Table Of ContentDeformation of SU(4) singlet spin-orbital state due to Hund’s rule coupling
Hiroaki Onishi
Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan
(Dated: January 5, 2007)
Weinvestigatetheground-statepropertyandtheexcitationgapofaone-dimensionalspin-orbital
modelwithisotropicspinandanisotropicorbitalexchangeinteractions,whichrepresentsthestrong-
coupling limit of a two-orbital Hubbard model including the Hund’s rule coupling (J) at quarter
7
filling, by using a density-matrix renormalization group method. At J=0, spin and orbital correla-
0
tions coincide with each otherwith a peak at q=π/2, corresponding tothe SU(4) singlet state. On
0
the other hand, spin and orbital states change in a different way due to the Hund’s rule coupling.
2
WithincreasingJ,thepeakposition oforbitalcorrelation changestoq=π,whilethatofspincorre-
n lation remains atq=π/2. Inaddition,orbital dimercorrelation becomesrobust incomparison with
a spin dimercorrelation, suggesting that quantumorbital fluctuationis enhancedbytheHund’srule
J
coupling. Accordingly, a relatively large orbital gap opens in comparison with a spin gap, and the
5 system is described by an effectivespin system on the background of theorbital dimer state.
] PACSnumbers: 75.45.+j,75.10.Pq,75.40.Mg
l
e
-
r I. INTRODUCTION beenrevealed.9,10,11,12,13,14,15 Aroundthe SU(4)point, a
t
s quantum critical phase extends in one side next to the
.
t SU(4)point,13,14,15 while inthe othersidethereoccursa
a It has been recognized that orbital degree of freedom
m playsacrucialroleintheemergenceofexoticmagnetism gappedphase,where spin- andorbital-singletdimers are
formed in an alternating pattern.9
- in strongly correlatedelectron systems. The interplay of
d spin and orbital degrees of freedom triggers off a variety In a more realistic situation, however,the Hund’s rule
n
oforbitalorderingthatstabilizesacertainspinstructure coupling causes anisotropic exchange interactions in the
o
on the orbital-ordered background, leading to complex orbitalpart, and the SU(4) symmetry is brokendownto
c
[ orderedphases,asfrequentlyobservedintransitionmetal SU(2)spin U(1)orbital.6,16,17,18,19 To clarify the effects of
oxides and f-electron compounds.1,2 such symm×etry breaking, an SU(4) Hubbard model per-
1
turbedbytheHund’srulecouplinghasbeenanalyzedby
v Inadditiontotheappearanceofdiversespin-orbitalor-
means of renormalization-groupand bosonization meth-
2 dered states, the combined quantum effects of spin and
ods.17Ithasbeenproposedthattheexcitationgapopens
9 orbital degrees of freedom yields a possibility of novel
0 quantumcriticalandquantumdisorderedstates.3 Inthis for an arbitrarily small Hund’s rule coupling, while the
1 context, a high symmetric SU(4) exchange model, which opening excitation gap is exponentially small as a func-
0 tionoftheHund’s rule coupling,correspondingtoagen-
describes a spin-1/2 system coupled with a pseudospin-
7 eralized type of Kosterlitz-Thouless transition.20
1/2 for two-fold orbital degeneracy, has been one of the
0
/ intriguing subjects from a theoretical viewpoint.4,5,6,7,8 Ontheotherhand,ithasbeencommonunderstanding
at In such a high symmetric case, spin, orbital, and com- that the Hund’s rule coupling in multi-orbital systems
m bined spin-orbital degrees of freedom play an identical playsanimportantrolein itinerantferromagnetism.21,22
role.8 In particular, the one-dimensional model is Bethe In fact, it has been shown that the Hund’s rule coupling
-
d ansatzsolvable,4andthequantumcriticalbehaviorofan induces a ground-state transition from a paramagnetic
n SU(4) singlet state has been revealed by analytical and (PM) state to a ferromagnetic (FM) state in the strong-
o numerical investigations.5,6,7 In fact, correlation func- couplingregion,23,24,25 butthere,thediscussionhasbeen
c
tionsshowpower-lawdecaywithacriticalexponent3/2, focused on the appearance of ferromagnetism itself and
:
v and the elementary excitation is gapless. the property of spin and orbital states in the PM phase
Xi The SU(4) spin-orbital exchange model represents the has not been clear yet. Thus, we believe that it is an
r strong-coupling limit of a two-orbitalHubbard model at intriguing issue to clarify how the SU(4) singlet state is
a quarter filling, in which electrons hop between the same deformed to a spin- and orbital-singlet state due to the
types of orbitals with equal amplitude and the Hund’s Hund’s rule coupling.
rule coupling is ignored. Indeed, such a simplified model Inthispaper,weinvestigatetheground-stateproperty
is anappropriatestarting point to gaindeep insight into and the excitation gap of a one-dimensional spin-orbital
thecomplexquantumfluctuationsofspinandorbitalde- modelwiththeSU(2) U(1) symmetry,byusing
spin orbital
×
greesoffreedom. However,inactualmaterials,the high- numerical techniques. When the Hund’s rule coupling is
estSU(4)symmetryislikelytobebrokendown,thusitis zero, the system has the SU(4) symmetry, and spin and
quite important to clarify the effects of possible symme- orbital correlations have a peak at q=π/2. On the other
try breaking. For instance, intensive studies have been hand,theHund’srulecouplinginducesFMandantiferro-
devoted to a one-dimensional SU(2) SU(2) ex- orbital (AFO) interactions, leading to the characteristic
spin orbital
×
change model and the ground-state phase diagram has changeofthepeakpositionoforbitalcorrelationtoq=π,
2
while that of spin correlation remains at q=π/2 in the (a) (b)
β β
PM phase. Moreover, we observe the stabilization of an
orbital dimer state with a significant orbital excitation α α
gap. Takingintoaccounttherobustorbitaldimerization,
the low-energy physics is described by an effective spin FIG. 1: Spin-orbital configuration in (a) one-electron state
and (b) two-electron state.
system on the backgroundof the orbital dimer state.
The organizationofthis paperis asfollows. InSec.II,
we describe the model Hamiltonian and the numerical
TABLE I: Eigenenergy and eigenstate of two-electron state
method. Starting from a two-orbital Hubbard model, a
intheascendingorderofeigenenergy,where|τσidenotesthe
spin-orbital exchange model in the strong-coupling limit
spin σ and orbital τ state.
is introduced. In Sec. III, we show our numerical results
of physical quantities such as correlation functions and eigenenergy eigenstate
excitation gaps. We discuss distinctive changes of spin U′−J |α↑i|β ↑i, |α↓i|β ↓i, √12(|α↑i|β ↓i+|α↓i|β ↑i)
andorbitalstatesfromtheSU(4)singletstateduetothe U′+J √12(|α↑i|β ↓i−|α↓i|β ↑i)
Hund’s rule coupling. Finally, we summarize the paper U−J′ √12(|α↑i|α↓i−|β↑i|β ↓i)
in Sec. IV. U+J′ √12(|α↑i|α↓i+|β↑i|β ↓i)
II. MODEL AND METHOD In order to consider an effective model in the strong-
coupling limit, as usual, we start from the ground state
Let us consider doubly degenerate orbitals on a one- of the atomic limit t=0, and treat the electron hopping
dimensionalchainwithN sitesincludingoneelectronper as a perturbation.28,29 In the groundstate of the atomic
site (quarter filling). The two-orbital Hubbard model is limit, each site is occupied by one electron, leading to
described by four-folddegenerate states, as shown in Fig. 1(a). Then,
theelectronhoppingcausesvirtualtwo-electronstates,as
H˜ = t (d† d +h.c.)+U ρ ρ showninFig.1(b). Thereappearfoureigenenergystates,
iτσ i+1τσ iτ↑ iτ↓
iX,τ,σ Xi,τ aslistedinTableI.Amongthem,thelowestenergystate
is spin triplet, and has energy U′ J. Thus, the Hund’s
+U′ ρiασρiβσ′ +J d†iασd†iβσ′diασ′diβσ rule coupling stabilizes spin polar−ized states. The other
iX,σ,σ′ iX,σ,σ′ three states are spin singlet. Note that the second- and
+J′ d†iτ↑d†iτ↓diτ′↓diτ′↑, (1) third-lowest energy states are degenerate because of the
i,Xτ6=τ′ relationU=U′+J+J′. Notealsothatallthesefourlevels
are degenerate when U=U′ and J=J′=0.
where d (d† ) is an annihilation (creation) operator Now we apply the degenerate perturbation theory by
iτσ iτσ
for an electron with spin σ(= , ) in orbital τ(=α,β) at considering the second-order processes of the electron
site i, ρ =d† d , and t ↑is↓the hopping amplitude. hopping, and obtain an effective spin-orbital exchange
iτσ iτσ iτσ
model, such as
Here, we assume that electrons hop between the same
typesoforbitalswithequalamplitude. Notethat,ingen-
H = H ,
eral,thehoppingamplitudesofmulti-orbitalsystemsare i,i+1
evaluatedfromtheoverlapintegralbetweenorbitals.26,27 Xi
3 1
The present simple form of the hopping amplitudes rep- H = A S S + T T
i,j i j i j
resents the system with, for instance, (p ,p ) orbitals or (cid:18) · 4(cid:19)(cid:18) · − 4(cid:19)
x y
(dyIzn,tdhzex)inotrebriatcatlisonontearmchsa,iUn,aUlo′n,gJ,thanedzJa′xdise.noteintra- +B(cid:18)Si·Sj − 41(cid:19)(cid:18)Ti·Tj −2TizTjz+ 41(cid:19)
orbitalCoulomb,inter-orbitalCoulomb,inter-orbitalex-
1 1
change(Hund’srulecoupling),andinter-orbitalpairhop- +C S S T T 2TxTx+
ping interactions, respectively. These Coulomb integrals (cid:18) i· j − 4(cid:19)(cid:18) i· j − i j 4(cid:19)
are all positive. Note that the spin diagonal part of the 1 1
+D S S T T 2TyTy+ (,2)
Hund’s rule coupling causes an attractive interaction, so (cid:18) i· j − 4(cid:19)(cid:18) i· j − i j 4(cid:19)
that the inter-orbital Coulomb interaction is effectively
reduced to U′ J. To ensure that the total inter-orbital where
−
Coulomb interaction is repulsive, we consider the region
of J<U′. We also note that the relation U=U′+J+J′ Si =(1/2) d†iτσσσσ′diτσ′ (3)
τX,σ,σ′
holds due to the rotational invariance in the local or-
bital space, and J=J′ due to the reality of the orbital is the S=1/2 spin operator and
wavefunction.27 When the Hund’s rule coupling is zero,
i.e., U=U′ and J=J′=0, the system possesses the SU(4) Ti =(1/2) d†iτσσττ′diτ′σ (4)
symmetry.17 τX,τ′,σ
3
indicates the T=1/2 pseudospin operator representing 0 0.5
(a) (b)
twoorbitals,withthePaulimatrixσ. Concerningtheex-
-0.1 0.4
fcrhoamngtehientceorrarcetsipoonnsdAin∼gDeiignenEeqn.e(r2g)y, eoafcthheofvtihrteumalartwiseos- K-0.2 y) / K0.3
electron state, and they are expressed as J / S-0.3 x, JT0.2
A = 4t2/(U′ J), (5) -0.4 z(JT0.1 JJTTyz x/ /K K
−
B = 4t2/(U′+J), (6)
-0.5 0
C = 4t2/(U J′)=B, (7)
0.3 30
− (c) (d)
D = 4t2/(U +J′)=4t2/(U′+3J). (8)
Note that the relation A>B>D>0 holds in the realistic K0.2 20
parameter region of 0 J<U′. Hereafter, we take t2/U′ y / K
as the energy unit. N≤ormalized by t2/U′, the exchange JST0.1 10
interactions are parametrized by J/U′. Since we focus
ontheregionof0 J<U′,wesetU′=1forsimplicityand
≤
consider the variation of J within 0 J<1. 0 0
0 0.5 1 0 0.5 1
≤
Here, let us discuss the symmetry of the spin-orbital J J
model. WhenU=U′andJ=J′=0,thespin-orbitalmodel
is reduced to FIG.2: Exchangeinteractionsinthespin-orbitalmodel(10):
H(0) = 8t2 S S + 1 T T + 1 , (9) (fuan)cJtSio/nKo,f(Jb.) JTzx/K, JTy/K, (c) JSyT/K, and (d) K, as a
i,j U (cid:18) i· j 4(cid:19)(cid:18) i· j 4(cid:19)
whichhas the SU(4) symmetry.6,8 Onthe otherhand, in are induced by the Hund’s rule coupling. We note the
the spin-orbitalmodel of Eq.(2), we canclearly see that relation of
exchangeinteractionsofthespinpartareisotropic,while
thoseoftheorbitalpartshowanisotropicformsaccording Jy +Jy /4=Jzx, (16)
T ST T
totheenergylevelofthevirtualtwo-electronstate. Note
that the second and third terms have the same energy among anisotropic orbital exchanges.
denominator due to the relation U=U′+J+J′, so that InFigs.2(a)-(d),exchangeinteractionsareplottedasa
exchangeinteractionsintheorbitalpartbecomeisotropic function of J. At J=0, exchange interactions other than
in the zx plane. In fact, Ttyot= iTiy commutes with H, K are equal to zero, and the system possesses the SU(4)
and becomes a good quantumPnumber. Concerning spin symmetry. On the other hand, the Hund’s rule coupling
degree of freedom, total spin Stot and Stzot= iSiz are induces FM and AFO interactions, which should cause
good quantum numbers. Thus, the SU(4) symPmetry is the deviation of the spin-orbital state from the SU(4)
broken down to SU(2)spin U(1)orbital due to the Hund’s symmetric case,as will be discussedin detailin the next
×
rule coupling. section.
In order to discuss the change of the spin-orbitalstate
We analyze the spin-orbital model (10), or equiva-
fromtheSU(4)symmetriccase,itisusefultorewritethe
lently (2), by a density-matrix renormalization group
HamiltonianbythesumoftheSU(4)symmetricpartand (DMRG)method.30,31Thefinite-systemalgorithmisem-
the other parts, such as
ployed with open boundary conditions. We keep up to
800statesintherenormalizationstepandthetruncation
H = K(S S +1/4)(T T +1/4)
i,j i· j i· j error is kept around 5 10−6 or smaller. We also use a
+JSSi·Sj +JTzx(TixTjx+TizTjz)+JTyTiyTjy Lanczos method for th×e analysis of a four-site periodic
+Jy S S TyTy, (10) chain.
ST i· j i j
where the exchange interactions are given by
III. RESULTS
K = A+D, (11)
J = ( A+B)/2, (12)
S − A. Four-site chain
Jzx = (A D)/2, (13)
T −
Jy = (A B)/2, (14) To grasp the fundamental property of the spin-orbital
T −
Jy = 2(B D). (15) configurationinthegroundstate,firstweconsiderafour-
ST −
site periodic chain, which is a minimal model to form a
We can easily see that J is negative, while Jzx and Jy unique SU(4) singlet ground state at J=0.8 The SU(4)
S T T
are positive, indicating that FM and AFO interactions singlet state for N=4 is composed of different four spin-
4
1 (a) 0.01
(a) (d)
β N=4 spin orbital J=0, N=96
(b) αβ ght00..68 wwwwabca+wb+wc orrelation 0
ei c
α w0.4
-0.01
(c) 0.2 (b) 0.01
β
spin J=0.2, N=96
α 0
0 0.1 0.2 0.3 0.4 0.5
J (r)s 0
C
FIG.3: (a)-(c)Threeclassesofrelevantspin-orbitalconfigu-
rationsinthegroundstateofthefour-sitesystem. (d)Weight
ofeachclassofFigs.3(a)-(c)inthegroundstateasafunction -0.01
of J. (c) 0.01
orbital J=0.2, N=96
orbital configurations at four sites, expressed as (r)o 0
C
SU(4) =(1/√24) d† d† d† d† 0 , (17)
| i iα↑ jα↓ kβ↑ lβ↓| i
i6=Xj6=k6=l
-0.01
0 20 40 60 80 100
where 0 isthevacuumstateandthesummationistaken distance r
| i
overallpermutationsintermsofthesiteindex. Notethat
SU(4) consists of 24 states with the same weight. For FIG. 4: (a) Spin and orbital correlation functions at J=0.
|finite Ji, however,the ground state is not represented by (b) Spin and (c) orbital correlation functions at J=0.2.
SU(4) itself. In fact, the ground state is changed to a
| i
spin-andorbital-singletstate,whichiswelldescribedby
thesuperpositionofthe24statesincludedin SU(4) ,but On the other hand, the class (b) also exhibits a parallel
the 24 states are split into three classes acco|rding tio the spin arrangement,but there occurs a partly ferro-orbital
weightinthegroundstate,18,19asshowninFigs.3(a)-(c). configuration, leading to relatively small wb comparing
Notethateachclasshaseightequivalentstatesduetothe withwa. The class (c)is more unfavorabledue to a fully
translation and the reversal of orbitals. In Fig. 3(d), we antiferromagnetic configuration.
show the J dependence of the weight of each class m in
the ground state, defined by
B. DMRG results
w = φ ψ 2, (18)
m i G
|h | i|
iX∈m Fromthe discussionwithinthe smallfour-sitechainin
the previous subsection, we can intuitively understand
whereψ isthegroundstateandφ denotesthebasis. At
G i the effect of the Hund’s rule coupling to cause the char-
J=0,eachclass contributes to the groundstate with the
acteristicchangeofthespin-orbitalstate,butitishighly
equal weight due to the SU(4) singlet ground state. On
required to clarify what types of correlations develop in
theotherhand,withincreasingJ,w graduallyincreases,
a thethermodynamiclimit. Forthispurpose,herewemove
whilew doesnotchangesomuchandw decreases. Note
b c on to our DMRG results for longer chains. In Fig. 4, we
that the total weight of the three classes remains almost
show spin and orbital correlation functions, defined by
unityevenwhenJ isincreased,butitabruptlydecreases
at around J=0.4, since the ground state turns from the
C (r) = (1/N ) SzSz , (19)
singlet state into a FM state. s r h i ji
Here, it is worth noting that each of the three classes |i−Xj|=r
of Figs. 3(a)-(c) is identified by a distinct peak position C (r) = (1/N ) TzTz , (20)
o r h i ji
in spin and orbital structure factors, denoted by (qs,qo): |i−Xj|=r
(π/2,π)fortheclass(a),(π/2,π/2)fortheclass(b),and
(π,π/2) for the class (c). In the class (a), a parallelspin where N is the number of site pairs (i,j) with r=i j ,
r
| − |
arrangement is stabilized by the FM interaction due to and denotes the expectation value. Note that we
h···i
theHund’srulecoupling,whilethepairsofparallelspins average over all pairs of sites separated by distance r in
point opposite directions to form a spin-singlet ground order to reduce the effect of open boundaries. At J=0,
state for small J. Concerning the orbital state, an AFO as shown in Fig. 4(a), C and C exhibit exactly the
s o
configuration appears so as to avoid the energy loss due same behavior with a four-site periodicity, which is in
to the double occupancy in the virtual hopping process. agreementwiththe previousinvestigationsforthe SU(4)
5
0.6 (a) 10-1
(a) S(q) N=64 (b) Tz(q) N=64 spin N=96
0.5
or 10-2
ct0.4
ure fa0.3 C(r)s10-3 JJ==00.1
ct J=0.2
stru0.2 J=0 J=0 10-4 J1=/r03./32
J=0.05 J=0.05 1/r
0.1
J=0.1 J=0.1
0 10-5
0 q0 ./5 π 1 0 q0 ./5 π 1 (b) 10-2
orbital N=96
1
10-3
(c) N=64 (d) N=64
actor00..68 SS((ππ/)2) TTzy((ππ//22)) TTzy((ππ)) C(r)o 10-4 JJ==00.1
cture f0.4 10-5 JJ1==/r003../232
u
str0.2 10-6 10 100
distance r
0
0 0.1 0.2 0.3 0 0.1 0.2 0.3 FIG. 6: Log-log plot of (a) spin and (b) orbital correlation
J J
functions at several values of J. The results are shown for
distances of four multiples.
FIG. 5: (a) Spin and (b) orbital structure factors at sev-
eral values of J. (c) Spin and (d) orbital structure factors at
q=π/2 and q=π as a function of J.
insharpcontrasttothecaseofS(q). Infact,wefindthat
Tz(π) becomes larger than Tz(π/2) at around J=0.05,
exchange model.6,7 For finite J, however, C and C are asshownin Fig.5(d). It is interesting to note thatthese
s o
not equivalent any longer due to the breakdown of the characteristic changes of the spin and orbital structure
SU(4) symmetry. In Fig. 4(b), it is found that C keeps factors are naturally understood from the discussion for
s
the four-site periodicity, while the amplitude of the os- the four-site chain in the previous subsection. Namely,
cillation becomes large. On the other hand, as shown in the spin-orbital configuration in Fig. 3(a) is stabilized
Fig. 4(c), we observea drastic changein the structure of by the Hund’s rule coupling at least for short distances,
C . Indeed, the four-site periodicity almost disappears, leading to the enhancement of S(π/2) and Tz(π) in the
o
andthe correlationdecaysmuchfasterforlongdistances whole system.
in comparison with that at J=0. We note here that Ty(q) shows qualitatively the same
In order to see clearly the changes in the spin and behaviorwithTz(q),butthereappearsaslightdifference
orbitalstructures,itisusefultomeasurespinandorbital between them. In Fig. 5(d), Tz(q) and Ty(q) at q=π/2
structure factors, defined by
and q=π are plotted as a function of J. With increasing
J, Ty(π/2) is suppressed and Ty(π) is enhanced in the
S(q) = SzSz eiq(j−k)/N, (21)
h j ki same way as Tz(q), while Tz(π) is much enhanced than
Xj,k Ty(π). This anisotropic behavior is naturally explained
Tz(q) = TzTz eiq(j−k)/N, (22) bytheanisotropyintheorbitalexchanges(Jzx,Jy,Jy )
Xj,kh j ki within a mean-field discussion. For the JSyTTtermTofStThe
spin-orbital model (10), replace S S by S S . In
Ty(q) = hTjyTkyieiq(j−k)/N. (23) thefullypolarizedFMstate, Si Sij·tajkesthheim·axjiimum
Xj,k value 1/4 at all bonds, andhthe· aniisotropy disappears,
InFigs.5(a)and5(b),weshowS(q)andTz(q)atseveral since the relation JTy +JSyT/4 = JTzx holds, as shown in
values of J, respectively. At J=0, S(q) and Tz(q) agree Eq. (16). However, in the spin-singlet state, Si Sj
h · i
with each other and have a peak at q=π/2 due to the shouldtakeasmallervalue than1/4,indicating thatthe
clear four-site periodicity. Evenwhen J is increased,the orbital correlation in the zx plane becomes larger than
peak position of S(q) remains at q=π/2, since S(π/2) is that along the y direction.
enhanced and S(π) is suppressed. The J dependence of InFigs.6(a)and6(b),weshowC andC inthelog-log
s o
S(π/2) and S(π) is plotted in Fig. 5(c). On the other scale to observe the decaying behavior. At J=0, C and
s
hand, the peak position of Tz(q) changes from q=π/2 to C exhibit power-law decay with the critical exponent
o
q=π,sinceTz(π/2)issuppressedandTz(π)isenhanced, 3/2, consistent with the previous analytical works5 and
6
(a) 0.02 (a) 0.1
J=0 spin N=96 spin J=0.2, N=96
(i=48,j)s0.01 JJJ===000...123 nn(i) 0
D s
r1) C -0.1
(-
0 -0.2
(b) 0.02 (b) 0.1
7,j) JJ==00.1 orbital N=96 orbital J=0.2, N=96
D(i=4o0.01 JJ==00..23 nnC(i)o 0
r1) -0.1
(-
00 10 20 30 40 50 -0.20 20 40 60 80 100
distance r (=j-i) site i
(c)
FIG.8: (a)Spinand(b)orbitalcorrelationfunctionsbetween
spin
nearest-neighbor sites at J=0.2.
orbital α β α β α β α β
defined by
FIG.7: (a)Spinand(b)orbitaldimercorrelation functions,
measuredfrom thecenterofthechain,atseveralvaluesofJ. D (i,j) = SzSz (SzSz Sz Sz ) , (24)
(c) Spin-orbitalconfiguration in thedimerstate. Dotted and s h i i+1 j j+1− j+1 j+2 i
D (i,j) = TzTz (TzTz Tz Tz ) . (25)
solid oval enclosures represent spin- andorbital-singlet pairs, o h i i+1 j j+1− j+1 j+2 i
respectively.
InFigs.7(a)and7(b),D andD areshown,respectively,
s o
forseveralvaluesofJ. Notethat,takingintoaccountthe
alternationofthespinandorbitaldimerizations,weshift
numerical investigations.6,7 With increasing J, as shown the starting point of Ds and Do by one, i.e., we set i=48
in Fig. 6(a), Cs gradually increases in accordance with forDs andi=47forDo with96sites. AtJ=0,wefindno
the enhancement of S(π/2), while the slope seems to be indication of dimer long-range order, since both Ds and
gentle. In particular, we observe that Cs decays as 1/r Do rapidly decay as the distance becomes large. With
at J=0.2, as denoted by the dotted line in Fig. 6(a). increasing J, Ds and Do show a tendency to growin the
At J=0.3, C comes to decay exponentially, indicating system,butwefindacleardifferencebetweenthem. Itis
s
the behavior of a quantum disordered spin state. On observedthatDsremainssmallevenwhenJ isincreased,
the other hand, as shown in Fig. 6(b), Co turns to show whileDoshowsasignificantdevelopment,indicatingthat
an exponential decay for smaller J in comparison with the orbital dimerization is stabilized by the Hund’s rule
C ,suggestingthataquantumdisorderedorbitalstateis coupling.
s
stabilized by the Hund’s rule coupling. In Fig. 7(c), we schematically depict the spin-orbital
configurationinthedimerstate. Here,wenoteagainthat
However, it is important to remark that, according
the Hund’s rule coupling induces FM and AFO interac-
tothegeneralizedKosterlitz-Thoulesstransition,17,20 the
tions. Indeed, with increasing J, the ground state turns
correlation length would diverge as ξ exp(const./J) for
∼ to be a fully polarized FM state and the orbital state is
small J. Unfortunately, from numerical calculations, it
described by a T=1/2 AFO critical state. However, the
is quite difficult to distinguish whether the correlation
ground state remains spin singlet for small J. It is con-
length is finite but very large or certainly infinite. In
sideredthatorbital-singletdimersareformedtostabilize
this sense, our results indicate that the spin correlation
aparallelspinconfigurationineachorbital-singletdimer,
develops at least within the present system size, and the
whilethecorrelationsbetweenthedimersaresuppressed.
variation of the power-law decay could be a crossover
Then, there occurs a staggered configuration of parallel
phenomenon from the SU(4) critical state to a quantum
spinpairs,leadingto the spin-singletgroundstate. Note
disordered spin state due to the finite size effect.
thatsuchaspinconfigurationisalreadyseeninthefour-
ConcerningtheorderparameteratfiniteJ,ithasbeen site problem, as shown in Fig. 3(a).
proposed that there occurs a dimer order with the alter- Wenotethatthedimerorderisassociatedwithaspon-
nating pattern of spin- and orbital-singletpairs,17 which taneouslybrokentranslationalsymmetry,andtheground
issimilartothedimerstateoftheSU(2) SU(2) statehastwo-folddegeneracywithafiniteexcitationgap.
spin orbital
exchangemodel.9 Inordertoclarifyhowspi×nandorbital The brokensymmetry is discrete rather than continuous
dimerizationsdevelopduetotheHund’srulecoupling,we rotationalsymmetry, sothata true long-rangeordercan
investigate spin and orbital dimer correlation functions, exist even in one dimension. On the other hand, in the
7
(a) 0.4 Indeed, it is observed that ∆ and ∆ at J=0 are well
s o
J=0 spin reproduced by Eq. (30), as denoted by the dotted curve.
J=0.05
0.3 JJ==00..12 With increasing J, we find that ∆s becomes small for
J=0.3 each system size up to N=192, as shown in Fig. 9(a).
Eq. (30)
∆s 0.2 Eventhoughthe groundstate is the gappeddimerstate,
the spin gap does not show a remarkable development,
since the formation of the spin dimer state is very weak.
0.1
This resultis consistent with the analyticalwork stating
that the spin gap in the thermodynamic limit is expo-
0 nentially small as ∆ exp( const./J).17
s
∼ −
(b) 0.4
On the other hand, as shown in Fig. 9(b), we observe
J=0 orbital
J=0.05 a quite different behaviorof ∆ . At the beginning of the
J=0.1 o
0.3 J=0.2 increase of J (J<0.1), ∆ keeps almost the same finite
J=0.3 o
Eq. (30) size spectrum. F∼or larger J, ∆o shows a drastic increase
∆o 0.2 foreachsystemsize,and∆o seemstoconvergetoafinite
valueinthethermodynamiclimit. Wementionthatsuch
0.1 a development of the orbital gap is accompanied by the
stabilizationofthe orbitaldimer stateduetothe Hund’s
rule coupling. Thus, the low-energy physics is described
0
0 0.01 0.02 0.03 0.04 by spin excitations below the orbital gap. We conclude
1/N
that the Hund’s rule coupling leads to the characteristic
difference between spin and orbital states, and the mag-
FIG. 9: System size dependence of (a) spin and (b) orbital
netic property is represented by an effective spin system
gapsat severalvaluesof J. Dottedcurvedenotestheanalyt-
on the background of the robust orbital dimer state.
ical result at J=0, given byEq. (30).
presentcalculations,oneofthetwopossibledimerization
patterns is favored due to open boundaries. In fact, we
canseecharacteristicspatialpatternsinspinandorbital IV. SUMMARY
correlation functions between nearest-neighbor sites,
Csnn(i) = hSizSiz+1i, (26) Inthis paper,we havestudied the effect ofthe Hund’s
Cnn(i) = TzTz . (27) rule coupling on the SU(4) spin-orbital exchange model.
o h i i+1i
The Hund’s rule coupling introduces isotropic FM and
AsshowninFig.8,wenoticethatCnnandCnnexhibitan
s o anisotropic AFO interactions as well as the spin-orbital
oscillation in an anti-phase manner. In particular, truly
coupled term, leading to the symmetry breakdown from
FM/AFO bonds appear around the edges. As shown
SU(4) to SU(2) U(1) . When the Hund’s rule
in Fig. 8(b), orbital-singlet dimers are robustly formed spin× orbital
coupling is zero, spin and orbital correlations coincide
at the edges and the whole system is efficiently covered
with each other and have a peak at q=π/2 due to the
with the orbital-singletdimers, so as to lowerthe energy
SU(4) singlet ground state. With increasing the Hund’s
makinguseofquantumorbitalfluctuation. Then,weakly
rule coupling, the peak position of orbital correlation is
formedspin-singletdimersarearrangedalternatelytothe
found to change to q=π, while that of spin correlation
orbital-singlet dimers, as shown in Fig. 7(c).
remainsatq=π/2,duetothechangeoftherelevantspin-
Nowwe turnourattentiontothe low-lyingexcitation.
orbital configuration.
To clarify how the distinct behavior of spin and orbital
states is reflected in the spin and orbital excitations, we On the other hand, the Hund’s rule coupling leads to
investigate spin and orbital gaps, the characteristic quantum behavior in a different way
between spin and orbital states. In fact, the orbital
∆ = E(1,0) E(0,0), (28)
s dimerstateisstabilizedbytheHund’srulecoupling,indi-
−
∆o = E(0,1) E(0,0), (29) cating the enhancement of quantum orbital fluctuation.
−
whereE(Sz ,Ty ) denotesthe lowestenergyinthe sub- In accordance with the robust formation of the orbital
spacewithtSotz taontdTy . InFigs.9(a)and9(b),weshow dimerstate,weobserveadevelopmentoftheorbitalgap,
tot tot while the spin gapis significantly suppressedevenin the
the system size dependence of ∆ and ∆ , respectively.
s o gapped dimer state. Thus, the low-energy physics is de-
Weshouldnotethat,attheSU(4)pointJ=0,theelemen-
scribedbyanearlycriticalspinstatewiththe smallspin
tary excitation is gapless, while the finite size spectrum
gapon the backgroundof the robustorbitaldimer state.
includes a logarithmic correction,14
We believe that the present study is useful to explore
π2 1 novel quantum states in actual low-dimensional materi-
∆ =∆ = 1 . (30)
s o N (cid:18) − lnN(cid:19) als with active orbital degree of freedom.
8
Acknowledgments ResearchResourcesofJapanAtomicEnergyAgencyand
the Grant-in-Aid for Scientific Researchin PriorityArea
The author is grateful to T. Hotta and K. Ueda for “Skutterudites”fromtheMinistryofEducation,Culture,
useful discussions and comments. He also thanks K. Sports, Science, and Technology of Japan.
Kubo for comments. He is supported by the HOUGA
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