Orbital ordering in frustrated Jahn-Teller systems

5 Pages·0.14 MB·

by  M. V. Mostovoy

Orbital ordering in frustrated Jahn-Teller systems
M. V. Mostovoy and D. I. Khomskii
Materials Science Center, University of Groningen,
Nijenborgh 4, 9747 AG Groningen, The Netherlands
(Dated: February 1, 2008)
We consider the superexchange in ‘frustrated’ Jahn-Teller systems, such as the transition metal
2 oxides NaNiO2, LiNiO2, and ZnMn2O4, in which transition metal ions with doubly degenerate
0 orbitals form a triangular or pyrochlore lattice and are connected by the 90◦ metal-oxygen-metal
0 bonds. We show that this interaction is much different from a more familiar exchange in systems
2 with the 180◦-bonds, e.g. perovskites. In contrast to the strong interplay between the orbital and
n spindegreesoffreedominperovskites,inthe90◦-exchangesystemsspinsandorbitalsaredecoupled:
a thespin exchangeismuchweaker thantheorbital oneandit is ferromagnetic forall orbital states.
J Due to frustration, the mean-field orbital ground state is strongly degenerate. Quantum orbital
3 fluctuations select particular ferro-orbital states, such as the one observed in NaNiO2. We also
2 discuss why LiNiO2 may still behaveas an orbital liquid.

] PACSnumbers: 75.30.Et,71.27.+a,75.10.Jm,71.30.+h
l
e
-
r There is a large class of compounds containing transi- systems orbitals and spins are essentially decoupled and
t
s tion metal (TM) ions with orbital degeneracy - the so- that orbital and magnetic orderings can be considered
.
t called Jahn-Teller (JT) systems. Orbital degrees of free- separately from each other. Considering in particular
a
m domand, in particular,their orderingorthe cooperative layeredmaterials with a triangular lattice, we show that
Jahn-Teller effect, give rise to a very rich physics [1, 2]. theorbitalinteractionsarestronglyfrustrated,whichre-
-
d The interplay between orbitals and spins often leads to sults in a large number of disordered mean-field ground
n rather peculiar magnetic structures, turning e.g. cubic states. However, we argue that this frustration is lifted
o perovskites like KCuF into quasi-one-dimensional spin- by quantum orbital fluctuations that stabilize a ferro-
3
c
chain materials [1]. It was recently suggested that in orbitalstate andinduce a gapin the spectrum of orbital
[
some JT systems orbitals remain disordered at all tem- excitations. We also show that the ordering of spins in

1
peratures,forminganorbitalliquidstate[3,4,5,6]. The the layers is ferromagnetic. These results agree with ex-
v
suppression of an orbital ordering was discussed for ma- perimental properties of NaNiO [8] and lead us to a
0 2
terials with a simple cubic structure, e.g. the perovskite conclusion that the puzzling absence of both spin and
2
4 LaTiO3,andwasattributedtoastronganisotropyofor- orbital ordering in LiNiO2 [9, 10] results from disorder
1 bital interactions and a strong coupling between orbital rather than from the superexchange.
0 and spin excitations. One would expect to find an even We first obtain the exchange Hamiltonian describing
2
stronger tendency to form an orbital liquid in JT com- interactions between TM ions with one electron or hole
0
/ pounds with more complicated crystal structures, espe- ondoublydegenerateeg-levels. Inthiscaseorbitalstates
t
a cially, in ‘geometrically frustrated’ systems [7], e.g. the are conveniently described by introducing isospins (or
m spinel ZnMn2O4, in which Mn3+ ions with the configu- pseudospins) Tj on each TM site j, which act on the
nd- rtweaitttrihoanhae(dtt32rrgaie,a1gno)rgfutohlraemrlalaayteptryiecrdeocmhoaflotrleoerwilaa-lstsptLiicnieNoNiOfi3c2+oarnniodenrNsihanNaritiOnhg2e teuiffpveecalytni,dvded3zoin2w−tenrr2astcaatnitodens,dxb|2eT−tzwy2e=eonr±bt21ihtiaelisdo.ernbQtitiufiaieltsdeawgneidnthes,rparilenlyssp,eoacnn-
o
(t6 e1) configuration. Orbital ordering in such systems pairs of neighboring TM sites i and j has the form [1]
c 2g g
: is an open issue, interesting from both theoretical and
v
i practical points of view, as some of these compounds HST = [JS(SiSj)+JTTiTj +JST(SiSj)TiTj] (1)
X (LiNiO2, NaNiO2) are now studied as promising mate- Xij
r rials for rechargeable batteries, and their performance
a
may be significantly affected by the JT effect. (for simplicity, we do not show here the full structure of
An important difference between the TM oxides with the orbital interactions, which are strongly anisotropic
‘frustrated lattices’ and perovskites is the angle between both in real and isospin space). In perovskites with the
the oxygen-metalbonds connecting twoneighboringTM 180◦-bonds the spin- and orbital-exchange constants J
S
ions. While in perovskites this angle is close to 180◦, and J , as well as the constant J , describing the cou-
T ST
in the geometrically frustrated JT systems it is typically pling between orbitals and spins, are all of the same or-
90◦. Though usually ignored, this difference has impor- der of magnitude. This makes the spin and orbital or-
tantconsequencesfororbitalandmagneticorderings. In derings dependent on each other [1]. The strong inter-
thisLetterwederivetheHamiltonianofthe90◦-exchange play between orbitals and spins is also crucial for the
and show that, unlike in perovskites, in frustrated JT mechanisms of suppression of these orderings used in
2
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) Ni O2 2 t, which is assumed to be much smaller than the energy
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) O ∆oftheelectrontransferfromOtoNiandtheHubbard
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) Na repulsion on oxygen Up. The orbital and spin exchange
(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) appearsinthefourthorderoftheperturbativeexpansion
z y(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) β 1 O1 and the higher-order terms are neglected here.
x α We firstneglectalsothe Hund’s rule couplingbetween
(a) (b) the spins of two holes on oxygens. In that case the 90◦-
exchangeisindependentofaspinconfigurationandonly
FIG. 1: Thecrystal structureof NaNiO2 (a) and a plaquet- involves orbitals. The orbital exchange Hamiltonian of
te in the αβ-plane (α,β = x,y,z) formed by two nearest-
the plaquette in the αβ-plane has the form
neighbor Niions, 1 and 2, and two oxygens, O1 and O2 (b).
H =J αβ +const, (3)
T T
I
Refs. [4, 11, 12] to explain an apparent absence of both
where
orbital and magnetic ordering in LiNiO [9, 10]. In par-
2
ticular,inthe SU(4)versionofthe modelEq.(1)[11,12] 3 3 3 3
αβ = Iα Iβ + Iβ Iα (4)
both the spin and orbital interactions are isotropic and I (cid:18)2 − 1(cid:19)(cid:18)2 − 2(cid:19) (cid:18)2 − 1(cid:19)(cid:18)2 − 2(cid:19)
J =J = 1J . Wefirstshowthattheactualsituation
S T 4 ST
inthe90◦-systemsisquitedifferent: theorbitalexchange and JT is the coupling constant specified below. The
is strongly anisotropic and J J J . factor 3 Iα counts the number of the oxygenelectrons
S ≪ ST ≪ T 2 − 1
We derive the exchange Hamiltonian using as an ex- that can hop to the Ni site 1 along the bond in the α-
ample the layeredmaterialNaNiO2, in which Ni and Na direction. This number equals 1, if the eg-electron is in
ions occupy consecutive [111]-planes of the NaCl struc- the state |I1α = +21i, and 2, if the electron occupies the
ture(seeFig.1(a)). Ineachplanethelow-spinNi3+ ions state |I1α = −21i. (For one hole on eg-level, as in Cu2+,
withone electrononthe doubly degeneratee levelform theexchangeinteractioncorrespondingtoEq.(4)involves
g
a triangular lattice. The exchange between two nearest- combinations 1 +Iα instead of 3 Iα .) The two
2 2 −
neighbor Ni ions in the planes occurs via common oxy- termsinEq.(4(cid:0))corresp(cid:1)ondtothetw(cid:0)oexcha(cid:1)ngepathson
gensandtheanglebetweenthe twooxygenp-orbitalsin- the plaquette through the oxygens O1 and O2.
volvedinthis exchangeis90◦. The exchangebetweenNi The orbital coupling
ions from two neighboring planes (separated by a plane
4t2 2t4 2t4U
of Na ions) is weak and will be neglected here. J = + = p (5)
Theelectronhoppinginthez-directioncanonlyoccur T −∆2(2∆+Up) ∆3 ∆3(2∆+Up)
between the pz oxygen and the d3z2−r2 nickel orbitals. is the sum of the contributions of two different mecha-
The corresponding hopping amplitude is denoted by t.
nisms of the orbital exchange. The first term in Eq.(5)
All other processes are forbidden by symmetries of the
is due to the exchange via the intermediate states with
oxygen p and nickel e orbitals. Thus, the hopping am-
g twoholes onone of the oxygensfromthe plaquette. The
plitude in the z-direction is t for the Tz = +1 nickel
| 2i mechanism resulting in the second term in Eq.(5) works
state,and0forthe Tz = 1 state. Moregenerally,the
hopping in the α-di|rection−(2αi = x,y,z) is only nonzero evenfor infinite Up, when the two-holeoxygenstates are
forbidden. To understand its origin consider first a sin-
for the oxygen pα and the nickel d3α2−r2 orbitals. It is gle Ni site inside the oxygen octahedron. The hopping
convenienttointroducetheisospinoperatorsIα =T nα,
· of electrons from the 6 oxygens to nickel results in the
where nα are three unit vectors in the (Tx,Tz) plane
energy decrease δE. For two neighboring Ni sites this
decrease is smaller than 2δE, as the surrounding octa-
1 √3 1 √3
nx = ˆz xˆ, ny = ˆz+ xˆ, nz =zˆ. (2) hedra share two oxygens and the two-hole oxygen states
−2 − 2 −2 2
are forbidden for infinite U . The effect of the blocking
p
The operators Ix and Iy play the role of Tz for the x of the two-hole states on common oxygens depends on
andy-directions,i.e. theydescribethe occupationof,re- orbital states of the eg-electrons, which gives rise to the
spectively, the d3x2−r2 and d3y2−r2 orbital. The hopping orbital exchange with the coupling constant 2∆t34.
amplitude in the α-direction is then t for the Iα =+1 The intermediate states with two holes on one oxygen
| 2i
state and 0 for the Iα = 1 state. alsoresultinaspinexchange. Asinsuchstatestheholes
| −2i
Figure 1(b) shows a plaquette in the αβ-plane (α,β = occupy two mutually orthogonal orbitals p and p , the
α β
x,y,z) formed by two neighboring Ni sites 1 and 2 and spin exchange is only possible due to the Hund’s rule
the two oxygens shared by the oxygen octahedra sur- coupling J between the spins of the holes (in systems
H
rounding the TM ions. An effective exchange Hamilto- with the 180◦-bonds the spin exchange occurs even for
nianfor twoe nickelelectronsis obtainedby expanding J = 0). The corresponding exchange Hamiltonian in-
g H
the plaquetteenergyinpowersofthe hoppingamplitude volvesbothspinandisospinoperatorsandtothe lowest-
3
z
x
exy
eyz ezx (a) (b)
FIG. 3: The self-energy diagrams due to the qubic (a) and
the quartic (b) interactions that result in a gap opening and
FIG.2: ThetriangularlatticeformedbyNiionsinthe[111]-
a two-dimensional dispersion.
plane. Shownalso isadisordered mean-fieldgroundstate,in
which the isospins form lines parallel to the unit vector exy,
such that hTzi is the same on all lattice sites, while the sign
j
of hTxi varies arbitrary from line to line. around the y-axis combined with the rotation of the tri-
j
angularlattice overthe sameangle. This transformation
is equivalent to a cyclic permutation of the indices x, y,
order in powers of JH it has the form and z of the vectors eαβ and the isospin operators Iα.
Thus the Hamiltonian Eq.(7) has the same symmetry as
3
H = J αβ +(S S ) , (6) theso-called‘compass’model,describingexchangeinter-
TS TS 1 2
− I (cid:20)4 (cid:21)
actions between TM ions on a cubic lattice [1, 13] (note,
where J = 4t4JH and αβ is given by Eq.(4). however,thatinour casethe three operatorsIx, Iy,and
TS ∆2(2∆+Up)2 I Iz are not independent).
TheinteractiondescribedbyEq.(6)isdifferentthe180◦-
The orbital exchange on a triangular lattice turns out
exchange in two important respects. First, since for
tobestronglyfrustratedandthemean-fieldgroundstate
any orbital state the expectation value αβ > 0, the
hI i is not unique. The simplest states with the minimal
spin exchange is effectively ferromagnetic independent energy E = 3J per site are ferro-orbital states, in
of an orbital state (the average spin-exchange coupling 0 −8
which T = Tm on all lattice sites, where m is an
J αβ < 0 is negative). Second, since J U , h ji
− TShI i H ≪ p arbitrary unit vector in the (Tx,Tz)-plane and T = 1.
the spin-isospin coupling in the 90◦-systems is weaker 2
The energy of these states is independent of orientation
than the pure orbital exchange, described by Eq.(3):
of m, even though the Hamiltonian (7) is not invariant
J J 2∆ underarbitraryrotationsinthe(Tx,Tz)-plane(thiscon-
TS = H .
tinuous ground-state degeneracy is similar to the one of
J U (2∆+U )
T p p
the ‘compass’ model [13]). Furthermore, there exist also
Thus, in90◦-exchangesystems spins and orbitalscannot disordered mean-field ground states, which can be ob-
strongly influence each other, i.e. they are essentially tained from the ferro-orbitalstates by inverting the sign
decoupled. (AsimilarconclusionwasreachedbyReitsma of Tx on an arbitrarily selected set of lines parallel to
h j i
andFeiner(unpublished). WearegratefultoL.F.Feiner e (see Fig. 2). Such states are ordered along the xy-
xy
for informing us about their results.) lines, but there are no long-range correlations between
We cannowobtainthe Hamiltoniandescribingthe or- the x-projections of isospins in the transverse direction.
bitalinteractionsintheNilayersofNaNiO2(seeFig.1a). By circular permutations of the x, y, and z indices one
In each layer Ni ions form a triangular lattice, the sites can obtain similar states, which are only ordered along
ofwhichlieonintersectionsofthe threesetsoflines par- the lines parallel to the e and e vectors.
yz zx
allel to the unit vectors exy, eyz, and ezx, as shown in This large ground-state degeneracy is lifted by quan-
Fig.2. Itwillbeconvenienttoidentifyeαβ withthethree tum orbital fluctuations (the so-called ‘order-from-
unit vectors in the isospace (see Eq.(2)) by nx = eyz, disorder’ mechanism [14, 15]). First, we have checked
ny =ezx,andnz =exy. Thebondsbetweenthenearest- numerically that disordered states (see Fig. 2) have a
neighborsites j andj+eαβ ofthe triangularlattice (the higherenergyofthezero-pointfluctuationsthanthecor-
lattice constant is put to 1) are diagonals of the Ni-O respondinguniformstates. Second,thezero-pointenergy
plaquettes lying in the αβ plane (cf. Fig. 1b). Hence, ofuniformstateshas6minimaattheisospinorientations
the Hamiltonian, describing orbital interactions on the m = nx, ny, nz. In the original notation these are
± ± ±
triangular lattice has the form: the ferro-orbital states with one of the orbitals d3z2−r2,
H =J IαIβ . (7) d3x2−r2, d3y2−r2, dx2−y2, dy2−z2, and dz2−x2 occupied at
T T j j+eαβ each site. Furthermore, the anharmonicity effects [16]
Xj Xαβ usually stabilize elongated octahedra, i.e. the orbitals
α6=β
d3z2−r2, d3x2−r2, or d3y2−r2. As was mentioned above,
The terms linear in the operators Iα (see Eq.(3)) are spins in layers order ferromagnetically.
j
canceled in (7), since Iα =0. We would like to point out that the standard descrip-
α j
The Hamiltonian EPq.(7) is invariant under the global tion of orbital excitations with non-interacting bosons
rotation of the isospin operators Tα over the angle 2π (‘isospin-wave approximation’) is insufficient for the cal-
3
4
culation of quantum corrections to the ground-state en- der,notincludedinthepresenttheory. Aswasarguedin
ergy of the frustrated Hamiltonian Eq.(7). In this ap- Refs. [8, 20], the ‘intrinsic’ disorder caused by the pres-
proximation the orbital excitation spectrum for the six ence ofmagneticNi ionsin the Li[111]planes,results in
ground states is one-dimensional astronginterlayercouplingthatfrustratestheferromag-
netic spin ordering in Ni layers. In NaNiO such disor-
(q m) 2
ωq =3√2TJT sin · , (8) dershouldbe weaker: asthe size differenceofNaandNi
(cid:12) 2 (cid:12)
(cid:12) (cid:12) ionsis largerthanthatofLiandNi ions,the alternating
(cid:12) (cid:12)
which is a direct consequen(cid:12)ce of the a(cid:12)bsence of long- stackingoftheNaandNi[111]layersismoreperfectthan
range correlations between chains in disordered mean- that of Li and Ni in LiNiO2. The larger charge radius of
field ground states. The gapless one-dimensional spec- the Na ion also implies stronger electron-lattice interac-
trum leads to infrared-divergentfluctuations. To get rid tions in NaNiO2. In general, such interactions suppress
of them, one has to take into account interactions be- quantumorbitalfluctuationsandstabilize aferro-orbital
tween the bosons, which can be consistently done in the ordering [21, 22]. This may explain why NaNiO2 is or-
large isospin limit, T 1. The interactions suppress bitally ordered, while LiNiO2 is not.
≫
quantum fluctuations by opening a gap and inducing a Inconclusion,wedevelopedatheoryoftheorbitaland
dispersion in the direction transverse to m. This can be spinexchangeinJTsystemswithdoublydegenerateelec-
understood from the fact that the self-energy diagrams, tronic orbitals and 90◦ metal-oxygen-metal bonds. We
showninFig.3(a)and(b), divergeunlessagapisintro- showed that in these systems spins and orbitals are es-
duced (in field theory a similar mechanism is known as sentially decoupled: spins interactmuch weakerthanor-
a ‘dynamical mass generation’ [17]). The physical origin bitals and the spin exchange is ferromagnetic for any
ofthe gapisthe breakingofthe continuousground-state orbital state. The orbital exchange between TM ions
degeneracy by quantum fluctuations [13]. In the large-T on a triangular lattice is strongly frustrated, resulting
limit the quantum fluctuations are relatively small and in infinite number of disordered classical ground states.
thegap∆ JT√T ismuchsmallerthanthebandwidth An even stronger frustration can be shown to occur in
∝
W 3√2JTT of the orbital excitations. The dispersion spinels, in which TM ions form a pyrochlore lattice (to
∼
in the transverse direction is of the order of ∆ [18]. bepublishedelsewhere). Weshowed,however,thatsmall
The mainconclusions ofour theory ofthe exchangein quantum orbital fluctuations remove the frustration and
frustrated systems (spins are coupled much weaker than induce a ferro-orbital ordering. Our theory is an agree-
orbitals, the ground state is ordered both ferro-orbitally ment with the orbital and magnetic structure observed
and ferromagnetically) are in agreement with the or- in NaNiO . We also argued that large quantum fluctu-
2
bital and magnetic structure of the layered compound ations and ‘intrinsic’ disorder may, in principle, turn a
NaNiO2,whichundergoestwotransitions: AtTo =480K frustrated system into an orbital liquid, which may ex-
the oxygen octahedra become elongated, which corre- plain the absence of orbital ordering in LiNiO .
2
sponds to the ferro-orbital ordering of the d3z2−r2-type, We are grateful to A. Stepanov and to G. Chouteau
and at the much lower temperature, T = 20K, the Ni
s andmembersofhisgroupforextensivediscussionsofthe
spins in the [111] layers order ferromagnetically [8].
experimentalsituationinLiNiO andNaNiO ,andtoG.
2 2
It is, therefore, very puzzling that the structurally
Khaliullin and L.-F Feiner for the useful discussions.
identical material LiNiO shows neither orbital nor spin
2
ordering. In principle, we cannot exclude that the quan-
tum orbital fluctuations, which for T = 1 are relatively
2
large, destroy the long-range ferro-orbital ordering and
stabilize an orbital liquid (such quantum melting of the [1] K. I. Kugel and D. I. Khomskii, Sov. Phys. JETP 52,
‘order-from-disorder’was discussed in the context of an- 501 (1981); Sov. Phys.Uspekhi25, 232 (1982).
other frustrated system in Ref. [19]). Since for T = 1 [2] Y. Tokura and N.Nagaosa, Science, 5465, 462 (2000).
2 [3] S. Ishihara, M. Yamanaka, and N. Nagaosa, Phys. Rev.
there is no small parameter in the problem, the ques-
B 56, 686 (1997).
tionwhetherthegroundstateofthe orbitalHamiltonian
[4] L.F.Feiner,A.M.Oles,andJ.Zaanen,Phys.Rev.Lett.
Eq.(7) is ordered or disordered, can only be resolved by 78, 2799 (1997).
numerical calculations that are beyond the scope of this [5] B. Keimer et al. Phys.Rev. Lett.85, 3946 (2000).
Letter. In any case, this possible explanation of the ab- [6] G.KhaliullinandS.Maekawa,Phys.Rev.Lett.85,3950
sence of orbital ordering in LiNiO is fundamentally dif- (2000).
2
ferent from those based on a strong interplay between [7] A. P. Ramirez, C. L. Broholm, R. J. Cava, and G.
R. Kowach, Physica B 280, 290 (2000) and references
orbitals and spins [4, 11, 12]. Actually, due to the de-
therein.
coupling of spins from orbitals in frustrated systems, an
[8] E. Chappel et al.,Eur. Phys. J. B 17, 609 (2000).
orbital liquid should still have a ferromagnetic order. [9] Y. Kitaoka et al.,J. Phys.Soc. Jap. 67, 3703 (1998).
WethinkthatthepuzzlessurroundingLiNiO2indicate [10] F. Reynaudet al.,Phys. Rev.Lett. 86, 3638 (2001).
an importance of electron-lattice interactions and disor- [11] Y.Q.Li,M.Ma,D.N.Shi,andF.C.Zhang,Phys.Rev.
5
Lett.81 3527 (1998). (1973).
[12] M. van der Bossche, P. Azaria, P. Lecheminant, and F. [18] Details of the calculation of thedispersion of orbital ex-
Mila, Eur. Phys.J. B 17, 367 (2000). citations, which also requires a vertex renormalization,
[13] G. Khaliullin, Phys.Rev.B 64 212405 (2001). will bepublished elsewhere.
[14] J. Villain, R. Bidaux, J. P. Carton, and R. Conte, J. de [19] P. Lecheminant et al.,Phys. Rev.B 56, 2521 (1997).
Physique 41, 1263 (1980). [20] D. Mertz et al., Phys.Rev.B 61, 1240 (2000).
[15] E. F. Shender,Sov.Phys. JETP 56, 178 (1982). [21] J. Kanamori, J. Appl.Phys., 31, 14S (1960).
[16] D. I. Khomskii and J. van der Brink, Phys. Rev. Lett. [22] R.EnglmanandB.Halperin,Phys.Rev.B2,75(1970).
85, 3329 (2000).
[17] S. Coleman and E. Weinberg, Phys. Rev. D 7, 1888