Table Of ContentInstabilities and Insulator-Metal transitions in Half-Doped Manganites induced by
Magnetic-Field and Doping
O. C´epas,a,b,∗ H. R. Krishnamurthy,a,c and T. V. Ramakrishnana,c,d
a. Department of Physics, Indian Institute of Science, Bangalore 560012, India.
b. Institut Laue Langevin, BP 156, 38042 Grenoble, France.
6 c. Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore 560 064, India.
0 d. Department of Physics, Banaras Hindu University, Varanasi 221005, India.
0 (Dated: February 6, 2008)
2
We discuss the phase diagram of the two-orbital model of half-doped manganites by calculating
n self-consistentlytheJahn-Teller(JT)distortionpatterns,charge,orbitalandmagneticorderatzero
a temperature. We analyse the instabilities of these phases caused by electron or hole doping away
J
from half-doping, or by the application of a magnetic-field. For the CE insulating phase of half-
0 doped manganites, in the intermediate JT coupling regime, we show that there is a competition
2 between canting of spins (which promotes mobile carriers) and polaronic self-trapping of carriers
by JT defects. This results in a marked particle-hole asymmetry, with canting winning only on
] the electron doped side of half-doping. We also show that the CE phase undergoes a first-order
i
c transition to a ferromagnetic metallic phase when a magnetic-field is applied, with abrupt changes
s in the lattice distortion patterns. We discuss the factors that govern theintriguingly small scale of
-
l the transition fields. We argue that the ferromagnetic metallic phases involved have two types of
r
charge carriers, localised and band-like,leading to an effectivetwo-fluid model.
t
m
.
t I. INTRODUCTION doped” side (x < 1/2), and is often accompanied by
a
m metalicity (except in the PrCa system, which stays in-
sulating for all x). This asymmetry between ”electron
- ”Half-doped” manganites, corresponding to the gen-
d doping” and ”hole doping” away from half-doping is a
eral formula Re A MnO with x = 1/2 where Re is
n 1−x x 3 striking feature of the majority of the half-doped man-
o a 3+ rare-earth metal ion and A a 2+ alkaline earth ganites.
c metalion,eg.,La1/2Ca1/2MnO3,havebeentheobjectof
[ extensive experimental and theoretical studies for many One simplifying feature of the half-doped manganites
years.1,2HereeachMnhasanaveragevalenceof3.5+i.e., is that the low temperature phase is generally regarded
1
v an average configuration of d3.5, corresponding to one as reasonably well characterised. Early neutron diffrac-
tion work by Wollan and Koehler4 suggested that the
9 Mn-eg electron for every two Mn sites hopping around
6 amongst the two [(x2 y2) and (3z2 r2)] e orbitals magneticstructureofLa1/2Ca1/2MnO3 canbeviewedas
g
4 on each Mn. The rema−ining three t −electrons on each a set of ferromagnetic zig-zag chains antiferromagneti-
2g
1 callyorderedrelativetoeachother,withan8-sublattice,
Mnarespin-alignedbystrongcorrelations(Hund’srules)
0 (2√2 2√2) unit cell, andis referredto as the CE mag-
to form ”core spins” with S =3/2. Similarly to the end
6 netic×order (Fig. 1). The structure was qualitatively
members,i.e.,LaMnO orCaMnO ,thehalf-dopedcom-
0 3 3 explained soon thereafter by Goodenough,5 who pro-
/ pounds, thanks to their commensurate filling, are sim-
t posedadditionallythatthe phasealsohasa2-sublattice,
a pler in some ways than the doped manganites for gen-
m eralx.1,2 Nevertheless,theyexhibitaveryrichvarietyof √2 √2 charge order with alternating Mn3+ and Mn4+
×
properties as well.1 Specifically, as the system is cooled, sites, and a 4-sublattice, 2√2 √2 striped orbital or-
- ×
d therearephasetransitionsinvolvingchangesinmagnetic, der, as indicated in Fig. 1. Since then, CE order has
n charge and orbital order, and, in some cases, metalicity. been found in several other half-doped systems, such as
o Nd Sr MnO 6,7 orNd Ca MnO 8,thoughsome,
The details vary from material to material, depending 1/2 1/2 3 1/2 1/2 3
:c systematically on the sizes of the ”A site” ions of the such as Pr1/2Sr1/2MnO3,6,7 show A-type antiferromag-
v perovskite structure. Generally, the lowest temperature netism, corresponding to [0,0,π] spin order, i.e, ferro-
Xi phaseisinsulating,withsimultaneouscharge,orbitaland magnetic planesofspins which areantiferromagnetically
CE type antiferromagnetic order (see below), and the aligned in the z-direction).
r
a charge/orbitalordersetsinfirst,athighertemperatures, The presence of charge and orbital order is, however,
compared to the antiferromagnetic order (e.g., for PrCa harder to establish directly experimentally because of
T 240K,whereas T 170K3). The NdSr and the lack of experimental probes that couple directly to
CO/OO N
∼ ∼
PrSrsystemsshowferromagneticmetallic orderatinter- them. Indeed, the perfect Mn3+/Mn4+ charge order-
mediate temperatures, but in the LaCa and PrCa sys- ing proposed by Goodenough.5 is currently regarded as
tems, the different phases obtained with increasing tem- controversial9,10,11,12. X-ray diffraction data do suggest
perature continue to be insulating. Typically,the charge the presence of large Jahn-Teller (JT) distortions of the
order and insulating behaviour at low temperatures per- oxygenoctahedra surrounding the Mn sites7,13 with two
sist on the ”over-doped” side (x > 1/2), whereas the inequivalent Mn sites, of effective valence 3.5 + δ and
charge order disappears rather quickly on the ”under- 3.5 δ, but δ is not really known, and is unlikely to be
−
2
closeto0.5. InPr0.6Ca0.4MnO3 (whichisslightlyunder- CE−CO
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shared by two Mn sites paired in dimer-like structures,
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0.5 0.5 3
claims indeed to confirm the picture of the original CE
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ions, substantial hole occupancy on the oxygen ions on a z(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)
the chains, and alternating ”O2−/O−” order.17 There
FIG.1: (color online). A depiction of theCE charge-ordered
havebeenX-raystudiesonorbitalorderandcorrelations
antiferromagnetic phase (CE-CO). The ”bridge sites” (1,3),
as well as charge and magnetic order using a variety of
at the centers of the arms of the zig-zag chains which are
methods such as soft x-ray resonantdiffraction,18 coher-
ferromagnetically ordered, have alternate occupancies of the
ent x-ray scattering,19 which explore the spatial extent 3x2 r2 and 3y2 r2 orbitals. At the ”corner sites” (2,4),
of orbital order, in particular. The resonant scattering − −
there is no orbital order, unless JT interactions are present.
experimentsinPr0.6Ca0.4MnO3concludethatthecharge δisthechargedisproportionationbetweentheoccupanciesof
disproportionationislessthancomplete,thatthereisor- thecorner sites and thebridge sites.
bital mixing, and that therefore the simple Goodenough
modelisnotright. Onthe otherhand, whenholesarein
excess (x>0.5),it has been suggestedthat chargeorder
been examined in the context of half-doped manganites
persists but becomes incommensurate.20,21
as well.1 The simplest model has mobile electrons mov-
A closely related family of manganites carefully stud-
ing amongst non-degenerate orbitals, coupled to the Mn
iedrecentlyisA A’ MnO whereAisarareearth(Y,
0.5 0.5 3 t core spins by a large Hund’s rule (double) exchange
Tb, Sm, Nd, Pr, La in order of ion size) and A’ is Ba.22 2g
coupling J . The latter promotes ferromagnetism, but
H
The phases have been studied as a function of A-A’ site
competes with a direct antiferromagnetic coupling J
AF
order/disorder. When there is order, the low tempera-
between the core spins. Even in this simple model, the
ture phase is charge-ordered(CO) and orbitally ordered
ferromagneticorCEtypesoforderarestabiliseddepend-
(OO) for ion size from Y to Nd. Beyond Nd, up to La,
ing upon the strength of J .26,27 Van den Brink et al.
AF
the phaseis aferromagneticmetal. However,whenA-A’
considered a more realistic model with the two types
sitesaredisordered,thereisnoCOphaseatall,butonly
of e orbitals of Mn, and found that the CE phase is
g
a spin glass (SG) phase, from Y to Sm, after which the
orbitally-ordered: the”bridgesites”ofthezig-zagchains
groundstateisferromagnetic(FM).Thismeansthatthe
have alternating preferred occupancy of (3x2 r2) and
CO/FMand CO/SGcompetition depends on ionsize as (3y2 r2) orbitals28 as indicated in Fig. 1. −They also
wellonAsiteorderingintheperovskiteABO structure. −
3 showed that a charge contrast δ (not bigger than 0.2)
Another interesting and intriguing property of the can be generated by including on-site Coulomb inter-
CE charge-ordered (CE-CO) phase is the magnetic-
action. This is because the ”corner sites” turn out to
field-induced insulator-metal transition first discovered have equal occupancy of (x2 y2) and (3z2 r2) or-
in (Nd,Sm) Sr MnO ,23,24 and later shown to be − −
1/2 1/2 3 bitals, and that costs Coulomb energy. To reduce this,
ubiquitous1. Though insulating at zero field, these ma- the system adopts a preferred occupancy of the bridge
terials become metallic by the application of magnetic- sites which are orbitally ordered. In later work, nearest
fields of the order of 10 - 40 Tesla via sharp, first- neighbour Coulomb interactions were also included.29,30
order, resistive transitions.23 The magnetic-field ener- However, the charge order due to long-range Coulomb
gies involved are much smaller than the thermal ener- interactions is generally of the Wigner-type with wave
gies (of order 200 K) needed to destroy the charge or- vector Q=(π,π,π), contraryto the (π,π,0) order,with
der, and orders of magnitude smaller than the charge chargestackingalongthez-direction,suggestedbyexper-
gap of 0.2 - 0.3 eV, as observed as a function of field iment. To stabilise the (π,π,0) order in a wider regime
by STM in Nd0.5Sr0.5MnO3.25 This can be viewed as a of parameters,JT interactions between the Mn ions and
differentmanifestationofthecolossalmagneto-resistance their surroundingoxygenoctahedra,whicharesupposed
seen at the metal-insulator transition of doped mangan- to be quite large,13 have to be included.31 The conse-
ites for x 0.25,1 and the microscopic understanding quent JT distortions further lower the energy of the CE
∼
of the above features poses similar difficult theoretical phasebecauseofthealreadypresent(3x2 r2)/(3y2 r2)
challenges.1,2 orbital order. Classical Monte Carlo simu−lations inc−lud-
Theory. A variety of models and mechanisms have ing static JT distortions on small clusters as well as self-
3
consistent mean field treatments of models including JT particularweobtainanalyticresultsforthephasebound-
and Coulomb interactions31 suggest that the CE charge aries at strong JT coupling.
stacked state has the lowest energy in an intermediate Next, we show that the periodic ferromagnetic phase
range of JAF, unless the nearest neighbour coulomb in- obtainedatsmallJAF bythemethoddiscussedabovecan
teraction V becomes much too large. become unstable with respect to a phase with two types
However, to our knowledge very few of these studies (ℓ b) of electrons when the JT coupling is lowered. We
−
have addressed the other issues, such as the magnetic- show indeed that it becomes energetically favourable to
field-inducedinsulatormetaltransition,andtheelectron- create single site JT defects, i.e. release the distortion
hole asymmetry. The first issue was tackled in refs. onafinite numberofsitesthatwerepreviouslydistorted
[26,27,32] by assuming model parameters very close to and promote previously trapped electrons onto a mobile
the phase boundary between the ferromagnetic and CE band,thussuggestingametallicphase. Theexactnature
states. Theresultingsmallenergydifferencebetweenthe of the phase can not be figured out by such an instabil-
two phases can then be overcome by an arbitrary small ity analysis. Nonetheless, it suggests an effective (ℓ b)
−
magnetic-field. But it is hard to justify why the sys- Hamiltonian with orbital degrees of freedom explicitly
tem parameters should be so finely tuned for so many included.
systems. As regards the second issue, band structure The observed phases at half-doping, such as the CE
arguments,28 andtreatmentsincludingJTdistortionson phase,areantiferromagnetic,correspondingto appropri-
small clusters31 necessarily lead to metallic phases upon ately larger values of J . But they show transitions to
AF
additionofelectronsorholes,incontrasttoexperiments. ferromagnetic metallic phases in an external magnetic-
Recently, a theory for doped manganites has been field or when electrons are added. To understand such
proposed33 where it is argued that due to strong JT in- transitions, in addition to considering changes in the JT
teractions the e electrons dynamically reorganisethem- distortions, canting of spins is important.
g
selves into two types. The majority of the electrons (la- Canted phases are expected to appear not only in a
belled ℓ) become localised polarons, trapped by large lo- magnetic-field, but also upon doping with carriers (and
calJTdistortions;andaminorityofthem(labelledb)can irrespective of their nature), following the original ar-
remain mobile and non-polaronic. Still virtual adiabatic gument by de Gennes.35 Here we show, however, that
transitions to empty neighbouring sites induces a ferro- canted metallic phases appear only when electrons (and
magneticexchangereferredasvirtual doubleexchange.33 notholes)areadded,becauseoftheunderlyingasymme-
The resulting Falicov-Kimballlike, ℓ b model Hamilto- tryofthedensityofstatesathalf-doping. Whenweallow
−
nian treated in a simple dynamicalmean-field treatment for JT distortions, we find a competition with a disor-
in the framework of an ”orbitalliquid” description, gave deredphasewheretheaddedelectronsaretrappedbyJT
a good accountof the properties of doped manganites.33 distortions, the latter phase winning only at small elec-
In this paper, we propose an extension of the above tron concentration. On the hole-doped side, added holes
theory to the half doped case, which has to include the aresimplytrappedbythelatticedistortionsandthesys-
possibilitiesfororbital,charge,andantiferromagneticor- tem remains insulating. Thus our work provides an ex-
der. We obtain pointers to this by studying the proper- planation for the particle-hole asymmetry near x 1/2,
∼
ties of electronic excitations coupled with JT defects in atintermediateJTcouplingswhichwearguearerelevant
the lattice distortion pattern. We find that such a study for the majority of the manganites.
suggests the incipient instabilities of the CE phase in- Similarly, we find that there is a strong interplay be-
dicative of the doping and magnetic-field induced phase tweenturning ona magnetic-fieldathalfdoping andthe
transitions seen experimentally, as well as the presence JTdistortionpattern. Thisisconsistentwithx-raymea-
of localised and mobile carriers. The localised states on surements in La Ca MnO in a field.38,39 Starting
1/2 1/2 3
the defectswhichweobtainaredifferentfromthemicro- from the distorted CE phase, we find in addition an in-
ferrons suggested at small x around a dopant,2 as they stability of the high-field ferromagnetic phase to the for-
are self-generated and could exist even in the absence of mation of JT defects. This suggests that the high-field
chemical disorder. In principle, the JT defects we are phase seen in experiments may need a two-fluid descrip-
considering could be mobile on a longer time-scale, al- tion.
though disorder may indeed pin them down. Interestingly,averysimilartwo-carrierhypothesiswas
More specifically, in this paper, we first determine the proposed based on phenomenological grounds to under-
zero temperature phase diagram of the 3d two-orbital stand the resistivity of La1−xCaxMnO3 (x 1/2).36
∼
model of half-doped manganites for periodic phases in More recently, a particle-hole asymmetric Ginzburg-
the thermodynamic limit, including JT distortions, but Landautheorywasproposedtoexplain37 theincommen-
ignoringCoulombinteractions. Wedothisbyminimising surate charge order20 seen for x > 0.5. We believe that
the energy assuming a periodic unit-cell of eight sites,34 our theory provides the microscopic basis for these facts
insidewhichstaticJTdistortionsandcorespindirections both.
are allowed to be arbitrary. This allows us to determine The rest of this paper is organised as follows. In sec-
them self-consistently without using finite-size clusters, tion II, we discuss the phase diagram of the half-doped
thereby extending and reinforcing earlier work.31,60 In manganites restricted to periodic groundstates with the
4
mostgeneral8-sublatticestructure. Wegiveinparticular α and β at the two nearest neighbour sites (i,a) and
ananalyticstrong-couplingdescription(IID). Insection (j,b) respectively, arising from their hybridisation with
III,westudytheinstabilitiesofsomeofthesephases: in- the O p orbitals (with 4t/3 being the hopping be-
σ
stability of the strong JT coupling ferromagnetic phase, tween (−3z2 r2) orbitals in the z-direction)44. The sec-
−
which defines a new phase (IIIA); instability upon dop- ond term J is the antiferromagnetic coupling of the
AF
ing to the canted phases (IIIB1) or to self-trapping of t core spins that comes from standard superexchange
2g
addedcarriers(IIIB2),andthecompetitionbetweenthe processes.45 It can be roughly estimated from the N´eel
two (IIIB3). The effect of the magnetic-field on the CE temperature of a system with only t core spins, such
2g
phase is studied in section IIIC, where we also discuss as CaMnO , although the structure of the half-doped
3
the nature of the high-field ferromagnetic phase. In sec- system is not exactly the same. The third term is the
tion IV, we summarise and discuss the possibilities for Zeeman energy where H is the external magnetic-field.
testing these ideas experimentally. A short account of ThelasttwotermsincludetheJahn-Teller(JT)phonons
some of these results has been presented elsewhere.40 and their coupling to the e electrons. We neglect the
g
P2/2M terms since ¯hω t (where ω is the typical
ia ia 0 ≪ 0
phononfrequency), but include their effects heuristically
II. OPTIMISED PERIODIC PHASES AND when we argue that JT defects lead to polaron formation.
PHASE DIAGRAM FOR HALF DOPED Q and Θ are, respectively, the amplitude (measured
ia ia
MANGANITES
in units of the typical JT distortions in these materials)
and the angle of the JT distortion at the site (i,a). The
A. Model Hamiltonian coupling matrix is given by:
OurdiscussionsarebasedonthefollowingHamiltonian
cosΘ sinΘ
for the manganites τ(Θ)= (3)
sinΘ cosΘ
(cid:18) − (cid:19)
[ S ,Q ,Θ ]= t˜αβ (S ,S )c† c K is the lattice stiffness of a simplified non-cooperative
H{ ia ia ia} − abij ia jb iaα jbβ model where distortions on neighbouring sites are not
ijαβab
X coupled. More detailed and realistic models would in-
+ JAFSia.Sjb gµB H Sia cludecooperativeJTcouplingsandcouplingtobreathing
− ·
<ijab> ia modes such as in the lattice model of Ref. [46].
X X
1 We have neglected the on-site Coulomb interaction,
+ K Q2 g Q τ (Θ )c† c (1)
2 ia− ia αβ ia iaα iaβ U iniaαniaα¯ betweendifferentorbitalstates. Although
Xia iXaαβ itisanimportantinteractioninthe problem,wecannot
P
where c† creates an e electron (in a low-energy- treat it using the methods used in this paper except in
iaα g a mean-field approximation. However, it is effectively
projected Wannier orbital with e symmetry41), on the
g taken into account when orbital order occurs. We com-
sublattice site a (Mn site) of the unit-cell i in the 3d
ment on the effects of its inclusion at appropriate places
cubic lattice, and in the orbital state α = 1,2, with
in the paper. When a local JT distortion occurs on a
c1om≡podsxe2d−yi2n,t2oe≡ightds3uz2b−lar2tt.iceTs,helaboerlilgeidnawlitlahttai.c3e4 Tishderee- site, the degeneracy of the eg orbitals is lifted. If only
one electron is present, there is a gain by occupying the
are N sites and cN =(1 x)N electrons (when x=1/2
− lowest energy level. If a second electron is added, how-
the number of electrons is denoted N N/2). The first
0 ≡ ever, it has to occupy the higher energy level because
term is the kinetic energy of the electrons. The hop-
of the strong Hund’s coupling. There is a compensation
ping parameters are taken to be of the usual Anderson-
and the energy gain vanishes. JT distortions therefore
Hasegawa form42 which takes care of the Hund’s cou-
suppress double occupancy of sites, mimicking the effect
pling, J S s in the limit of large J /t, with
Sia, theHSP=i3/i2ac·orieaspin formed from the MnHt2g elec- iotfiUs.imWphoerntagnt≪tot,exthpelicdiitsltyorinticolnusdearUe,smwahlilchordzoeersopalnady
tronsbeingapproximatedasaclassicalspin. Asaconse-
a role. For instance, it induces a charge-ordering in the
quence only the electrons with spin projections parallel
CE phase,28 just as a finite g/t does.31 When g t (see
to the core spins are present, and their hopping ampli- ≫
section IID), JT distorted phases appear naturally, and
tudes are functions of the polar angles of the core spins
the inclusion of U is not crucial. Similarly, the inclusion
given by:42,43
ofthetermU n n isunimportant(completelyirrel-
i i↑ i↓
t˜αβ (S ,S )=tαβ evantwhen JH )because the largeHund’s coupling
abij ia jb abij × prevents doubPle→oc∞cupancy of this type. We have also
θ θ θ θ
cos ia cos jb +sin ia sin jbei(φia−φjb) (2) neglected the longer range coulomb interactions as they
× 2 2 2 2 are expected to be weak because of the large dielectric
(cid:18) (cid:19)
constant of the manganites, and we do not consider is-
Here tαβ is the usual, anisotropic and symmetry de- sues (such as macroscopic phase separation) which are
abij
termined, hopping amplitude31 between the e orbitals sensitive to their presence.
g
5
Note also that regardingthe direct coupling of the t done calculations using up to 6912 blocks.] The compu-
2g
spins,werestrictourselvestoapureHeisenbergsuperex- tational effort of such an approach compared with that
change coupling. This may not be absolutely accurate ofRef. [31]is,ononehand,muchreducedbecausewedo
for S = 3/2 spins. Further couplings, such as single-ion not have to equilibrate a large number of variables. On
anisotropies, are certainly present in the real materials, theotherhand,thecalculationoftheenergyofeachcon-
but are not of crucial importance for the issues we focus figuration takes more time because we sum over a large
on in this paper. We therefore restrict ourselves to the number of k-points (equal to the number of blocks) in
Hamiltonian of eq. (1). the Brillouin zone corresponding to the periodic struc-
ture. We have carefully studied the finite-size effects on
these Brillouin zone sums. The error on the total en-
B. Method ergy due to the truncation of the sums is of the order of
10−2t where t is the typical energy scale of the problem.
Thethermodynamiclimitisthereforereachedwithinthis
TheHamiltonianofeq. (1)representsmobileelectrons
accuracy,i.e.,forallthephasescompatiblewiththesub-
coupledto localclassicaldegreesoffreedomthatactlike
latticestructureweexpectthatourresultsarewithin1%
annealed disorder. The probability weight of a partic-
of the thermodynamic limit results.
ular configuration of the classical variables is given by
exp[ F /(k T)] where F is the electronic free energy
el B el
−
in the presence of that configuration; their distribution
C. Results
thus hastobe determinedself-consistently. Atzerotem-
perature,itisreasonabletoassumethatthe classicalde-
grees of freedomare frozenand have well-defined values.
On a finite lattice these can, in principle, be determined
10
as follows. One can diagonalise the Hamiltonian exactly
K/t=10
foragiven(arbitrary)configurationoflatticedistortions
(Q ,Θ ) and polar angles of the spins θ (for simplicity 8
i i i G-CO
A-CO
weareignoringtheazimuthalanglesofthespinsφi). For FI-CO
each configuration, one can thus find the electronic en- FM-CO
6
ergy levels,fill up the states up to the Fermi energy,and
FM
calculate the total energy. To obtainthe groundstate of /t d CE-CO
g
the system, one then needs to minimise this energy with 4
respecttoallthepossibleconfigurationsofclassicalvari-
FM Inc. ?
ables. Such a procedure can be implemented, for exam-
A
ple, using a Monte-Carlo technique47 for a finite lattice, 2 d
but becomes a more and more difficult task as the num-
ber of lattice sites, and hence the number of variables,
0
increases.
0 0.1 0.2 0.3
Since our aim is to explore the physics of the exper- 2
J S /t
imentally observed CE state, we adopt a simpler ap- AF
proach. Weassumea8-sublatticeperiodicstructurethat
FIG. 2: Phase diagram of the 3D two-orbital model (T =0,
iscompatiblewiththe periodicityoftheCEstate,which
x=0.5, K/t=10). FM(resp. FM ): ferromagnetic metallic
permits us to tackle the problem in a lower dimensional d
phase with no distortions (resp. small uniform distortions).
space of the classical variables. [Needless to say, this
FI-CO(resp. FM-CO):charge-orderedferromagneticinsulat-
rulesoutthepossibilityofincommensurate(withrespect
ing (resp. metallic) phase with distortions that favour occu-
to the assumed eight sublattice structure) or inhomoge- pancy of the x2 y2 orbitals (Fig. 3). A : ferromagnetic
d
neous phases.] We have implemented a simulated an- planes AF aligne−d with uniform distortions. A-CO: A with
nealingroutinetominimisethetotalenergywithrespect charge order. CE-CO: Ferromagnetic zig-zag chains AF or-
to (essentially all possible) distortions and spin angles dered, charge and orbital ordered (3x2 r2/3y2 r2) [Fig.
− −
on the 8-sublattices. Thus our approach differs from 1]. G-CO: N´eel AF phase with charge-order. Inc.: Possible
and is complementary to earlier numerical approaches incommensurate states that interpolate between CE and G.
which have considered small clusters and done a full Dotteddashedlinescomefromanalyticalexpressionsderived
in the strong-coupling limit (section IID) . Solid (dashed)
classical Monte-Carlo simulation for the spin and lattice
lines show first-order (second-order) phase transitions.
variables.31 For us, the only limitation is the number of
sublattices, which we fix to be eight; the system size is
vastlylarger(wearetreatingthereal3dcase),andprac- Figure 2 shows the phase diagram as a function of
tically in the thermodynamic limit. [The system size, the JT coupling, g/t, and the antiferromagnetic cou-
i.e, the total number of sites, is 8 times the number of pling, J S2/t, at zero temperature. [We choose units
AF
periodic repetitions ofblocksof8 sites(the spinanddis- such that the JT distortions are dimensionless, whence
tortions being the same in all the blocks), and we have K and g both have dimensions of energy, which we spec-
6
FI-CO FM-CO
havejust describedexceptthatsuccessivelayersarenow
3.5+δ antiferromagnetically ordered. For g/t up to 5.1, the
∼
phase noted A is uniformly distorted with a distortion
d
amplitude given in Fig. 4. It is metallic in this regime
3.5−δ (see the charge gap in Fig. 6). For larger values of g/t,
the A phasebecome charge-orderedandinsulating,as in
case of the FI-CO phase (Fig. 3).
The CE-CO phase, the CE phase with charge stacked
order and orbital order (Fig. 1), is the stablest over a
widerangeofparametersforintermediateJ ,asisclear
AF
from (Fig. 2). As pointed out in Refs. [28,31], there is
FIG. 3: (color online). A depiction of the ferromagnetic in- orbital ordering even at g = 0, but no charge ordering;
sulating (resp. metallic) charge-ordered phase (FI-CO [resp. the ”bridge sites” having an average occupancy of 0.5,
FM-CO]) stable at strong JT coupling and small antiferro- but only of (3x2 r2) and (3y2 r2) orbitalsalternately
− −
magnetic coupling (see fig. 2). Alternate sites have charge (Fig. 1). The ”corner sites” on the other hand, have
disproportionation δ (given in Fig. 5 as a function of g/t). equal occupancy (0.25 each) of (x2 y2) and (3z2 r2)
The sites with higher occupancies also have strong JT dis- − −
orbitals. The sites are undistorted at g =0, but get dis-
tortions (see fig. 4), of such orientation as to promote the
torted as soon as g >0 (Fig. 4, bottom-left panel). The
occupancy only of (x2 y2) on thesesites, leading to orbital
− distortionsonthebridgesitesarethelargest,andareori-
order as well. Note that the lattice is rotated by 45 degrees
entedinsuchawayastofurtherstabilisethealternating
with respect to fig. 1.
occupancy of the (3x2 r2) and (3y2 r2) orbitals that
− −
already exists at g = 0, since distortions that precisely
favour this alternation lower the energy of the system.
ify in units of t. We use a fixed K/t = 10. To com-
In addition, small distortions get generated also on the
pare with earlier work, the JT energy is then E /t =
(g/t)2/(2K/t) = (g/t)2/20.] We basically find thJeTsame corner sites that favour the (x2 y2) orbital (Fig. 4).
−
As a further consequence, a charge disproportionation δ
phases that were found before either by comparing the
energies of selected phases at g =0,28,29,48 or by Monte- between the bridge and corner sites develops, favouring
Carlo simulations at finite g;31 except that now we have a higher occupancy of the former. The variation of δ
with g/t is shown in Fig. 5. The system is an insulator
provided confirmation that they are indeed the optimal
whateverthechargedisproportionation,asshownbythe
8-sublattice structures in the thermodynamic limit.
finite charge gap in Fig. 6.
We now describe the different phases shown in Fig. 2,
For strong JT coupling (g/t 1), the CE-CO phase
including the amplitudes of the JT distortions in them ≫
is degenerate energetically with the C-CO phase which
and the corresponding electronic properties.
consistsofstraightferromagneticchainsantiferromagnet-
For small values of J an undistorted metallic phase
AF
icallyorderedwith respectto eachother. The chargeor-
with 3-dferromagneticorder(FM) is stable up to a crit-
derisaccompaniedbyorbitalorderofthe(3z2 r2)type
ical value of g/t 5. Above this threshold, there is −
∼ if the chains are oriented along the z-direction. This de-
a ferromagnetic phase with very small uniform distor-
generacyis discussedin explicit detail in sectionIID. In
tions (Fig. 4), notedFM . There is also a narrowregion
d
this limit, itis easy to show(see sectionIID) thatthe G
(5.6 < g/t < 5.9) where the solution displays many in-
phase (completely 3d-AF phase with localised electrons)
equivalent sites. As discussed later, we believe that this
isalwaysthestablestforlargevaluesofJ . Forsmaller
is consistent with the instability that we find in section AF
III. For g > 5.9, the stable phase is the FM-CO fol- valuesofg/t(andlargeJAF)wefindsolutionswithmany
inequivalent(canted)sites,suggestingthatthetransition
lowed by th∼e FI-CO. These phases have the structure
from the CE phase to the G phase in this regime might
depicted in Fig. 3; i.e., a layered structure with large
proceedviaintermediatestatesthatareincommensurate
JTdistortionsontwositesoutoffourinachecker-board
relativetotheperiodicityoftheunit-cellwehaveconsid-
pattern in each layer, and oriented in such a way as to
ered (Fig. 2).
favourthe(x2 y2)orbitalonthestronglydistortedsites
− We consider next the strong JT coupling limit, g/t
(Fig. 3). There is a charge disproportionation δ that is ≫
1, whence we can calculate the energetics of the phases
given in Fig. 5. The structure is metallic (FM-CO) up
and the phase boundaries discussed above analytically.
to g/t 6.3, and is insulating (FI-CO) for larger g/t,
∼
as is clear from Fig. 6. This structure has been found
before,31 and is known to compete with a similar struc-
ture which prefers a (3x2 r2)/(3y2 r2) orbital order, D. Localised description for g/t 1
− − ≫
when strong anharmonic and cooperative JT couplings
are taken into account.32 At large g/t, it is energetically favourable for all the
TheAphases,whicharemorestableatlargerJ (the electronsinthe systemtobe self-trappedbylocallattice
AF
larger the g/t, the smaller the coupling J required for distortions since the JT energy gain is large. We hence
AF
thetransition)aresimilartotheferromagneticphaseswe start with Wannier-type wave-functions with electrons
7
6
1 1
2 F A 5 CE
0.8 3 FM
4 4 A
0.6 g/K
Q
∆/t 3
0.4
FM 2
0.2 d FI-CO A A-CO
d
FM
1
0
1
CE-CO G-CO 0
0.8 0 2 4 6 8
g/t
0.6
Q
FIG.6: Chargegapvs. g/tfortheCE,FandAtype-phases.
0.4
TheCEphaseisalwaysinsulating,whiletheAandFphases
0.2 are insulating beyond gcA/t = 5.1, and gcF/t 6.3. The
∼
dottedlineisthegapobtainedwithnon-optimiseddistortions
0
(eq. (11) in IIIB).
0 2 4 6 8 0 2 4 6 8
g/t g/t
orbital state, as long as the orbital state correlates with
FIG.4: Amplitudesofdistortionsofthefourinequivalentsites
the orientation Θ of the JT distortion as
i
oftheunit-cellasfunctionofthecouplingparameterg/t. The
differentpanelsrepresentthevariousphasesfoundpreviously: Θ Θ
i i
F,A,CE,G. The last three phases do not exist for all values |Ψ(Θi)i=cos 2 |dx2−y2i+sin 2 |d3z2−r2i (4)
of g/t; the curves are then obtained by fixing the magnetic
structureand optimising with respect to thedistortions. For
(for instance, Θ = π/3 for 3x2 r2, and Θ =π/3 for
the G-CO phase, the distortions are exactly given by g/K 3y2 r2). i − − i
since theelectrons are completely localised. −
This degeneracy is lifted at second-order in pertur-
bation theory in the kinetic energy of the electrons.
0.5 Consider an electron localised on a site with an empty
0.5
neighbouring site and with the corresponding core spins
aligned. Then, in the adiabatic limit (t ¯hω where ω
0.4 0 0
≫
δ is the frequency of the JT phonons) appropriate here, it
canhopvirtuallyontoanyofthetwoorbitalstatesofthat
0.3
site andback, without giving the lattice distortions time
δ 0 1 2 3 4 to relax (the relevant energy denominator being 2E ),
JT
0.2 E /t
JT and hence lower its energy. This energy lowering is less
if the core spins are misaligned whence the hopping am-
CE
0.1 F plitude is reduced (and even fully suppressed in case of
A anti-alignment). It is also less if the neighbouring site
0 is occupied, whence the energy denominator is larger,
0 2 4 6 8 equal to 4E (4E +U in the presence of U, so that
JT JT
theprocessgetssuppressedaltogetherforlargeU). Such
g/t
a process hence gives rise to a effective double exchange
FIG. 5: Charge disproportionation defined by the valence of term in the Hamiltonian as pointed out in ref. [33] and
thetwoinequivalentMnions,Mn3.5+δ andMn3.5−δ (seeFigs. labelled virtual double exchange. The dominant term in
1 and 3), as a function of g/t (or EJT/t [inset] defined by the effective Hamiltonian is then49:
2EJT =gQmax,whereQmax isthedistortionofthesitewith
thelargest distortion) for theCE, F and A type-phases.
˜ = E n + J S .S gµ H S
JT i AF i j B i
H − − ·
i <ij> i
fully localised on strongly distorted sites. The local en- X X X
J
ergy per electron is the sum of the elastic energy 1KQ2 (S S +S2) n (1 n )C2 +(i j) (5)
2 −2 i· j i − j i,j ↔
and the electronic energy gain gQ which is minimal at <i,j>
Q = g/K with a net energy g−ain of E = g2/(2K). X (cid:2) (cid:3)
JT
In the limit of large g/t, there is a large degeneracy be- Here E = g2/(2K), n is the electronic occupancy
JT i
cause electrons can be trapped on any site and in any on site i, J = t˜2/(2E S2), t˜ = 4t/3. C
JT i,j
≡
8
cos[(Θ +Ψ )/2] with Ψ = Ψ = +π/3, Similarly we can compare the energies of the different
i ij i,i+x i+x,i
Ψ =Ψ = π/3 and Ψ =Ψ =π. phases as a function of the magnetic-field and draw the
i,i+y i+y,i i,i+z i+z,i
−
TheeffectiveHamiltonianofeq. (5)isaclassical spin- correspondingphasediagraminthe (J /J,gµ H/JS)
AF B
charge-orbital model with no quantum fluctuations. If plane (Fig. 7). The degeneracy between the C and the
the charges are assigned specific positions (so as to min- CE phases is lifted and the C phase wins at finite fields.
imise the energy), the model reduces to a spin-orbital This is because the fourth term of (5) favours Wigner-
model. It is different from the spin-orbital model pro- crystal type of ordering. For the C phase, for instance,
posed for undoped LaMnO3 obtained by projecting out the critical field is gµBHc =8JAFS−JS.
double occupancies50 because double occupancy is irrel-
evant in the limit being explored here. The orbital (and
4
JTdistortionorientation)variablesonneighbouringsites
are not directly coupled in this model (as C involves g>>t
ij
only one orbital angle Θi). Such a coupling would arise 3
if one takes into account a direct coupling between JT
S
dJTistmorotdioenl.sNonevenretihgehlbeossu,reinvgensiintetsh,iasssiimnptlhifieedcoaoppperroaatcivhe, H/J 2 FI-CO
B G-CO
the virtualdouble exchangelifts the degeneracybetween µ
g
different orientations of the JT distortions and the cor-
1 C-CO
responding orbital degeneracy. In addition, it clearly
favours ferromagnetic bonds and charge disproportion-
A-CO CE-CO; C-CO
ation. 0
We can now understand the strong-coupling limit of 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
the phase diagram(Fig. 2), and furthermore even in the
J /J
AF
presence of a magnetic field, by estimating the energies
of the various phases using equation (5). For the fully
FIG. 7: Phase diagram when g/t 1 at T = 0 (x = 0.5).
cohbatargineeddisbpyrompionrimtioisniantged˜pwhiatshesretshpeecetnteorgtiheespcaenrtsiintgeaanre- The phases are the same as in Fig.≫2. JS2 =t˜2/2EJT. The
H CE-COandC-COphasesaredegenerateatzero-field,butthe
gle (at finite fields) andorbitalangle (the latter depends
latter wins at finite fields.
on the field for the CE phase, and only the leading term
in H 0 is given):
→ As noted earlier, in the limit of large g/t much of
the physics is insensitive to the inclusion in the model
of the on-site Coulomb interaction between different or-
E = 3J S2 3JS2/2 gµ HS
FI−CO AF − − B bitalstates,Uniαniα¯. Thetotalenergyishardlyaffected
E = J S2 3JS2/2 (gµBH)2 since double occupancy is much reduced. We emphasise
A−CO AF − − 8J again that this is contrary to what happens in the other
AF
(gµ H)2 limit g/t 1 where the electron density is uniform. In
EC−CO = JAFS2 JS2 B the latter≪case, it is known that U by itself will induce
− − − 16J 2J
AF − charge-ordering in the CE phase,28 at least if the latter
E = J S2 JS2 (gµBH)2 (H 0) is not destabilised by other phases.51 In the CE phase
CE−CO AF
− − − 16JAF J → at large g/t, U will slightly modify the charge contrast
−
(gµ H)2 by pushing the electrons further off the corner sites. We
E = 3J S2 B
G−CO − AF − 24J 6J haveperformedaself-consistentHartree-Fockcalculation
AF
− to confirmthis point. At g =0, the calculationgives the
At zero field, there is only one free parameter, J /J, same results as the slave-boson approach.28 At g/t = 7
AF
which determines the relative energies. At J =0, the and for the optimised lattice distortions,the chargecon-
AF
FIphase(whichisorbitallydisorderedinthislimit)isde- trast increases, with respect to U = 0, by a very small
generatewiththeA-typephase(with(x2 y2)orbitalor- amount of order 0.05 for U/t evenas large as 25. There-
−
der),butthelatterisfavouredassoonasJ >0. There fore, we conclude that the effect of U is small and does
AF
is a succession of first-order phase transitions as J /J notchangethenatureoftheinsulatingphasesinthelimit
AF
is increased, first to the CE-CO phase (degenerate with of large g/t.
the C-COphase)atJ /J =1/4andthentothe G-CO We next consider the interesting question as to what
AF
phase(see Fig. 7). Intermsofthe originalvariables,the the appropriatelow energy effective Hamiltonian replac-
firsttransitionat1/4islocatedatJ S2/t=4tK/(9g2); ing(5)iswhent/E becomessufficientlylargethatper-
AF JT
the second transition is at J S2/t = 8tK/(9g2). The turbation theory in t/E and the Hamiltonian (5) are
AF JT
phaseboundariesgivenby these equationsinthe strong- not valid anymore. We argue below, by studying the
coupling regimearedisplayedas dash-dottedlines inthe excitations and instabilities of the original model (1) in
phase diagram (Fig. 2) and are in good agreement with the ferromagnetic phase, that the effective model that
the phase boundaries obtained numerically. replaces (5) when t/E gets largertakes a similar form
JT
9
1 1
except that mobile electrons have to be included. + K(Q Q )2 KQ2 (6)
d
2 − − 2
Here E and E (N ,Q,0) [N = N/2 is the number of
III. INSTABILITIES OF THE PERIODIC 0 el 0 0
electrons] are the total and electronic ground state ener-
PHASES
giesoftheoptimalFIdistortedphase,obtainedasafunc-
tion of g/t by minimising with respect to Q as discussed
We have discussed in the previous section the vari-
in the previous section. E (N ,Q,Q ) is the electronic
ous phases stable in the thermodynamic limit that are el 0 d
ground state energy of the defective state. One expects
periodic with a 8-sublattice unit-cell. We will discuss
a gain in lattice energy and a loss in electronic energy,
in this section several instabilities that point to non-
because one energy level has been raised at one site. To
periodicphases,athalf-doping(IIIA)andalsoupondop-
evaluatethelatter,wefirstsolvenumericallytheproblem
ing(IIIB)oradditionofamagnetic-field(IIIC). Wewill
of the one electron eigenvalues in the presence of the ex-
show that the ferromagneticinsulating phase (FI-CO) is
tra single site potential for a finite-size system. Then we
infactunstablewheng/tisloweredbelowacriticalvalue
calculate E (N ,Q,Q ) by filling the N/2 lowest one-
g /t 6.8. Thisinstabilityoccursbeforeanyofthetran- el 0 d
c ∼ electron levels. We have so far considered 3d systems
sitionsdiscussedabove(atg /t 6.3and5.9)takeplace.
c ∼ with up to N =1728 sites.
For this purpose, we study the energetics of defects
Fig. 8 shows the energy E E plotted vs. Q
in the lattice distortion pattern of the periodic phases. − 0 d
for different values of g/t. We have checked that finite
We consider particle and hole excitations accompanied
size effects are negligible (the curves corresponding to
by single site JT defects. We consider both types of de-
N = 216,1000,1728 are given for g/t = 6.7 in Fig. 8).
fects, one where we add a distortion on a site that was
For large g/t, the energy is positive but there is a lo-
previously undistorted, and the other where we remove
cal minimum at large Q which can be described as a
the distortion of a site that was distorted. Without the d
particle-hole excitation with reduced distortion on one
lattice distortion defect, the lowest energies of the par-
site. When g/t decreases, this excitation softens and
ticle or hole excitations accessible are the appropriate
vanishes at g /t 6.8. We believe that this signals the
gapsdeterminedbytheband-structuresdiscussedinsec- c ∼
onsetofanewphasewheresuchdefectsareenergetically
tion IIC (Fig. 6). The defect modifies locally the JT
favourable and thus proliferate in the system.
energylevel,andhence constitutesascatteringpotential
for the particle and hole excitations. The problem lacks
lattice translation invariance, and we have solved it by 1
exact numerical diagonalisation of Hamiltonian (1) rep-
resented in real-space. We consider a problem of N sites 0.8 g/t=8
(up to N =1728)with one special site, and we calculate
0.6
all the eigenvalues and the total energy. A key question
iswhetherboundstateswithenergieslowerthanthatal- 0 0.4
E
lowedby band-structure canappear near the defect. We E- g/t=7
find that they do in several cases, and when their bind- 0.2
ingenergyexceedsthegap,itsignalsaninstabilityofthe g/t=6.8
0
periodic phase.
g/t=6.7 (N=216,1000,1728)
-0.2 g/t=6.5
A. Instability of the Ferromagnetic Insulating 0 0.2 0.4 0.6
Phase when g/t is decreased
Q
d
We consider firstthe FI-COphase athalf-doping(pic- FIG.8: Energychangewhen asingle JTdefect isintroduced
turedinFig. 3). ItisstableforverysmallJAF andlarge intheFI-COphase,vs. Qd. Q Qd istheJTdistortionona
−
g/t (see the phase diagram in Fig. 2). Out of the two defectsite;alltheotheroccupiedsiteshavingthesamedistor-
sitesintheunit-cell,onesiteisdistortedwithadistortion tion Q. Wesee that there is an excitation with Qd Q that
∼
orientation that favours the dx2−y2 orbital. softens when g/t decreases. The excitation corresponds to a
bandparticle-holeexcitationwiththeremovalofalatticedis-
We now consider the problem when one introduces a
tortionofonesite,whiletheQ =0minimumisthepolaron.
singlesiteJTdefect: theamplitudeofthedistortionQof d
theFI-COphaseismaintainedatN/2 1sitesexceptat The softening for gc/t ∼ 6.8 signals a phase transition with
− proliferation of mobile electrons and defects. Finite-size ef-
one site where the distortion is now reduced to Q Q .
− d fectsaresmallandshownforg/t=6.7(N =216,1000,1728).
Q takes all values from 0 (no defect) to Q (the lattice
d
distortionhasbeencompletelyremovedonthissite). The
From the calculation of the energy levels in the pres-
excess energy of such a state is given by:
ence of the defect, we find that there is no bound state
withinthegapfortheQ thatminimisestheenergy. The
d
E E =E (N ,Q,Q ) E (N ,Q,0) electron occupies a higher-energy band-like state and is
0 el 0 d el 0
− −
10
mobile. The instability therefore corresponds to the en- more that the above Hamiltonian does not include ℓ b
−
ergy of this mobile electroncrossing the chemical poten- hybridisationeffects, which must be included in orderto
tial (i.e., E ). This suggests that the proliferation of describepropertiessensitivetoℓ bcoherencewhichcan
JT
− −
thedefects leadstothe conversionofsomesmallfraction develop at sufficiently low temperatures in the metallic
of the localised electrons into mobile electrons moving phases.33 Itis straightforwardto generalisethe Hamilto-
on weakly distorted sites, resulting in a metallic phase. nian to include these effects, as well as cooperative JT
Such a state wouldnotbe accessible in the minimisation effects.
procedure ofsectionII (which has a maximalunit cellof
8sites)evenifthedefectsitesweretoarrangethemselves
in a super-lattice. B. Instability of the CE Phase upon Doping
The situation is rather similar to that described by
Ramakrishnan et al. at the metal-insulator transition in
In the band picture of the CE phase28,52, doping with
holedopedmanganitesintheorbital-liquidregime;33 ex-
electrons, corresponding to x < 1/2 (resp. holes, corre-
cept that we have here a calculation in the context of a
spondingtox>1/2)providesmobilecarriersinthecon-
microscopicmodelthatexplicitlysuggestssuchapicture
duction (resp. valence) band. In either case, the system
even when g/t is not very large. Thus we can identify
will be metallic. This is contrary to experiment, where,
the high-energymobileelectronsasthe broad-band b-like
inmostcases,thesystemremainsinsulatingforx>1/2,
electrons of ref. [33], and the low-energy localised states
but typically becomes metallic quickly for x<1/2. The
as the ℓ polarons. In the present context, at half-doping
transition to the ferromagnetic metal for x < 1/2 has
and above the transition (g > g ), all the electrons oc-
c beendescribedasbeingduetothecrossingoftheenergies
cupy the ℓ states, which form a regular checker-board
ofthe CEand ferromagneticmetallic states.28 The tran-
array (Fig. 3). The sites are singly occupied and U
sitionisthennaturallyfirst-order. However,asdiscussed
does not play a crucial role. This is no longer the case
in ref. [33] evenfor x<1/2 a simple bandpicture of the
belowthetransitionwhenwestarttotransfersomeelec-
ferromagnetic metallic state is severely limited. Apart
trons from the ℓ states to the b states. The b states are
from that, the band picture fails to describe the insulat-
delocalised over the empty sites but also visit the sites
ingcharacterofthe regimex>1/2andtheparticle-hole
occupied with ℓ electrons. Double occupancies become
asymmetry around x=1/2. We discuss this issue next.
inevitableandU hastobetakenintoaccountinorderto
It was pointed out a long time ago by de Gennes35,
determineaccuratelythepropertiesofthemetallicstate.
in the context of slightly doped LaMnO , that adding
The question of what kind of new metallic state arises 3
carriers to the antiferromagnetic phase of LaMnO may
for JT couplings just below the instability is clearly in- 3
favour canted structures. The qualitative argument is
teresting. The mobile electrons, for instance, may be
that at small concentration the carriers gain kinetic en-
able to destroy the orbital and charge order. While a
ergywhichis linearinthe cantinganglewhereasthe loss
study of such issues is beyond the scope of the present
of magnetic energy is quadratic in the canting angle. By
article, the above results suggest than one should add
the same token, adding carriers to the CE phase should
mobile electrons to the strong-couplingHamiltonian (5),
lead to canting of the core-spins. As such phases inter-
in order to describe metallic phases with possible partial
polate betweenthe CE and FMphases, the transitionto
orbital/chargeorder:
ferromagnetism should be naively second-order.
Inviewofthis,wehavecalculatedtheenergyofhomo-
˜ = E n + J S .S geneous CE canted phases (defined in Fig. 9) for differ-
JT ℓi AF i j
H −
ent carrier concentrations on either side of x=1/2 (i.e.,
i <ij>
X X
J retaining the 8-sublattice periodic structure even when
−2 (Si·Sj +S2) nℓi(1−nj)Ci2,j +(i↔j) x 6= 1/2). We find that canting is favourable for adding
<i,j> electrons to the half-doped system but not for adding
X (cid:2) (cid:3)
tαβb† b + Unb nℓ (7) holes, as de Gennes’s general argument is valid only for
− ij iα jβ iα iα¯ verysmallcarrierconcentrationandbreaksdownquickly
<i,j> i
X X on the hole side, due to special features of the CE state.
where the orbital index α of the mobile b electrons takes We have calculated, in addition, the energy in the pres-
both values on the undistorted sites, but is constrained ence of a single-site defect in the JT distortion as in the
to be orthogonal to the ℓ orbital on the occupied sites, previous subsection. We find that, when g/t is sizeable,
and the other quantities have the same meanings as in canting is in competition with self-trapping of the carri-
eq. (5). For infinite U the mobile electrons can-not hop ers in JT defects. We find that the energy gain due to
to the ℓ sites at all, and the last pair of terms can sim- trapping is linear in the carrier concentration (and thus
ply be replaced by tαβb† b (1 n )(1 n ). dominates at low concentration) whereas it is quadratic
− <i,j> ij iα jβ − ℓi − ℓj
This Hamiltonian needs to be studied in a framework for the canting. For intermediate values of g/t, this re-
P
that can handle the strong interaction effects, such as sultsinafirst-ordertransitiontoacantedmetallicphase
the dynamicalmean-fieldtheory,in a similarway aswas when electrons are added to the half-doped system (i.e.,
done before for the orbital-liquid state.33 Note further- for x < 1/2), and the persistence of a CE-type phase
