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Seillac, 31 May 2006 1
Spin-Orbital Entanglement and Violation of the Kanamori-Goodenough Rules
Andrzej M. Oleś
Max-Planck-Institut für Festkörperforschung, Stuttgart M. Smoluchowski Institute of Physics, Jagellonian University, Kraków
Self-organized Strongly Correlated Electron SystemsSeillac, 31 May 2006
•Peter Horsch, Max-Planck-Institut FKF, Stuttgart
•Giniyat Khaliullin, Max-Planck-Institut FKF, Stuttgart
•Louis-Felix Feiner, Philips Research Laboratories, Eindhoven
Institute of Theoretical Physics, Utrecht University
oo
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Outline
• Spin-orbital superexchange models• Goodenough-Kanamori rules in transition metal oxides
• Example: magnetic and optical properties of LaMnO3
• Violation of Goodenough-Kanamori rules in t2g systems due to spin-orbital entanglement
• Continuous orbital transition
• Spin-orbital fluctuations in LaVO3
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Orbital physics in transition metal oxides
Current status:
Focus on Orbital Physics
New Journal of Physics
2004-2005
http://www.njp.org
LaVO3
t2g orbitals
LaMnO3
eg orbitals
C-AF A-AFGoodenough-Kanamori rules:
AO order supports FM spin order
FO order supports AF spin order
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Electron interactions and multiplet structure
[AMO and G. Stollhoff, PRB 29, 314 (1984)]
)(
2)(
,
,,25
int
iiiiiiii
iH
iiiH
iiiH
iii
ddddddddJ
SSJnnJUnnUH
Two parameters: U – intraorbital Coulomb interaction, JH – Hund’s exchange
Anisotropy in Hund’s exchange:
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Seillac, 31 May 2006 5[AMO et al., PRB 72, 214431 (2005)]
Multiplet structure of transition metal ions
Follows from three Racah parameters (Griffith, 1971):
single parameter: η=JH /U
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orbji
jijijiji
orbn HKSSJJHijHJH
)()()(1)(
Magnetic and optical properties of Mott insulators (t<<U)
Spin-orbital superexchange model for a perovskite, γ=a,b,c (J=4t2/U):
contains orbital operators:
By averaging over orbital operators one finds effective spin model:
c abij ij
jiabjics SSJSSJH
Here spin and orbital operators are disentangled.
Superexchange determines partial optical sum rule for individual band n:
0
)(2
20)()( )(
2)(2
d
e
aijHK nnn
[G. Khaliullin, P. Horsch, and AMO, PRB 70, 195103 (2004)]
)()( ijij KandJ
)( ijJJ
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Exchange constants and optical spectral weights in LaMnO3
Jc and Jab for varying orbital angle spectral weights for increasing T
[ AMO, G. Khaliullin, P. Horsch, and L.F. Feiner, PRB 72, 214431 (2005) ]
AF
FM
S=2 spins and eg orbitals are disentangled (MF can be used)
A-AF phase
xz
xz
B
A
|sin|cos|
,|sin|cos|
22
22
orbital order:
exp: F. Moussa et al., PRB 54, 15149 (1996) exp: N.N. Kovaleva et al., PRL 93, 147204 (2004)
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Spin waves in La1-x SrxMnO3 and in bilayer manganites
Isotropic spin waves in La1-xSrxMnO3
[ AMO and L.F. Feiner, PRB 65, 052414 (2002); 67, 092407 (2003) ]
Double exchange and superexchange explain Jab and Jc
FM phase
Anisotropic spin waves in La2-2xSr1+2xMn2O7
[ T.G. Perring et al., PRL 87, 217201 (2001) ] [ T.G. Perring et al., PRB 77, 711 (1996) ]
2Dqq
x=0.30 x=0.35
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Charge transfer insulator: KCuF3
Jc and Jab for varying orbital angle
Valid if S=1/2 spins and eg orbitals disentangle (MF can be used)
spectral weights for increasing T
Parameters: J =33 meV, η =0.12, R=2U/( 2Δ+Up ) =1.2
One of the best examples of a 1D AF Heisenberg model
optical properties would help to fix the parameters
[ AMO et al., PRB 72, 214431 (2005) ]
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Spin-orbital models with entanglement
• d1 model – titanates (LaTiO3, YTiO3), S=1/2, t2g orbitals;
• d2 model – vanadates (LaVO3, YVO3), S=1, t2g orbitals, (xy)1(yz/zx)1 configuration;
• d9 model – KCuF3, S=1/2, eg orbitals.
Spin-orbital models were derived in:
d1 model [G. Khaliullin and S. Maekawa, PRL 85, 3950 (2000)]
d2 model [G. Khaliullin, P. Horsch, and AMO, PRL 86, 3879 (2001)]
d9 model [L.F. Feiner, AMO, and J. Zaanen, PRL 78, 2799 (1997)]
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Orbital degrees of freedom
In t2g systems (d1,d2) two flavors are active, e.g. yz and zx along c axis – described by pseudospin operators:
},,{ ziiii TTTT
At finite η the orbital operators contain:zj
zijijiji TTTTTTTT )(
21
Pseudospin operators for eg systems (d9) with 3z2-r2 and x2-y2:zi
ci
xi
zi
bai TT
21)(
41),( ,)3(
GdFeO3-type distortions induce orbital interactions leading to FO order:
ij
zj
ziorb TTVH
)()( j
ijiorb TTVH
Jahn-Teller ligand distortions favor AO order:
eg orbitals t2g orbitals.,,21
21
21 z
izi
yi
yi
xi
xi TTT
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Spin-orbital superexchange at JH=0
=> chain along c axis
=> 2D model in ab planes
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Intersite spin, orbital and spin-orbital correlations
Spin correlations:
Orbital and spin-orbital correlations for t2g (d1 and d2) systems:
,)(ji
tij TTT
2)(2STTSSTTSSC jijijiji
tij
Orbital and spin-orbital correlations for eg (d9) model:
,)()(
21)(
jijieij TTTTT
.)()( )()(
21)( e
ijijjijijieij TSTTTTSSC
2)2( SSSS jiij
• Definitions follow from the structure of the spin-orbital SE at JH0;
• Method: exact diagonalization of four-site systems.
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Intersite correlations for increasing Hund’s exchange η
V=0 V=J
Sij – spin correlations
Tij – orbital correlations
Cij – spin-orbital correlations
[AMO, P. Horsch, L.F. Feiner, G. Khaliullin, PRL 96, 147205 (2006)]
d1
d2
d9
• all correlations identical in d1 at η=0: Sij =Tij =Cij = 0.25 [SU(4)];
• regions of Sij<0 and Tij<0 both at V=0 and V=J in d1(2) models;
• Cij<0 in low-spin (S=0) states;
• different signs of Sij and Tij in d9
GK rules violated in d1, d2
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Seillac, 31 May 2006 15[AMO, P. Horsch, L.F. Feiner, G. Khaliullin, PRL 96, 147205 (2006)]
V=0 V=J
Spin exchange constants Jij for increasing Hund’s exchange η
d1
d2
d9
In the shadded areas
Jij is negative FM
Sij is negative AF
for d1 and d2 t2g models
=> GK rules are violated
In d9 eg model
spin correlations Sij
follow the sign of Jij
=> GK rules are obeyed
)(ijij JJ
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Dynamical exchange constants due to entanglement
Fluctuations of Jij are measured by
2122)( ijij JJJ Fluctuations dominate the behavior of t2g systems at η=0, V=0:
1,0 JJ ij d1 model:
d2 model: 247.0,04.0 JJ ij
[ SU(4) symmetry ]
Fluctuations large but do not dominate for eg system at η=0, V=0:
d9 model: 50.0,56.0 JJ ij ,i.e., ijJJ
for a bond <ij> fluctuations: ( S=0 / T=1 ) ( S=1 / T=0 )
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Seillac, 31 May 2006 17
Quantum corrections in spin-orbital models
[AMO, P. Horsch, L.F. Feiner, G. Khaliullin, PRL 96, 147205 (2006)]
Large corrections beyond MF due to spin-orbital entanglement
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Continuous orbital phase transition in d2 model
zj
zijijiji TTTTTTTT )(
21
with full t2g orbital dynamics:V=J
continuous transition
0,02,2 zz TTTT
when only Ising term:zj
ziji TTTT
sharp transition0,12,2 zz TTTT
orbital transitions are continuous
S=0 S=4
quantum numbers T and Tz nonconserved
T and Tz conserved
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Optical spectral weights for the C-AF phase of LaVO3
mean-field approach orbital and spin-orbital dynamics
[G. Khaliullin, P. Horsch, and AMO, PRB 70, 195103 (2004)]
spin-orbital fluctuations important at T>0!
orbital disorder unlike in LaMnO3Data: S. Miyasaka et al.,
[ JPSJ 71, 2086 (2002) ]
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Conclusions1. Spins and orbitals disentangle in eg systems ( LaMnO3 )
[AMO, G. Khaliullin, P.Horsch, and L.F. Feiner, PRB 72, 214431 (2005)]
2. In systems with t2g degrees of freedom
3. Dynamic spin and orbital fluctuations in t2g systems: spin triplet
orbital singlet
spin singlet
orbital triplet
[AMO, P. Horsch, L.F. Feiner, and G. Khaliullin, PRL 96, 147205 (2006)]
4. Joint spin-orbital fluctuations in LaVO3
magnetic and optical properties [G. Khaliullin, P. Horsch, and AMO, PRL 86, 3879 (2001); PRB 70, 195103 (2004)]
spins and orbitals are entangled
static Goodenough-Kanamori rules are violated
Any other experimental manifestations of entanglement?
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Seillac, 31 May 2006 21
Thank you
for your attention!