arXiv:1010.4424v2 [cond-mat.str-el] 5 Dec 2010
Onset and melting of local orbital order
Avinash Singh and Dheeraj Kumar Singh∗
Department of Physics, Indian Institute of Technology Kanpur - 208016
The onset and melting of locally staggered charge/orbital correlations is inves-
tigated within a two-orbital correlated electron model with inter-orbital and inter-
site Coulomb interactions. Beyond the critical doping concentration xc≈ 0.2, the
(π,π,π) staggered orbital order in the ferromagnet re-emerges sharply at finite tem-
perature. The CE-type orbital correlation exhibits a sharp onset close to the Curie
temperature and rapid thermal melting thereafter, which provides quantitative un-
derstanding of the (π/2,π/2,0) feature observed in neutron scattering experiments
on La0.7(CaySr1−y)0.3MnO3 single crystals. In the zig-zag AF state, the CE-type
orbital correlations are found to be even more readily stabilized, but only within a
narrow doping regime around x = 0.5.
PACS numbers: 75.30.Ds,71.27.+a,75.10.Lp,71.10.Fd
The orbital degree of freedom of the electron has attracted considerable attention in
recent years due to the rich variety of electronic, magnetic, and transport properties ex-
hibited by orbitally degenerate systems such as the ferromagnetic manganites, which have
highlighted the interplay between spin and orbital degrees of freedom in these correlated elec-
tron systems.1,2Orbital fluctuations, correlations, and orderings have been observed in Ra-
man spectroscopic studies3of orbiton modes in LaMnO3, polarization-contrast-microscopy
studies4of La0.5Sr1.5MnO4, magnetic susceptibility and inelastic neutron scattering studies5
of La4Ru2O10, and resonant inelastic soft X-ray scattering studies6of YTiO3and LaTiO3.
A new detection method for orbital structures and ordering based on spectroscopic imag-
ing scanning tunneling microscopy is of strong current interest7in orbitally active metallic
systems such as strontium ruthenates and iron pnictide superconductors.
A composite charge-orbital ordering is exhibited by half-doped narrow-bandwidth man-
ganites such as La0.5Ca0.5MnO3, with nominally Mn3+and Mn4+atoms on alternate lattice
sites in a checkerboard pattern, a staggered orbital ordering of the Mn3+electron between
the two egorbitals corresponding to wavevector (π/2,π/2,0), and a CE-type AF ordering
of the Mn core spins arranged in ferromagnetic zig-zag chains.8,9
Such CE-type orbital correlations persist in the metallic ferromagnetic phase of the colos-
sal magnetoresistive (CMR) manganites, as revealed in neutron and x-ray scattering exper-
iments in ferromagnetic bilayer10,11and pseudo-cubic manganites.12,13Short-range (10-20˚ A)
charge and orbital correlations associated with the CMR in the orthorhombic paramagnetic
phase were observed as diffuse peaks at wave vector (π/2,π/2,0) in the same positions as
superlattice peaks in the CE-type charge-orbital structure.
Sharp onset of CE-type dynamical charge/orbital correlations near the Curie tempera-
ture were also indicated in Raman scattering studies of Sm1−xSrxMnO3(x = 0.45) single
crystals.14Systematic enhancement of this feature was observed with bandwidth reduction,
along with the enhancement of CMR. CE-type dynamical charge/orbital correlations and
accompanying collective and dynamical lattice distortions were suggested as being respon-
sible for the steep metal-insulator transition and CMR near the charge/orbital ordering
Recent neutron scattering studies of La1−x(CaySr1−y)xMnO3crystals have also revealed a
sharp onset of the (π/2,π/2,0) diffuse peak near the Curie temperature,15which gradually
diminishes in intensity and sharpness with decreasing Ca concentration. This behaviour
of CE-type orbital correlations is remarkably similar to the resistivity temperature profile
of these crystals with varying Ca concentration,16and the sharp rise in resistivity and the
metal-insulator transition observed near the Curie temperature.
CE-type orbital correlations are also important for magnetic couplings and excitations in
the ferromagnetic manganites. Due to a spin-orbital coupling effect, low-energy staggered
orbital fluctuation modes, particularly with momentum near (π/2,π/2,0), generically yield
strong intrinsically non-Heisenberg (1 − cosq)2magnon self energy correction, resulting in
no spin stiffness reduction, but strongly suppressed zone-boundary magnon energies in the
Γ-X direction,17,18and can quantitatively account for the several zone-boundary anomalies
observed in spin-wave excitation measurements on ferromagnetic manganites.15,19–24
Although the Jahn-Teller electron-phonon coupling is considered to be important in man-
ganites, especially in the low and intermediate doping range,25the inter-orbital Coulomb in-
teraction has been suggested to be much stronger than the electron-phonon coupling in order
to account for the observed insulating behaviour in undoped manganites above the Jahn-
Teller transition and the bond length changes below it.26,27Generally, Coulombic and Jahn-
Teller-phononic approaches for manganites have been shown to be qualitatively similar.28
Indeed, a mean-field treatment of the Jahn-Teller term29yields an electronic exchange field
in orbital space proportional to the orbital magnetization ?niσα− niσβ?, exactly as would
be obtained from the inter-orbital interaction term. Since orbital correlations are driven
by both inter-orbital Coulomb interaction as well as Jahn-Teller electron-phonon coupling,
local orbital correlations are therefore generally accompanied with local lattice distortion.
While the role of inter-orbital Coulomb interaction has been investigated for the undoped
(x = 0) parent compound LaMnO3,26,27and at half doping (x = 0.5),28surprisingly the ther-
mal onset and melting of different types of local orbital correlations in the full doping range
0 < x < 0.5 for ferromagnetic manganites has not been studied within correlated electron
models. This should be of much interest in view of the importance of orbital correlations
on spin dynamics, their observation close to Tcin neutron, X-ray, and Raman scattering ex-
periments, and their likely role in the observed finite-temperature metal-insulator transition
and the CMR effect.
In this paper, we will therefore investigate the doping dependence of local orbital correla-
tions and their sensitivity on the inter-orbital (V ) and inter-site (V′) Coulomb interactions
within a two-orbital interacting electron model. Due to band narrowing in the correlated
ferromagnet, a strong sensitivity on the ratio V/t would result in sharp onset of local orbital
correlations near the Curie temperature. A detailed comparison of the overall shape with
observed neutron scattering features can provide fundamental insight into onset and melt-
ing of local orbital correlations in manganites. We will consider staggered (π,π,π) orbital
correlations, CE-type orbital correlations in the ferromagnetic state, and CE-type orbital
correlations in the zig-zag AF state, which are of interest in different doping regimes of
We will be concerned here only with local orbital correlations and not long-range orbital
order. We will therefore confine our investigation to the Hartree-Fock (HF) level which
correctly describes the local orbital moment formation associated with the short time scale
correlations corresponding to the high interaction energy scale V . Earlier HF studies of
two-orbital interacting electron models have obtained qualitatively correct phase diagram
for electron-doped (x ≥ 0.5) manganites.30
II.STAGGERED ORBITAL ORDER
The undoped parent compound LaMnO3has a staggered (π,π,0) orbital structure with
antiferro-orbital ordering in the ferromagnetic plane and ferro-orbital ordering in the per-
pendicular direction. Weakly doped manganites, which are ferromagnetic insulators, also
exhibit an orbitally ordered state for x<
∼0.2, as inferred from x-ray diffraction and neutron
In this section, we will examine the temperature dependence of the local staggered orbital
ordering in the low-doping regime. We therefore consider the two orbital model:
H = −t
corresponding to the two eg orbitals µ = α,β per site.An inter-orbital (V ) Coulomb
interaction is included along with the local exchange interaction J between the localized Mn
core spins Siand itinerant electron spins σi.
For simplicity, we consider an orbitally-ordered ferromagnetic state having staggered
orbital ordering in all three directions, with (spin-↑) electronic densities:
?nα?A = ?nβ?B= n + δm/2
?nβ?A = ?nα?B= n − δm/2 (2)
for the two orbitals α and β, where δm is the staggered orbital order and characterizes the
density modulation on the two sublattices A and B. The spin-↓ electronic densities vanish as
the spin-↓ electronic bands are shifted above the Fermi energy by the exchange splitting 2JS
in the ferromagnetic state. In the pseudo-spin space of the two orbitals, this orbital density
wave (ODW) state is exactly analogous to the antiferromagnetic state of the Hubbard model
with staggered spin ordering.32The self-consistent field approximation results in identical
effective single-particle Hamiltonian in the two-sublattice basis:
in terms of the self-consistently determined orbital exchange field ∆ = V δm/2, with µ = ±
for the two orbitals α/β.
A similar expression is obtained in the mean-field approximation of the Jahn-Teller part
involving the coupling of eg electrons with JT phonon modes,28,29with the equivalence
g2/k = V/2. For a value λ ≡ g/√kt ∼ 1.4 of the dimensionless coupling constant, which
lies in the range considered for manganites, the corresponding effective interaction energy
V/t = 2g2/kt = 2λ2∼ 4, as considered in our recent detailed comparison of calculated spin
dynamics for manganites with experiments.18The eigenvalues and eigenvectors of the above
Hamiltonian matrix yield the bare-level band-electron energies and amplitudes:
Ekµ = ±
for orbitals µ = α,β, where ⊕ and ⊖ refer to the two eigenvalue branches (±). Orbital
ordering splits the electron bands with energy gap 2∆ = V δm.
An important characteristic of the ODW state, as seen from equation (4), is that states
near the top of the band (ǫk = 0) contribute dominantly to orbital ordering (strongly
”orbitic”), with densities 1 and 0 on the two sublattices for orbital µ = α, whereas states
deep in the band (|ǫk| ≫ ∆) are nearly non-orbitic, with densities 1/2 on both sublattices.
Similarity with the AF state of the Hubbard model also results in identical self-consistency
(f−− f+) (5)
where f−(T) and f+(T) are the temperature-dependent Fermi functions corresponding to
the two branches, and the chemical potential is determined from the total fermion density
n = 1 − x =
(f−+ f+) .(6)
Coupled equations (5) and (6) then self consistently determine the magnitude of the local
orbital order δm = 2∆/V as a function of the effective interaction strength V and temper-
ature T. When only nearest-neighbor hopping is present, orbital ordering at quarter filling
sets in for any positive V due to Fermi-surface nesting, whereas a finite critical interaction
strength is required when frustrating next-nearest-neighbor hopping terms are included.
Fig. 1 shows the reduction of local staggered (π,π,π) orbital order with doping (x) at
different temperatures. Here the hopping energy scale t = 200meV ≈ 2000K. At low temper-
ature, as the strongly orbitic electronic states at the top of the lower band are progressively
emptied upon doping, local orbital order decreases rapidly and eventually vanishes at a crit-
ical doping concentration xc≈ 0.22, shown by a vertical line. Beyond this critical doping,
local orbital order re-emerges sharply at finite temperature due to thermal occupation of
these strongly orbitic states, as shown in Fig. 2, and then gradually melts away at high
temperature. This onset of orbital order will be further enhanced if the spin-disordering
induced bandwidth reduction near the Curie temperature is included, as discussed in the
The sharp onset of staggered (π) orbital correlations below xc∼ 0.2 results in sharply
reduced magnon energies and Curie temperature, as obtained in investigation of spin dynam-
ics in the orbitally ordered state, which is in good agreement with experiments.18Also, the
measured spin stiffness15in the low bandwidth (high V/t) compound La1−xCaxMnO3shows
a jump at xc= 0.22 separating the ferromagnetic insulating and metallic phases, whereas
0.1 0.14 0.18 0.22 0.26 0.3
Staggered orbital order
FIG. 1: Reduction of local staggered (π,π,π) orbital order with doping, shown at different temper-
atures. Beyond the zero-temperature critical doping concentration xc≈ 0.22 (shown by a vertical
line), orbital order reappears sharply at finite temperature.
0 500 1000
V/t = 5
FIG. 2: Temperature dependence of the local staggered (π,π,π) orbital order at fixed doping x
and interaction strength, showing sharp onset and melting with temperature.
the high bandwidth (low V/t) compound La1−xSrxMnO3shows a much weaker discontinuity
at much lower xc= 0.17.
Although long-range orbital order will be susceptible to orbital fluctuations at finite
doping in the same way as for the staggered spin ordering in the doped antiferromagnet,33
this HF result does provide a measure of the local orbital order. Fig. 1 also yields the
doping dependence of the energy gap 2∆ = mV corresponding to the local staggered orbital
order. The gap decreases rapidly with doping and vanishes at critical doping concentration
xc, is roughly 1000K for m = 0.1, V/t = 5, t = 200meV, and will be reduced significantly on
including the electron-orbiton coupling and multiple orbiton emission/absorption processes.
Beyond xc, this gap reappears at finite temperature, leading to a metal-insulator transition
and enhanced resistivity. This behaviour of the gap is similar to the observed pseudogap
behaviour in cuprates and manganites.
III.CE-TYPE ORBITAL ORDER IN THE FERROMAGNETIC STATE
At half doping, narrow bandwidth manganites such as La0.5Ca0.5MnO3 exhibit a CE-
type antiferromagnetic order consisting of ferromagnetic zigzag chains with staggered charge
and orbital ordering. Local CE-type orbital correlations also emerge in the ferromagnetic
phase near the Curie temperature, coincident with the metal-insulator transition and the
CMR effect, as mentioned earlier. While CE-type charge and orbital ordered ferromagnetic
phase was studied at half doping using cooperative phonons with large electron-phonon
coupling,34–37and recently at hole density x = 1/4 using Monte Carlo investigations,38and
at several fractional hole densities using numerical simulations on finite 3d clusters,39the
doping and temperature dependence of CE-type orbital correlations within an interacting
electron model has not been investigated in the ferromagnetic phase.
In order to investigate the onset and melting of combined charge-orbital orderings in the
ferromagnetic state, we therefore consider the following two-orbital model corresponding to
the two egorbitals µ = α,β per site:
H = −t
including inter-orbital (V ) and inter-site (V′) Coulomb interactions which will induce local
orbital and charge correlations. For simplicity, we consider a staggered charge-orbital order-
FIG. 3: CE-type staggered charge/orbital correlations corresponding to ordering wavevectors π
and π/2 for charge and orbital orders, respectively.
ing as shown in Fig. 3, corresponding to ordering wavevector π for charge order and π/2 for
orbital order in all directions, with (spin-↑) electronic densities:
?nα?A = ?nβ?C= (n + δn/2 + δm)/2
?nα?C = ?nβ?A= (n + δn/2 − δm)/2
?nα?B = ?nα?D= (n − δn/2)/2
?nβ?B = ?nβ?D= (n − δn/2)/2 (8)
where δn and δm represent charge and orbital density modulation. The spin-↓ electronic
densities vanish at T = 0 as the spin-↓ bands are shifted above the Fermi energy by the
exchange splitting 2JS. In a four-sublattice basis (ν = A,B,C,D), with the electron field
operator Ψkµ≡ (aA
kµ), the HF-level Hamiltonian for spin-↑ electrons:
where the effective charge and orbital exchange fields (z is the lattice coordination number):
∆ch = (V/2 − zV′)δn/2
∆orb = V δm/2 (10)
and the NN hopping term (which mixes AB,BC,CD and DA sublattices):
δk= −t(eikx+ eiky+ eikz) .(11)
The Hamiltonian matrix (9) was numerically diagonalized to obtain the four eigenvalues
Ekλcorresponding to the four sub-bands λ. The four-component eigenvectors φν
electronic amplitude on sublattice ν. Evaluation of the new staggered charge and orbital
order in terms of the electronic densities obtained by summing over occupied states yields
the self-consistency conditions:
where the Fermi energy EFis also determined self-consistently in terms of the average (spin-
↑) electronic density n = 1−x. The above self-consistent procedure provides an unrestricted
HF scheme, with independent determination of charge and orbital order.
Fig. 4 shows the behaviour of the self-consistent CE-type orbital order in the ferromag-
netic state with hole doping x for different inter-orbital interaction strength V . The orbital
order is optimal around x = 0.4 and decreases sharply towards half doping (x = 0.5). Due
to the small overlap between the two lowest-energy sub-bands as shown in Fig. 5, the lowest
sub-band is fully occupied not at x = 0.5 but at slightly lower x. As the Fermi energy
moves down with increasing x towards 0.5, the maximally orbitic states at the top of the
lowest sub-band get emptied, leading to a sharp suppression of the CE orbital order. For
0.3 < x < 0.35, the orbital order rises sharply for a small change in V/t from 4 to 4.4,
rendering the system extremely sensitive to small changes in bandwidth and temperature.
The strong presence of CE-type orbital correlations found here in the ferromagnetic
state near x = 0.45 is consistent with the observation of anomalous zone-boundary magnon
softening24, which has been ascribed to correlation-induced magnon self energy correction
arising from coupling of spin fluctuations with specifically CE-type orbital correlations.17,18
0.3 0.35 0.4
CE orbital order
FIG. 4: Behaviour of the CE-type orbital order with hole doping x for different inter-orbital
interaction strength V , showing its strong sensitivity to small changes in V/t. Here V′/t = 0.73.
-2-1.5 -1 -0.5 0 0.5 1 1.5 2
FIG. 5: The density of states showing the four sub-bands in the self-consistent CE-type orbitally
ordered ferromagnetic state.
A strong electron-hole asymmetry around half doping (x = 0.5) directly follows from the
strongly asymmetric band structure and orbitic character about the Fermi energy. While
the CE-type orbital order is strongly enhanced for x < 0.5 due to added electrons going in
0.55 0.60 0.65 0.70 0.75 0.80
CE orbital order
V/t = 5
x = 0.35
FIG. 6: The sharp sensitivity of the CE-type orbital order to the inter-site charge interaction
the maximally orbitic states at the top of the lowest sub-band, it is rapidly suppressed due
to further depletion of these states for x > 0.5. The CE-type orbital order peaks at x<
when the lowest sub-band gets completely filled, and further electron filling of opposite
sublattice state at the bottom of second sub-band results in slow decrease of orbital order
with decreasing x.
Fig. 6 shows the sharp onset of CE-type orbital order with increasing V′, showing its
strong sensitivity to the inter-site interaction strength. Here the inter-orbital interaction
strength is fixed at V/t = 5 and hole doping x = 0.35.
Fig. 7 shows the temperature dependence of the local CE order. Due to the extreme
sensitivity of CE correlations seen above, a slight enhancement in the ratios V/t and V′/t
due to reduction in hopping t(T) with temperature in the correlated ferromagnet results
in a sharp onset of CE-type orbital order close to the Curie temperature, which is seen to
rapidly melt away with increasing temperature. Both the shape and the temperature scale
are in good agreement with observed behaviour of local CE correlations in neutron scattering
studies of La0.7(CaySr1−y)0.3MnO3single crystals,15as shown in Fig. 8 for y = 1. Here we
have used the same hopping energy scale t(0) = 200meV for manganites as in our recent
comparison of spin dynamics results with experiments.18
The CE-type orbital order shown above was obtained using the unrestricted HF scheme
0 100 200 300
400 500 600
CE orbital order
FIG. 7: The sharp onset and melting of the CE-type orbital order close to the Curie temperature
in the correlated ferromagnet. The shape and temperature scale are quantitatively very similar to
the observed behaviour of the (π/2,π/2,0) intensity in neutron scattering experiments.
FIG. 8: Behaviour of the (π/2,π/2,0) intensity observed in neutron scattering experiments on
La0.7Ca0.3MnO3single crystal (from ref. ).
3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5 5.2 5.4
CE orbital order
V’/t = 0.8
x = 0.4
FIG. 9: Onset of CE-type orbital order shows hysteresis behaviour on increasing/decreasing V/t
of Eq. (12), extended to finite temperature by summing over all states with appropriate
Fermi functions. The hopping reduction was included approximately as t(T)/t(0) = 1 −
(1/3) ∗ (1 − ?Sz?/S) in terms of the temperature-dependent magnetization ?Sz?; this yields
a reduction from 1 in the ferromagnetic state to 2/3 in the paramagnetic state, which is
similar to the hopping reduction by the factor ?cos(θij/2)? in the double-exchange model.25
The corresponding thermal enhancements for the ratios V/t and V′/t considered in our
self-consistent analysis were from 3.3 to 5.0 and 0.5 to 0.75, respectively.
Bandwidth (hopping) reduction in a correlated ferromagnet follows from electronic self
energy correction due to electron-magnon interaction, which becomes important near Tcdue
to thermal excitation of local zone-boundary magnons. Due to this band narrowing near the
Curie temperature, finite Jahn-Teller distortion and orbital correlations were shown to be
self-consistently generated in a two-orbital FKLM.29However, only ferro orbital correlations
Fig. 9 shows hysteresis in the onset of CE-type orbital order with increasing/decreasing
V/t. For the same value of V/t (in the range between 4.0 and 4.6), there are two dis-
tinct self-consistent states with and without orbital ordering, depending on the history.
Bandwidth reduction with temperature will translate this into a hysteresis behaviour with
temperature, indicating metastability and coexisiting regions with and without local orbital
correlations. This behaviour of CE-type orbital correlations should be important in view of
recent observations of spin-glass behaviour, phase separation, and evidence of metastability
in manganites near half doping.40
IV.CE-TYPE ORBITAL ORDER IN ZIG-ZAG AF STATE
Long-range charge and orbital ordering sets in at half doping in bilayer and pseudo-cubic
manganites where equal numbers of nominal Mn3+/Mn4+ions form a checkerboard arrange-
ment in a plane with ferromagnetic zigzag chains coupled antiferromagnetically, which is re-
peated in the perpendicular direction. Evidence for such ordering comes from superstructure
reflections at wavevector (π,π,0) and (π/2,π/2,0) in diffraction experiments corresponding
to charge and orbital order respectively.
To investigate the onset of local CE-type orbital ordering in such zig-zag AF states in two
dimensions, we consider a 4+4 sublattice basis for the spin up and spin down ferromagnetic
zig-zag chains. In the HF approximation, the effective single-particle Hamiltonian for spin-↑
electrons is extended to:
where the intra-chain term for the spin-σ ferromagnetic chain (σ = up/down) is given by:
in the four-sublattice basis introduced earlier, and the inter-chain term:
-4 -2 0
zigzag AF state
FIG. 10: The spin-↑ electronic DOS showing four low-energy sub-bands corresponding to CE-type
orbital ordering in the zig-zag AF state. A similar structure shifted up by the exchange splitting
2JS corresponding to the spin-down ferromagnetic zig-zag chains is not shown.
where δkxetc. are the corresponding components of the NN hopping term given in Eq.
(11). The resulting 8 × 8 Hermitian matrix was diagonalized to obtain the eigenvalues and
eigenvectors, and the charge-orbital ordering was obtained self-consistently as described in
the previous section.
As the CE-type orbital order involves a four-sublattice structure, Fig. 10 shows the
corresponding four sub-bands in the low-energy part of the egelectronic DOS. There is a
similar structure shifted up by the exchange splitting 2JS corresponding to the spin-down
ferromagnetic chains. Out of the four sub-bands, the lowest one is fully occupied at x = 0.5
(corresponding to quarter filling in the spin-↑ electron sector).
Fig. 11 shows that the CE-type orbital order in the zig-zag AF state is stabilized within
a narrow doping range around x = 0.5. With increasing Hund’s coupling J and Coulomb
barrier in the AF state, the dimensionality of the mobile egelectrons decreases effectively
from two to one. Due to this reduced dimensionality and delocalization, the CE-type orbital
near half doping is found to be more readily stabilized even for smaller interaction strength.
Similar doping behaviour of the CE-type orbital order was recently obtained using the Jahn-
0 0.2 0.4 0.6 0.8 1
CE orbital order
zig-zag AF state
FIG. 11: The CE-type orbital order in a two-dimensionsal zig-zag AF state is stabilized only within
a narrow doping range around x = 0.5.
A strong sensitivity of local orbital order on doping concentration, interaction strengths,
and temperature is revealed in our investigation of onset and melting of local orbital order
within a two-orbital model with effective inter-orbital and inter-site Coulomb interactions.
In the ferromagnetic state, the CE-type orbital order was found to be optimal around
x = 0.4 and to diminish rapidly at x = 0.5. In the doping regime 0.3 < x < 0.4, the orbital
order was found to rise sharply around V/t = 4 and V′/t = 0.75, rendering the system
extremely sensitive to small changes in bandwidth and temperature. A sharp thermal onset
and melting of this orbital ordering was found, remarkably similar in shape and temperature
scale to neutron-scattering observations, when a small temperature-dependent reduction of
the hopping term was included within a self-consistent treatment of the local orbital order,
with similar values of t and V as used in a recent detailed comparison of calculated spin
dynamics for manganites with experiments.18
The onset of CE-type orbital order was found to exhibit hysteresis behaviour on increas-
ing/decreasing V/t (temperature), with two distinct self-consistent states with and without
orbital ordering. Indicating metastability and coexisiting regions with and without local
orbital correlations, this hysteresis behaviour should be important in view of recent observa-
tions of spin-glass behaviour, phase separation, and evidence of metastability in manganites
near half doping.40
In a two-dimensional CE-type AF state, with alternating zig-zag ferromagnetic chains,
the CE-type orbital order was found to be stabilized only within a very narrow doping range
around x = 0.5. Due to reduced dimensionality and delocalization of mobile egelectrons in
the zig-zag AF state, the CE-type orbital order near half doping is stabilized for even smaller
interaction strengths. This strong sensitivity and its rapid melting away on either electron
or hole doping implies that CE-type orbital order would be highly susceptible in a strong
magnetic field as well, which is of interest in context of magnetic field induced melting of
CE-type orbital order in half-doped manganites.41
While only local orbital correlations (local moments in pseudo-spin space) were considered
here, which are associated with short time scale correlations corresponding to the high-energy
scale V , including low-energy orbital fluctuations would allow for investigation of long-
range orbital ordering features (orbital correlation length, orbital disordering temperature)
and reduction of orbital order due to quantum and thermal excitation of orbitons. Also,
renormalization of the CE-state electron spectral properties due to electron-orbiton coupling
self energy resulting from multiple orbiton emission/absorption processes would provide
insight into correlated lattice polarons.
∗Electronic address: email@example.com
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