-
tonic orderparameteris nonzero,as shownin Fig. 3(d).
perature T =0.06 is summarized in Fig. 4(b). For small
3/2,±1/2
In addition, in the EXI phase, F (ω) are found
1/2,±1/2 U(.8),wehavedemonstratedthattheSOCinducesthe
to be non zero. This is a strong evidence for the exis-
transition from the metallic to the j =1/2 AFI state, in
tence of a stable EXI state formed by an electron-hole
good agreement with the previous numerical study [20].
pair between the j =1/2 and j =3/2 based bands with
ForlargeU(&8),themulti-orbitalAFIphaseappearsfor
m = 1/2. In the limit of λ = 0, this EXI state is re-
intermediatevaluesofλ,wherethesingle-orbitaldescrip-
±
placed by the orbital ordered insulating state in the t2g tion breaks down. Moreover,we have found that further
orbitals reported previously [42].
overlapofthenon-interactingj =1/2andj =3/2based
We shall now argue that there are two important in- bandsfavorsthe EXI phase,either withorwithoutmag-
gredients for the emergence of the EXI state. First, λ is neticorder,whichdominatesthephasediagramforsmall
strongly renormalized by the Coulomb interactions and λ(. 0.1). The EXI state is formed by an electron-hole
the renormalized λ becomes sizably large even for small pair in the j = 1/2 and j = 3/2 based bands with the
λ in the metallic phase at the vicinity of the EXI phase. same m= 1/2. We have argued that the strong renor-
To show this, we calculate the first moment of Aj,m(ω), malization±of the SOC near the MIT as well as the full
defined as Wj,m = dωωAj,m(ω), and evaluate the ef- Hund’s coupling and pair hopping are essential for the
R
fectiveSOCλ∗ =W1/2,1/2 W3/2,1/2 [43]. TheU depen- emergence of the EXI state.
−
dence of λ∗ for the metallic phase is shown in Fig. 4(a). Many experimental and theoretical studies support
Indeed, λ∗/λ is strongly renormalized with increasing U thatthe5dtransitionmetalIroxidessuchasSr2IrO4and
andcanbesignificantlylarge,aslargeas∼9atU =8.75 Ba2IrO4 are the j = 1/2 AF insulators [14–24]. These
near the transition to the excitonic insulator. materialsarecharacterizedwiththelargeSOCandmod-
The other key factors are the Hund’s coupling J and erate Coulomb interactions, thus in good qualitative ac-
thepair-hoppingJ′. Wehaveperformedthestaticmean- cordance with our phase diagram. Possible materials for
filed analysis andfound that the EXI state emerges only theexcitonicinsulatorfoundhereshouldbecharacterized
for a small λ region with non zero magnetic order. We with i) larger Coulomb interactions, ii) relatively small
have also found that the excitonic order is never stabi- SOC, and iii) the low spin configuration with no e or-
g
lized when J = J′ = 0. This implies that the particle bitals involved. For 4d electron systems, relatively large
number fluctuations are essentialfor the EXI state since Coulomb interactions and relatively small SOC are ex-
the particle number of each j = 1/2 and j = 3/2 based pected, while 3d electron systems exhibit large Coulomb
bands is conserved separately when J = J′ = 0. These interactions and very small SOC with the high or in-
results suggest that not only the SOC but also the full termediate spin configuration. We expect that the ideal
Hund’scouplingandthepairhoppingplaytheimportant systemsforthestableexcitonicinsulatorwouldbesome-
role for stabilizing the EXI state. The similar tendency where between 3d and 4d electron systems. This can
is found in the DMFT calculations. be achieved in experiments by, for example, 3d and 4d
Finally,wehavealsoexaminedthestabilityoftheEXI transitionmetalintermixedoxidesor4dtransitionmetal
5
oxide thin films grown on a compressive substrate with [19] H. Jin, H. Jeong, T. Ozaki, and J. Yu,
largedistortionofoxygenoctahedra. Furtherexperimen- Phys. Rev.B 80, 075112 (2009).
tal studies in this direction is highly desired. [20] H. Watanabe, T. Shirakawa, and S. Yunoki,
The authors are grateful to K. Seki and H. Watan- Phys. Rev.Lett. 105, 216410 (2010).
[21] T. Shirakawa, H. Watanabe, and S. Yunoki,
abeforvaluablediscussion. Numericalcomputationhave
J. Phys.:Conf. Ser. 273, 012148 (2011).
been performed with facilities at Supercomputer Center
[22] C. Martins, M. Aichhorn, L. Vaugier, and S. Biermann,
in ISSP, Information Technology Center, University of Phys. Rev.Lett. 107, 266404 (2011).
Tokyo,and with the RIKEN Cluster of Clusters (RICC) [23] R. Arita, J. Kuneˇs, A. V. Kozhevnikov, A. G. Eguiluz,
facility. This workhas beensupportedin partby Grant- and M. Imada, Phys. Rev.Lett. 108, 086403 (2012).
in-Aid for Scientific Research from MEXT Japan under [24] H. Onishi, J. Phys.: Conf. Ser.391, 012102 (2012).
the Grant No. 25287096and by RIKEN iTHES Project. [25] F. Wang and T. Senthil,
Phys. Rev.Lett. 106, 136402 (2011).
[26] H. Watanabe, T. Shirakawa, and S. Yunoki,
Phys. Rev.Lett. 110, 027002 (2013).
[27] Y.Yang,W.-S.Wang,J.-G.Liu,H.Chen,J.-H.Dai,and
Q.-H. Wang, Phys.Rev.B 89, 094518 (2014).
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Phys.Rev.Lett. 93, 167202 (2004). ′ ′
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Phys.Rev.B 72, 064401 (2005). spond to the anomalous single-particle excitations for a
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[44] To examine the effect of a tetragonal crystal field, we
[15] B. J. Kim, H. Ohsumi, T. Komesu, S. Sakai, T. Morita,
H.Takagi, T. Arima, Science323, 1329 (2009). have added Hδ =δPi,σndiσxy into H.In theDMFT cal-
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K. Ohashi, T. Takayama, J. Matsuno, and H. Takagi, eters, Mα,s(l) = 21Ps′=±ssign(s′)hb†lαs′blαs′i, and the
Phys.Rev.B 83, 115121 (2011). excitonic order parameters, ∆αα′,s,s′(l) = hb†lαsblα′s′i with
[17] S. Fujiyama, H. Ohsumi, T. Komesu, J. Matsuno, (α,s)6=(α′,s′),whereblαs istheeigenstates ofH0+Hδ
B.J. Kim, M. Takata, T. Arima, and H. Takagi, intheatomiclimit.Thesebasesarerelatedtothejbases,
[18] GPPhh.yyss..RReevvJ..aLLceekttettl..i110082,, 2041a77n22d1025 ((22001029))G... Khaliullin, ai.ne.d,bbll13ss==Aal123a−l2213ss2,−wshAer2ealA2312s,anbld2sA=2saAre2aclo21n2sst+anAt.1al322s,