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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
Seillac, 31 May 2006 2
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
Seillac, 31 May 2006 3
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
Seillac, 31 May 2006 4
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:
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
Seillac, 31 May 2006 6
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
Seillac, 31 May 2006 7
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)
Seillac, 31 May 2006 8
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
Seillac, 31 May 2006 9
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) ]
Seillac, 31 May 2006 10
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)]
Seillac, 31 May 2006 11
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
Seillac, 31 May 2006 12
Spin-orbital superexchange at JH=0
=> chain along c axis
=> 2D model in ab planes
Seillac, 31 May 2006 13
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.
Seillac, 31 May 2006 14
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
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
Seillac, 31 May 2006 16
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 )
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
Seillac, 31 May 2006 18
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
Seillac, 31 May 2006 19
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) ]
Seillac, 31 May 2006 20
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?
Seillac, 31 May 2006 21
Thank you
for your attention!
