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

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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


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


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)]


Multipletstructureof transition metal ions

Follows from three Racah parameters (Griffith, 1971):

single parameter: η=JH /U

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


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)]


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 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)]


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 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)]


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 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:

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.


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 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)]


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.,


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)]


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


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?


Thank you

for your attention!


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