Multipole correlations of $t_{\rm 2g}$-orbital Hubbard model with spin-orbit coupling

0.19 MB·English

by  Hiroaki Onishi

Proc.Int.Conf.HeavyElectrons(ICHE2010)
J.Phys.Soc.Jpn.80(2011)SAonishi-iche2010
c 2011ThePhysicalSocietyofJapan
(cid:13)
Multipolecorrelations oft -orbital Hubbard model withspin-orbit coupling
2g
HiroakiOnishi∗
AdvancedScienceResearchCenter,JapanAtomicEnergyAgency,Tokai,Ibaraki319-1195,JAPAN
Weinvestigatetheground-state propertiesofaone-dimensional t -orbitalHubbardmodel including
2g
anatomicspin-orbitcouplingbyusingnumericalmethods,suchasLanczosdiagonalizationanddensity-
matrixrenormalizationgroup.Asthespin-orbitcouplingincreases,wefindaground-statetransitionfrom
1 a paramegnetic state to a ferromagnetic state. In the ferromagnetic state, since the spin-orbit coupling
1
mixes spin and orbital stateswith complex number coefficients, an antiferro-orbital statewith complex
0
orbitalsappears.Accordingtotheappearanceofthecomplexorbitalstate,weobserveanenhancementof
2
Γ octupolecorrelations.
4u
n
a KEYWORDS: t orbitals,spin-orbitcoupling,multipole,density-matrixrenormalizationgroup
2g
J

1
1 Thecompetitionandcooperationbetweenspinandorbital the magnitude of the total spin. In the ferromagnetic phase,
degrees of freedom in strongly correlated electron systems antiferro-dipolecorrelationsdevelopevenwhenthespinstate

]
l manifest itself in the emergence of various types of spin- isferromagneticduetotheorbitalcontribution.Ontheother
e orbital ordered and quantum liquid phases.1–3) In general, hand,thespin-orbitcouplinginducesacomplexorbitalstate,
-
r amongcompetinginteractionsinvolvingspinandorbital,the inwhichrealxy,yz,andzxorbitalsaremixedwithcomplex
t
s spin-orbitcoupling is supposed to be weak in 3d transition- number coefficients. Accordingto the complex orbitalstate,
.
at metaloxidessuchascupuratesandmanganites,whileaswe Γ4uoctupolecorrelationsareenhanced.
moveto4dand5delectrons,thespin-orbitcouplingbecomes Let us consider triply degenerate t orbitals on a one-
m 2g
strong and responsible for magnetic, transport, and optical dimensional chain along the x direction with five electrons
-
d properties. When the spin-orbit coupling is dominant, spin persite.Theone-dimensionalt2g-orbitalHubbardmodelwith
n andorbitalarenotindependent,butinstead thetotalangular thespin-orbitcouplingisdescribedby
o momentumgivesagooddescriptionofthemany-bodystate.
c Infact,ithasbeensuggestedthatSr IrO ,inwhichIr4+ions H = X tττ′(d†iτσdi+1τ′σ+h.c.)+λXLi·Si
[ 2 4
have five electrons in triply degenerate t orbitals, exhibits i,τ,τ′,σ i
2g
v1 a novel Mott-insulating state with an effective total angular +UXρiτ↑ρiτ↓+(U′/2) X ρiτσρiτ′σ′
momentumJ =1/2 due to a strong spin-orbitcoupling.4–7)
5 eff i,τ i,σ,σ′,τ6=τ′
In the limit of strong spin-orbit coupling, the ground-state
5
0 Kramersdoubletatalocalioncanbedescribedbyanisospin +(J/2) X d†iτσd†iτ′σ′diτσ′diτ′σ
2 with Jeff=1/2.8,9) The exchange interaction among isospins i,σ,σ′,τ6=τ′
01. coafnspliena-dortboitaalveanriteatnygloefdosrtdaetersin.gand fluctuation phenomena +(J′/2) X d†iτσd†iτσ′diτ′σ′diτ′σ, (1)
1 Whenwemovetoheavy-elementf-electronsystemssuch i,σ6=σ′,τ6=τ′
1 asrare-earthandactinidecompounds,thespin-orbitcoupling where d (d† ) is an annihilation (creation) operator for
: iτσ iτσ
v is large comparing with other energy scales. In such a case, an electron with spin σ (=↑,↓) in orbital τ (=xy,yz,zx) at
Xi we usually classify the complicated spin-orbital state from sitei,andρiτσ=d†iτσdiτσ.Thehoppingamplitudeisgivenby
the viewpoint of multipole, which is described by the total t =t =t and zero for other combinationsof orbitals.
r xy,xy zx,zx
a angular momentum. Indeed, the multipole physics has been Hereafter, t is taken as the energy unit. L and S represent
i i
activelydiscussedinthefieldofheavyelectrons.10) Arecent orbital and spin angular momentum operators, respectively,
trendistounveilexotichigh-ordermultipoleordering.Asan andλisthespin-orbitcoupling.U,U′,J,andJ′denoteintra-
attemptto clarifymultipolepropertiesoff-electronsystems orbitalCoulomb,inter-orbitalCoulomb,exchange,and pair-
from a microscopic viewpoint, we have numerically studied hopping interactions, respectively. We assume U=U′+J+J′
multipolecorrelationsofan f-orbitalHubbardmodelonthe due to the rotation symmetry in the local orbital space and
basisofthej-j couplingscheme.11) Webelievethatitisalso J′=J duetotherealityoftheorbitalfunction.12)Throughout
importanttoclarifymultipolepropertiesind-electronsystems thispaper,weset~=k =1.
B
undertheeffectofthespin-orbitcoupling. Weinvestigatetheground-statepropertiesofthemodel(1)
In this paper, we investigate multipole properties in the by exploiting a finite-system density-matrix renormalization
groundstateofaone-dimensionalt2g-orbitalHubbardmodel group (DMRG) method with open boundary conditions.13)
includingthespin-orbitcouplingbynumericalmethods.With The number of states kept for each block is up to m=120,
increasing the spin-orbit coupling, the ground state changes and the truncation error is estimated to be 10−4∼10−5. We
fromaparamagneticstatetoaferromagneticstateintermsof remarkthatduetothethreeorbitalsinonesite,thenumberof
basesforthe singlesite is64,andthe size ofthesuperblock
∗E-mailaddress:[email protected]
SAonishi-iche2010-1
J.Phys.Soc.Jpn.80(2011)SAonishi-iche2010 SAonishi-iche2010-2
(a) 1 U=U'+J+J', J=J' Γγmultipole multipoleoperator
0.8 U'=10 Γ4udipole Jx,Jy,Jz
Γ3gquadrupole Ou=(1/2)(2Jz2−Jx2−Jy2)
λ0.6 ferromagnetic Γ5gquadrupole OOyvz==(√(√33/2/2)()JJx2y−JzJy2)
0.4 Ozx=(√3/2)JzJx
Oxy=(√3/2)JxJy
0.2 paramagnetic Γ2uoctupole Txyz=(√15/6)JxJyJz
0 Γ4uoctupole Txα=(1/2)(2Jx3−JxJy2−Jz2Jx)
0 2 4 J 6 8 10 Tyα=(1/2)(2Jy3−JyJz2−Jx2Jy)
(2Sb tot) 234 ( Sc ocloc) 12 Γ5uoctupole TTTTyxzzββαβ====((((√√√1/2111)555(///2666J)))(((z3JJJ−xyzJJJJzxzy222J−−−x2JJJ−yxz222JJJJyz2yxJ)))z)
1 U'=10, J=2 L l U'=10, J=2 Table I. Definitionofmultipoleoperatorsuptorank3.Theoverlineonthe
0 0 productdenotestheoperationoftakingallpossiblepermutationsinterms
0 0.2 0.4 λ0.6 0.8 1 0 0.2 0.4 λ0.6 0.8 1 ofcartesiancomponents,e.g.,JxJy=JxJy+JyJx.
6 2
(d) (e)
magnitude of the total angular momentum in the single site
5
2 J2 isshown.Atthetransitionpoint,J2 exhibitsa sudden
J loc 4 n τ1.5 xy inlcorcease,sincethespin-orbitcouplingstlaocbilizesalargetotal
3 U'=10, J=2 yz U'=10, J=2 angularmomentumstateateverylocalsites.Noteagainthat
zx
2 1 J2 doesnotreachthemaximumvalue 5(5 +1)=35 in the
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 loc 2 2 4
λ λ limit of large λ, since the total angular momentum is not a
conservedquantity.Regardingthe orbitalstate, we show the
Fig. 1. Lanczosresultsforthefour-siteperiodicchain.(a)Theground-state chargedensityineachorbitalinFig.1(e).Dueto thespatial
phasediagraminthe(J,λ)planeforU′=10.(b)Themagnitudeofthetotal anisotropyof orbitals, one hole is preferablyaccommodated
spininthewholesystem.(c)Thecorrelationbetweenspinandorbitalin
initinerantxy orzxorbitalsineachsite, whilelocalizedyz
thesinglesite.(d)Themagnitudeofthetotalangularmomentuminthe
orbitalsaredoublyoccupied.Measuringchargecorrelations,
singlesite.(e)Thechargedensityineachorbital.
wefindthatholesoccupyrealxyorzxorbitalalternatelyfor
smallλ(notshown).Namely,thegroundstateisarealorbital
Hilbert space grows as m2×642. To reduce the size of the state.Forlargeλ,however,xy,yz,andzxorbitalsaremixed
Hilbertspace,weusuallydecomposetheHilbertspaceintoa withcomplexnumbercoefficientsbythespin-orbitcoupling,
block-diagonalformbyusingsymmetriesoftheHamiltonian. leadingtoacomplexorbitalstate.
In the present case, however, the spin-orbit coupling breaks Now we move on to the analysis of multipole properties
thespin SU(2)symmetry,sothatwecannotutilizeSz asa to clarify the ground-state properties from the viewpoint of
tot
goodquantumnumber,whereSz isthez componentofthe multipole.Wemeasuremultipolecorrelationfunctions
tot
total spin. Since we ignore e orbitals among d orbitals, the
totalangularmomentumisnogtaconservedquantity.Thetotal χΓγ(q)=XhXjΓγXkΓγieiq(j−k)/N, (2)
j,k
numberofelectronscanbeusedasagoodquantumnumber.
Thus, since DMRG calculations consume much CPU times, where XiΓγ is a multipole operator with the symbol X of
wesupplementallyuseaLanczosdiagonalizationmethodfor multipole for the irreducible representation Γγ in the cubic
theanalysisofafour-siteperiodicchaintoaccumulateresults symmetryatsitei.Here,weconsider15typesofmultipoles
withrelativelyshortCPUtimes. includingthreedipoles(X=J),fivequadrupoles(X=O),and
LetusfirstlookatLanczosresultsforthefour-siteperiodic sevenoctupoles(X=T),aslistedinTableI.14)Weevaluatethe
chain. In Fig. 1(a), we show the phase diagram in the (J,λ) multipole correlation functions by DMRG calculations with
plane for U′=10. The phase boundary is determined by the chainsof16sites.
magnitudeofthetotalspinS2 .AsshowninFig.1(b),S2 Figure2showsDMRGresultsofthemultipolecorrelation
tot tot
is almost zero for small λ, indicating a spin-singlet ground functionsatU′=10,J=2,andλ=0fortheparamagneticphase.
state. Asλ increases,we find a transitionto a ferromagnetic Regarding dipoles, as shown in Fig. 2(a), the Jx correlation
state with finite S2 . Note that even in the limit of large λ, has a peak at q=π, which signals an antiferromagneticstate.
tot
S2 doesnotapproachthe maximumvalue 2(2+1)=6,since Here,wenoticethateachofthedipolecorrelationsexhibitsa
tot
the spin-orbit coupling mixes spin up and down states and kink at q=π/2, while the kink correspondsto a peak for the
thecompleteferromagneticstateisdisturbed.InFig.1(c),we Jy and Jz correlations. This kink structure originates in the
plotthecorrelationbetweenspinandorbitalin thelocalsite spin-orbitalSU(4)symmetrywhichrealizesataspecialpoint
L ·S .Atλ=0, thereisnocorrelationbetweenspinand J=λ=0.15–19) At the SU(4) symmetric point, correlations of
loc loc
orbital. As λ increases, the spin-orbital correlation develops spinSi andorbitalpseudospinTi=21Pτ,τ′,σd†iτσσττ′diτ′σ,
andapproachesonein the limitoflargeλ, indicatingtotally whereσarePaulimatrices,coincidewitheachotherandhave
parallel spin and orbital angular momenta. In Fig. 1(d), the a peak at q=π/2. With increasing J, the spin correlation of
q=π/2growsandthepeakofthespincorrelationremainsat
J.Phys.Soc.Jpn.80(2011)SAonishi-iche2010 SAonishi-iche2010-3
0.4 2 0.4 2
(a) (b) (a) (b)
U'=10, J=2, λ=0 U'=10, J=2, λ=0 U'=10, J=2, λ=0.5 U'=10, J=2, λ=0.5
0.3 J x O u 0.3 J x O u
J y O v J y O v
J z O yz J z O yz
χ0.2 χ 1 O zx χ0.2 χ 1 O zx
O xy O xy
0.1 0.1
0 0 0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
q / π q / π q / π q / π
30 4 30 4
(c) (d) (c) (d)
U'=10, J=2, λ=0 U'=10, J=2, λ=0 U'=10, J=2, λ=0.5 U'=10, J=2, λ=0.5
20 TTT xyz ααα 3 TTT xyz βββ 20 TTT xyz ααα 3 TTT xyz βββ
χ χ 2 T xyz χ χ 2 T xyz
10 10
1 1
0 0 0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
q / π q / π q / π q / π
Fig. 2. DMRGresultsofmultipolecorrelationsatU′=10,J=2,andλ=0: Fig. 3. DMRGresultsofmultipolecorrelationsatU′=10,J=2,andλ=0.5:
(a)Γ4udipoles;(b)Γ3gandΓ5gquadrupoles;(c)Γ4uoctupoles;and(d) (a)Γ4udipoles;(b)Γ3gandΓ5gquadrupoles;(c)Γ4uoctupoles;and(d)
Γ5uandΓ2uoctupoles. Γ5uandΓ2uoctupoles.
q=π/2.Ontheotherhand,thepseudospincorrelationofq=π thespinstateisferromagnetic.Moreover,thecomplexorbital
isenhanced,andthepeakpositionofthepseudospincorrela- state appears, since the spin-orbit coupling yields the linear
tionchangestoq=π.Notethatfortheorbitalangularmomen- combinationsofspinandorbitalstateswithcomplexnumber
tumLi,thecorrelationoftheq=πcomponentisenhancedas coefficients.Accordingly,weobserveanenhancementofthe
well.Thus,theorbitalcontributiontodipoleleadstothepeak Γ octupolecorrelations.Itisaninterestingissuetoexplore
4u
oftheJx correlationatq=πratherthanq=π/2. possiblemultipoleorderingin5d-electronIrcompoundswith
In Fig. 2(b), we find that for the Ou and Ov correlations, strongspin-orbitcoupling.
asharppeakappearsatq=0,sincehOuiandhOviareturned TheauthorthanksG.Khaliullin,S.Maekawa,andM.Mori
out to be finite. We also find a peak at q=π for the Ou, Ov, forusefuldiscussions.ThisworkwassupportedbyGrant-in-
andOyz correlations,implyinganantiferro-orbitalstate.For AidforScientificResearchofMinistryofEducation,Culture,
all quadrupoles, there occurs a kink at q=π/2 in similar to Sports,Science,andTechnolotyofJapan.
the case of dipoles, which is a trace of the SU(4) symmetry
atJ=λ=0.AsshowninFig.2(c)and2(d),wealsoobservea
kinkatq=π/2foroctupoles. 1) Proc.Int.Conf.StronglyCorrelatedElectronswithOrbitalDegreesof
In Fig. 3, we present the multipole correlation functions Freedom(ORBITAL2001),J.Phys.Soc.Jpn.71(2002)Suppl.
atU′=10,J=2, andλ=0.5 for the ferromagneticphase.At a 2) Y.TokuraandN.Nagaosa:Science288(2000)462.
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forallmultipoles.Concerningdipoles,asshowninFig.3(a),
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i
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