Spin-Orbital Entanglement and Violation of the Kanamori-Goodenough Rules

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


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


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


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


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


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)


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


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) ]


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)]


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


Spin-orbital superexchange at JH=0

=> chain along c axis

=> 2D model in ab planes


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.


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


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 )


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


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


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) ]


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?


Thank you

for your attention!

Spin-Orbital Entanglement and Violation of the Kanamori-Goodenough Rules

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