Spin orbital entanglement and violation of the kanamori goodenough rules
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Andrzej M. Oleś Max-Planck-Institut f ü r Festk ö rperforschung, Stuttgart M. Smoluchowski Institute of Physics, Jagellonian University , Kraków Self-organized Strongly Correlated Electron Systems Seillac, 31 May 2006. Spin-Orbital Entanglement and Violation of the Kanamori-Goodenough Rules.

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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 Systems

Seillac, 31 May 2006

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

  • 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
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


Spin orbital entanglement and violation of the kanamori goodenough rules

Orbital physics in transition metal oxides

Goodenough-Kanamori rules:

AO order supports FM spin order

FO order supports AF spin order

C-AF

A-AF

Current status:

Focus on Orbital Physics

New Journal of Physics

2004-2005

http://www.njp.org

LaVO3

t2g orbitals

LaMnO3

eg orbitals


Spin orbital entanglement and violation of the kanamori goodenough rules

Electron interactions and multiplet structure

Two parameters: U – intraorbital Coulomb interaction, JH – Hund’s exchange

Anisotropy in Hund’s exchange:

[AMO and G. Stollhoff,PRB 29, 314 (1984)]


Spin orbital entanglement and violation of the kanamori goodenough rules

Multipletstructureof transition metal ions

Follows from three Racah parameters (Griffith, 1971):

single parameter: η=JH /U

[AMO et al., PRB 72, 214431 (2005)]


Spin orbital entanglement and violation of the kanamori goodenough rules

Magnetic and optical properties of Mottinsulators (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:

Here spin and orbital operators are disentangled.

Superexchange determines partial optical sum rule for individual band n:

[G. Khaliullin, P. Horsch, and AMO, PRB 70, 195103 (2004)]


Spin orbital entanglement and violation of the kanamori goodenough rules

spectral weights for increasing T

AF

FM

Exchange constants and optical spectral weights in LaMnO3

Jc and Jab for varying orbital angle 

A-AF phase

orbital order:

exp: F. Moussa et al., PRB 54, 15149 (1996)

exp: N.N. Kovaleva et al., PRL 93, 147204 (2004)

S=2 spins and egorbitals are disentangled (MF can be used)

[ AMO, G. Khaliullin, P. Horsch, and L.F. Feiner, PRB 72, 214431 (2005) ]


Spin orbital entanglement and violation of the kanamori goodenough rules

Spin waves in La1-x SrxMnO3 and in bilayer manganites

Isotropic spin waves inLa1-xSrxMnO3

Anisotropic spin waves in La2-2xSr1+2xMn2O7

FM phase

x=0.30

x=0.35

[ T.G. Perring et al., PRL 87, 217201 (2001) ]

[ T.G. Perring et al., PRB 77, 711 (1996) ]

Double exchange and superexchange explain Jab and Jc

[ AMO and L.F. Feiner, PRB 65, 052414 (2002); 67, 092407 (2003)]


Spin orbital entanglement and violation of the kanamori goodenough rules

Charge transfer insulator: KCuF3

One of the best examples of a 1D AF Heisenberg model

Parameters: J =33 meV, η=0.12, R=2U/( 2Δ+Up ) =1.2

Jc and Jab for varying orbital angle 

spectral weights for increasing T

optical properties would help to fix the parameters

Valid if S=1/2 spins and egorbitals disentangle (MF can be used)

[ AMO et al., PRB 72, 214431 (2005)]


Spin orbital entanglement and violation of the kanamori goodenough rules

Spin-orbital models with entanglement

  • d1 model – titanates (LaTiO3, YTiO3), S=1/2, t2gorbitals;

  • 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)]

d9model [L.F. Feiner, AMO, and J. Zaanen, PRL 78, 2799 (1997)]


Spin orbital entanglement and violation of the kanamori goodenough rules

eg orbitals

t2g orbitals

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:

At finite η the orbital operators contain:

GdFeO3-type distortions induce orbital interactions leading to FO order:

Pseudospin operators for eg systems (d9) with 3z2-r2and x2-y2:

Jahn-Teller ligand distortions favor AO order:


Spin orbital entanglement and violation of the kanamori goodenough rules

Spin-orbital superexchange at JH=0

=> chain along c axis

=> 2D model in ab planes


Spin orbital entanglement and violation of the kanamori goodenough rules

Intersite spin, orbital and spin-orbital correlations

Spin correlations:

Orbital and spin-orbital correlations for t2g (d1 and d2)systems:

Orbital and spin-orbital correlations for eg (d9) model:

  • Definitions follow from the structure of the spin-orbital SE at JH0;

  • Method: exact diagonalization of four-site systems.


Spin orbital entanglement and violation of the kanamori goodenough rules

Intersite correlations for increasing Hund’s exchange η

V=0

V=J

d1

Sij – spin correlations

Tij –orbital correlations

Cij– spin-orbital correlations

d2

  • all correlations identical in d1at 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 Sijand Tij in d9

d9

GK rules violated in d1, d2

[AMO, P. Horsch, L.F. Feiner, G. Khaliullin, PRL 96, 147205 (2006)]


Spin orbital entanglement and violation of the kanamori goodenough rules

Spin exchange constants Jij for increasing Hund’s exchangeη

V=0

V=J

d1

In the shadded areas

Jij is negative FM

Sij is negative AF

for d1 and d2t2gmodels

=> GK rules are violated

d2

In d9eg model

spin correlations Sij

follow the sign of Jij

=> GK rules are obeyed

d9

[AMO, P. Horsch, L.F. Feiner, G. Khaliullin, PRL 96, 147205 (2006)]


Spin orbital entanglement and violation of the kanamori goodenough rules

Dynamical exchange constants due to entanglement

Fluctuations of Jijare measured by

Fluctuations dominate the behavior of t2g systems at η=0, V=0:

for a bond <ij> fluctuations: ( S=0 / T=1 )  ( S=1 / T=0 )

d1 model:

[ SU(4) symmetry ]

d2 model:

Fluctuations large but do not dominate for eg system at η=0, V=0:

d9 model:

,i.e.,


Spin orbital entanglement and violation of the kanamori goodenough rules

Quantum corrections in spin-orbital models

Large corrections beyond MF due to spin-orbital entanglement

[AMO, P. Horsch, L.F. Feiner, G. Khaliullin, PRL 96, 147205 (2006)]


Spin orbital entanglement and violation of the kanamori goodenough rules

when only Ising term:

sharp transition

S=0

S=4

Continuous orbital phase transition in d2 model

with full t2gorbital dynamics:

V=J

continuous transition

quantum numbers T and Tznonconserved

orbital transitions are continuous

T and Tz conserved


Spin orbital entanglement and violation of the kanamori goodenough rules

Optical spectral weights for the C-AF phase of LaVO3

mean-field approach

orbital and spin-orbital dynamics

orbital disorder unlike in LaMnO3

Data: S. Miyasaka et al.,

[ JPSJ 71, 2086 (2002) ]

spin-orbital fluctuations important at T>0!

[G. Khaliullin, P. Horsch, and AMO, PRB 70, 195103 (2004)]


Conclusions

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)]

Conclusions

  • 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 t2gdegrees of freedom

  • 3. Dynamic spin and orbital fluctuations in t2g systems:

spins and orbitals are entangled

static Goodenough-Kanamori rules are violated

Any other experimental manifestations of entanglement?


Spin orbital entanglement and violation of the kanamori goodenough rules

Thank you

for your attention!