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Octupole Moment as a Hidden Order Parameter in Orbitally
Degenerate f-Electron Systems
Yoshio Kuramoto and Hiroaki Kusunose
0
0 Department of Physics, Tohoku University, Sendai, 980-8578
0
(Received )
2
n Possibility of a novel pseudo-scalar (octupole) order is studied theoretically for orbitally de-
a generatesystemswithstrongspin-orbitcouplingsuchasCexLa1−xB6. Itisdiscussedthatcoex-
J
istence of an octupole order parameter and antiferromagnetic fluctuation should lead to drastic
8 softening of the elastic constant by a mode-mixing effect. Nonlinear coupling between dipole,
1 quadrupoleandoctupolefluctuationsistakenintoaccountintermsofaGinzburg-Landau-type
functional which is derived microscopically through path integral.
]
l
e KEYWORDS: octupole moment, CexLa1−xB6, ultrasound, elastic constant, orbital degeneracy, time reversal,
-
multipole,
r
t
s
.
t
a
m In this paper we study possibility of a new pseudo- The four-fold degeneracy can be broken in a variety of
- scalar order in orbitally degenerate systems with strong ways depending on types of symmetry-breaking fields:
d spin-orbitinteraction. Ourmotivationisastrangephase (a) magnetic order — In this case there remains no de-
n
calledIVfoundinCe La B withx 0.75. Thephase generacy since the Zeeman splitting caused by internal
o x 1−x 6 ∼
IV has the following properties: magnetic field is different between the two orbitals.
c
[ (i) The transverse elastic constant C shows a drastic (b) orbital order — The two-fold Kramers degeneracy
44
softening ( 20%).1) remains since the time-reversalsymmetry is not broken.
1 ∼
v (ii) There is almost no magnetoresistance in contrast In the phase II of CeB6 the antiferro-orbital order (or
8 with other phases.2) thequadrupoleorder)isrealized. InthephaseIIIsimul-
3 (iii) The phase IV is isotropic magnetically in contrast taneous presence of (a) and (b) have been invoked for
2 with phases II and III.3) explanationoftheneutronscatteringexperiment.5) One
1
Although the magnetic susceptibility shows a cusp at can then ask whether it is possible to have
0
the phase transition to the phase IV,3) preliminary ex- (c) breakdown of the time-reversal symmetry without
0
0 periment of neutron scattering4) has found no magnetic breaking the orbital degeneracy.
/ Braggscatteringalonghighsymmetryaxessuchas(100), Inthispaperweassertthatthequestioncanbeanswered
t
a (110) and (111). The experimental fact (iii) suggests in the affirmative.
m
that the order parameter is a scalar or a pseudo-scalar To describe the spin and orbital degrees of freedom
- instead of a dipole or a quadrupole. A pseudo-scalaror- in a concise way, we introduce the pseudo-spin opera-
d
der may bring a gap or a pseudo-gap in the magnetic torswhicharerepresentedbytwosetsofPaulimatrices:
n
o excitation spectrum with zero momentum since there is σx,σy,σz and τx,τy,τz . The formeroperatesonthe
{ } { }
c no Goldstone mode in contrast with the N´eel order. It Kramerspartners,andthelatterontheorbitalpartners.
:
v is conceivable that this is related to the fact (ii). Namely we have
i Thesefactstogetherwiththemostconspicuousfact(i)
X σz ψ = ψ , τz ψ =( 1)α−1 ψ , (2)
suggestthatanew type oforderisinvolvedinthe phase | α±i ±| α±i | α±i − | α±i
r
a IV.Wehypothesizethatthepseudo-scalarcomponentof with α=1,2. The x and y components of pseudo-spins
theoctupolemomentistheorderparameterofthephase change from one state to another in the Γ subspace.
8
IV, and explore some consequence of the hypothesis. In We can express the physical operators adapted to the
particularwe providepossible mechanismto explainthe point-groupsymmetryusing the pseudo-spins. They are
fact (i) in terms of the octupole order. enumerated as follows:6,7)
In order to characterize the new order we begin with
the symmetry analysis of the Γ8 level which is the Γ2u : {τy },
crystalline-electric-field ground state of each Ce3+ ion Γ : τz,τx ,
3g
in Ce La B . The excited level Γ lies about 500 K { }
x 1−x 6 7
from the Γ8 level, and can be safely neglected for our Γ(41u) : {σx,σy,σz },
purpose. The four states in the Γ8 level are represented Γ(2) : η+σx,η−σy,τzσz ,
with use of the basis Jz of J =5/2 states as follows: 4u
| i Γ5u : (cid:8)ζ+σx,ζ−σy,τxσz (cid:9),
5 5 1 3 1
|ψ1±i= 6 ±2 + 6 ∓2 , |ψ2±i= ±2 . (1) Γ5g : (cid:8)τyσx,τyσy,τyσz (cid:9),
r (cid:12) (cid:29) r (cid:12) (cid:29) (cid:12) (cid:29) { }
(cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12)
2 YoshioKuramotoandHiroakiKusunose
where we haveintroducedlinear combinationsof τx and quadrupole order occurs first as temperature is lowered
τz as inpureCeB ,substitution ofLa mayfavorabreakdown
6
1 of the time-reversal as the first instability. We interpret
η± = ( √3τx τz), (3)
the phase IV as being realized in this way.
2 ± −
The pseudo-spin representation is introduced just to
1
ζ± = (τx √3τz). (4) reproduce the matrix elements of multipole operators.
−2 ±
Hence any approximation to decouple σ and τ is not
The subscript u represents the odd property under the
meaningful physically. Instead we should consider on
time reversal, and g does the even one.
equal footing each fluctuation with a point-group sym-
Amongtheseoperators,τy inΓ hasthesamematrix
2u metry. To deal with coupled multipole fluctuations
element as the symmetrized product of JxJyJz, and is
from the high-temperature side, we work with the path-
regarded as a component of the octupole moment ten-
integralrepresentationofthepartitionfunction. Namely
sor.6,8) This component commutes with discrete rota-
weuse the Stratonovich-Hubbardidentity to replacethe
tions of the cubic group, and is odd under the time re-
intersiteinteractionbythelocalinteractionbetweenaux-
versal. The latter property is consistent with τy being
iliary fields and multipole moments.9) The most impor-
pure imaginary. Hence τy is regardedas a pseudo-scalar
tanttechnicalpointisthatwecantranslatethefullnon-
operator. On the other hand the operators belonging to
linearity represented by the pseudo-spin operators into
Γ andΓ describethequadrupoleoperatorswhichare
3g 5g the explicit coupling term in the Ginzburg-Landau-type
even under the time reversal. The dipole operators un-
(GL-type) functional. After this translation we can in-
derthecubicsymmetryisdecomposedintoΓ(1)andΓ(2).
4u 4u troduce a suitable approximationsuch as the mean-field
For example the x-component of the magnetic moment
theory with respect to these auxiliary fields.
is given in units of the Bohr magneton by
The local interaction at each site i is of the form
4
Mx =σx+ η+σx, (5) φ σ +ξ σ τy +ψ τy. (7)
7 i· i i· i i i i
where the first term on the right-hand side belongs to These auxiliary fields φ ,σ and ψ obey the Gaussian
i i i
Γ(1) andthesecondonetoΓ(2). Wenotethateachthree- distribution. In this paper we confine ourselves to the
4u 4u
dimensional odd representation is a linear combination static approximationwhere dynamical fluctuation is ne-
of dipole and octupole operators. In other words,dipole glected. Although some quantum effects already escape
and a part of octupole operators mix under the point- atthisstage,ourapproximationkeepsfaithfullythenon-
groupsymmetry. The remainingrepresentationΓ cor- trivial commutation property of pseudo-spins. Hence
5u
responds to pure octupole operators other than τy, and we expect that interesting consequences due to coupling
describes a part of third-rank tensors composed of Jα. between different multipoles can be understood qualita-
We take the simplest possible model to describe the tively within the static approximation.
coupling among dipole, quadrupole and octupole mo- Incomputingthe thepartitionfunction,wefirstcarry
ments. The conduction electrons which give rise to the out the trace over orbital part taking such basis that
Kondoeffectarenottreatedexplicitly. Forthemagnetic makes τy diagonal. Then we are left with another trace
moment, we keep only the representation Γ(1) and ne- over the Kramers partners. We note that an effective
4u
glect the orbital dependent part Γ(42u) for simplicity. The magnetic field which couples with σi is given by φi ±
effectoftheneglectedpartwillbediscussedintheendof ξi. Then the latter trace is performed most easily by
the paper. For the quadrupole moment, we include only rotating the quantization axis so that φi ±ξi is along
the Γ5g component which is known to be dominant in the z-axis9,10) of σi. In this way we obtain
CeB . In addition to these we include the pseudo-scalar
6
Z =Trexp( βH)= φ ξ ψexp( β ), (8)
component Γ of the octupole moment. The Hamilto-
2u − D D D − F
nian is given by Z
where the functional consists of three parts: =
1 F F
H = [J(m)σ σ +J(e)τyσ σ τy +J(8)τyτy], NTln4+ 0+ 1. The first part is the entropy term
−2 ij i· j ij i i· j j ij i j −and the secoFnd onFe, which describes the Gaussian fluc-
i6=j
X
(6) tuation, is given by
where J(α) with α = m,e,8 describes intersite interac- 1
tionofeiitjhermagnetic,electric(quadrupole)oroctupole F0 = 2 {[Jm(q)−1−β]φq ·φ−q +
q
degrees of freedom. X
theSidnicpeoltehσe qaunaddoructpuopleoloepτeyraotpoerraτtyoσrs,isthtehreepisroindturcintsoicf [Je(q)−1−β]ξq ·ξ−q +[J8(q)−1−β]ψqψ−q}, (9)
where J (q) with α = m,e,8 is the Fourier transform
frustration if all of them favor the antiparallel arrange- α
of J(α). On the other hand the interaction part is
ment between nearest-neighbor pairs. The complicated ij F1
magnetic structure of phase III seems to be realized as derived in the closed form from
a compromise between these competing interactions. If
exp[ β( + )]= 2[cosh(β φ +ξ )exp( βψ )
there is a delicate balance in realizing the actual struc- − F1 F02 | i i| − i
i
ture, slight change of the balance with substitution of Y
+cosh(β φ ξ )exp(βψ )], (10)
Ce by La might lead to another structure. Although a | i− i| i
OctupoleMomentasaHiddenOrderParameter 3
where consists of the entropy term and the second- pled to quadrupole moments by
02
F
order part which is already accounted for as the terms
H = g(Γ)ǫ (R )O (R ), (16)
proportional to β in eq.(9). By expanding up to ext Γγ i Γγ i
1
fourth order with respect to auxiliary fields weFobtain Xi XΓγ
where g(Γ) is the coupling constant of a representation
=β2 (φ ξ )ψ
F1 i· i i Γ, which runs over Γ3g and Γ5g, and the strain ǫΓγ(Ri)
Xi at site i and the quadrupole moment OΓγ(Ri) belong
+β3 (φ2)2+(ξ2)2+ψ4+2(φ ξ )2 .(11) to the same representation. For the quadrupolemoment
12 i i i i× i withtheΓ symmetry,thecomponentsO correspond
i 5g Γγ
X(cid:2) (cid:3) to Cartesian ones O as
αβ
This GL-type functional provides us with the starting
point for discussing growth of the order parameter and (Oxy,Oyz,Ozx) (τyσz,τyσx,τyσy). (17)
∝
couplingamongmagnetic,quadrupoleandoctupolefluc-
The change ∆C of the elastic constantandthe change
tuations. 44
∆v of the transverse sound velocity are related to the
The relation between the octupolar susceptibility s
homogeneous quadrupolar susceptibility χ by
χ8(q) and the fluctuation ψqψ−q is given by e
h i ∆C 2∆v g(Γ )2
J8(q)2χ8(q)=βhψqψ−qi−J8(q). (12) C4444 = vss =− M5vgs2 χe, (18)
We have analogous relations also for the magnetic sus-
where M denotes the mass of the unit cell. Therefore
ceptibility χ (q) and for the quadrupolar susceptibility
m the enhanced χ leads to softening of the C mode.
e 44
χ (q). If one neglects in taking the thermal average,
e 1 The Γ order breaks the time-reversalinvariance but
F 2u
one obtains the RPA result given by
leaves the orbital degeneracy in contrast with the mag-
netic order. With a finite octupole order parameter,
χ (q)=β/[1 J (q)β]. (13)
8 8
− theorbitalfluctuationhybridizeswiththemagneticfluc-
Letusassumethatthehigh-temperaturephasebecomes tuation. Then the quadrupolar susceptibility measured
unstablefirstagainstformationofanoctupoleorderwith by ultrasound probes the magnetic fluctuation with the
the wavevectorQ. ThetransitiontemperatureT8 is de- wave vector Q. Note that the wave number of ultra-
termined in the RPA (or the mean-field approximation) sound is negligible as compared with Q. We consider
| |
as the z-component of φQ and ξ0, and omit writing the
componentindexsinceothercomponentsfollowthesame
T =J (Q). (14)
8 8
equation. IntheRPA,weobtainthefollowingequations
One could include fluctuation corrections coming from to determine χ :
e
. However such sophistication is not necessary to our
F1 J (0)2χ =β ξ2 J (0), (19)
purpose of demonstrating the mode mixing in the pres- e e h 0i− e
ence of an octupole order. β hξ02i, hξ0φQi =
detTehrme imneadgnbityudtheeofmtehaen-ofiredldertphaeroarmyewtehrichhψQreiduccaens btoe hφQξ0i, hφQφ−Qi !
thestandardGLtheorynearthetransitiontemperature. J (0)−1 β, β2 ψ −1
e Q
We assume that Q is given by (1/4,1/4,1/4)in units of β2 ψQ ,− Jmh(Q)−i1 β ! , (20)
2π/awhereaisthelatticeparameterofthecubiccrystal. h i −
ThischoiceofQismotivatedbytheknownmagneticand where we take the order parameter real. This equation
quadrupolepatternsinthephaseIII.5) Namelythemag- appliestothetemperaturerangewhereneithermagnetic
netic supercell in the (001) plane contains eight Ce ions nor quadrupole order is present. Solving eq.(20) we ob-
inthe2√2 2√2structureandorbitalsupercellcontains tain
×
two Ce ions in the √2 √2 structure. To be consistent
withthesestructuresth×eoctupoleordershouldalsohave χ = 1 1−jm 1 ,
the 2√2 2√2 structure which correspondsto the wave e Je(0)"(1 jm)(1 je) jmjeβ2 ψQ 2 − #
− − − h i
×
vector (1/4,1/4)in the plane. Assuming the cubic sym-
where j = J (Q)β and j = J (0)β. It is easily seen
m m e e
metry at T we expect the wave vector (1/4,1/4,1/4)
8 that the result reduces to the conventional mean-field
(together with its stars) as a reasonable candidate.
one above the transition temperature T .
8
In the mean-field theory we replace fluctuating fields
Figure 1 shows an example of numerical results with
in by their averages. Then the stationary condition
F tentative values of interactions: Jm(Q)/T8 = 0.6 and
δ /δ ψ =0 leads to the result
F h Qi Je(0)/T8 = 0.2. We have taken the negative value
−
ψQ 2/N =4T2(1 T/T8), (15) of Je(0) to be consistentent with the nearest-neighbor
h i − antiferro-quadrupolar interaction. It is seen that the
where N is the total number of unit cells. and we take quadrupole susceptibility increases significantly below
intoaccountthefactthatthereareeightequivalentQ’s. T . The reasonforthe increaseis the increasedcoupling
8
In ultrasonic measurement, the external strain is cou- with growing antiferromagnetic correlation with the
wavevectorQ. Notethatthisantiferromagneticcorrela-
tioniscloselyrelatedtothewavevector(1/4, 1/4,1/2)
±
4 YoshioKuramotoandHiroakiKusunose
ofthemagneticorderinthephaseIII.Hence thegrowth tions. The drastic softening of C is interpreted as a
44
of this correlation is naturally expected. consequence of the coupling effect. We now discuss pos-
sible directions of further development. One can com-
pute the magnetic susceptibility χ in a manner similar
m
5 to what we have done for χe. The homogeneous mag-
J(0) = - 0.2T netic susceptibility is influenced by the quadrupole fluc-
e 8
tuation with the wave vector Q. It turns out that the
4 J (1/4,1/4,1/4) = 0.6 T susceptibility has a cusp at T provided that J (Q) is
m 8 8 e
negative. This behavior is similar to the one observed
experimentally.3) If J (Q) is positive, on the contrary,
χTe8 3 χm increases below T8elike χe. Unfortunately we do not
havefurtherinformationonJ (Q)orχ (Q). Thelackof
e e
information is in contrast with χ (Q) which is related
m
2
to the magnetic order in the phase III as we discussed
above.
Experimentally, the ordered magnetic moment in the
1
phase III lies in the (001) plane. This anisotropy is a
consequence of the orbital order with the Γ symmetry
5g
wherethewavefunctionextendstoward(110)or( 110)
−
1 1.5 2 depending on the sublattice.5,15) It should be possible
toidentify the octupole orderifone canobserveinduced
T/T
8
magnetic moment under uniaxial stress. Namely if the
stress is applied along the (110) direction, we expect
Fig. 1. The quadrupolar susceptibility plotted as a function of that a magnetic moment with Q is induced along the
temperature. The energy is normalized by T8 = J8(Q). The (001) direction. Recently large change of the magnetic
interactionsaretakentobeJm(Q)=0.6T8andJe(0)=−0.2T8. anisotropy was found in phases III and IV by applica-
tion of uniaxial stress.16) Such anisotropy is taken into
account only when one includes the Γ(2) component in
4u
With the wave vector Q = (1/4,1/4,1/4) the transi- additiontoσ includedinthis paper. We planto include
tion between the phases IV and III should be of first (2)
the Γ component and the quantum fluctuation effects
4u
order since the phase III has different wave vectors
in a future publication.
(1/4, 1/4,1/2) for the superstructure. This is consis-
±
tent with experimental observation.3) The transition to
Acknowledgement
thephaseIIwithincreasingmagneticfieldshouldbealso
ThisworkissupportedbyaGrant-In-AidforScientific
of first order with the present Q, since the phase II has
Researchfromthe MinistryofEducation,Science,Sport
the wave vector (1/2,1/2,1/2) as the quadrupole order,
and Culture, Japan.
which should mix with the octupole order in magnetic
field. Experimentally the phase boundary is extremely
narrow;theboundariestophasesIandIIIseemtomerge [1] S. Nakamura, O. Suzuki, T. Goto, S. Sakatsume, T. Mat-
withtheII-IVboundaryinthe(T,H)planewithH par- sumuraandS.Kunii,J.Phys.Soc.Jpn.66552(1997).
allel to (001). Our model is probably too simple to be [2] M. Hiroi, M. Sera, N. Kobayashi and S. Kunii, Phys. Rev.
B558339(1997).
appliedto suchdetails ofthe phasediagram. IfQofthe
[3] T. Tayama, T. Sakakibara, K. Tenya, H. Amitsuka and S.
octupole order were the same as that of the quadrupole Kunii,J.Phys.Soc.Jpn.662268(1997).
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IVshouldhavedisappeared.11) Thisissimilartotheab- [5] J.M.Effantin,J.Rossat-Mignod,P.Burlet,H.Bartholin,S.
senceofphaseboundarybetweentheN´eelphaseandthe Kunii and T. Kasuya, J. Magn. Magn. Mater. 47&48 145
(1985).
antiferro-quadrupolar phase in TmTe under finite mag-
[6] R.Shiina, H. Shibaand P.Thalmeier, J.Phys. Soc. Jpn. 66
netic field.12,13)
1741(1997).
Recently a phase diagram analogous to that of [7] H.KusunoseandY.Kuramoto,Phys.Rev.B591902(1999).
Ce La B hasbeenfound14)inatetragonalcompound [8] L.I. Korovin and E.K. Kudinov, Sov. Phys. Solid State 16
x 1−x 6
HoB C for magnetic field along (110). It appears that 1666(1975).
2 2 [9] Y. Kuramoto and N. Fukushima, J. Phys. Soc. Jpn. 67 583
the phase IV in this compound has a N´eel order. It
(1998).
should be interesting to see whether there is an elas- [10] N.FukushimaandY.Kuramoto,J.Phys.Soc. Jpn.672460
tic anomaly in the phase IV of HoB2C2. We note that (1998).
the angular momentum of the Hund-rule ground state [11] H.Shiba,privatecommunication.
of Ho3+ is as large as J = 8, and that 4f electrons here [12] T.Matsumuraetal.,J.Phys.Soc.Jpn.67612(1998).
[13] R.Shiina,H.ShibaandO.Sakai,J.Phys.Soc.Jpn.682105
aremorelocalizedthaninCe La B . Hencequantum
x 1−x 6 (1999).
fluctuations should be less significant in HoB C .
2 2 [14] H. Onodera, H. Yamauchi and Y. Yamaguchi, J. Phys. Soc.
In this paper we have proposed the simplest theory Jpn.682526(1999).
that can describe the coupling effect between the oc- [15] M.SeraandS.Kobayashi,J.Phys.Soc.Jpn.681664(1999).
tupole order and dipole as well as quadrupole fluctua- [16] T.Takikawaetal,preprint(1999).
