Optical Properties of Strongly Correlated Electrons: A Dynamical Mean Field Approach

1
Optical Properties of Strongly Correlated
Electrons A Dynamical Mean Field Approach

• G. Kotliar
• Physics Department and Center for Materials
  Theory Rutgers University

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Outline

• Correlated Electrons and the Dynamical Mean Field
  Theory (DMFT) framework.
• Restricted Sum Rules and Transfer of Optical
  Spectral Weight.
• Optics near the temperature driven Mott
  transition.
• The Cerium alpha-gamma transition, Mott
  transition or Kondo collapse ? A perspective from
  optics.

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References, Collaborators.

• DMFT Reviews A. Georges G. Kotliar W. Krauth
  and M. Rozenberg RMP68 , 13, 1996 Gabriel Kotliar
  and Dieter Vollhardt Physics Today 57,(2004).
• Optical transfer or spectral weight near the
  Mott transition. M. Rozenberg G. Kotliar and H.
  Kajueter PRB 54, 8452, (1996).
• DMFT Optics V. Udovenko S. Savrasov K. Haule and
  G. Kotliar Cond-matt 0209336.
• Alpha-Gamma Cerium. K. Haule V. Udovenko S.
  Savrasov and G. Kotliar. Cond-matt 0403086.

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MAIN MESSAGE

• DMFT is a working tool (under constant
  development).
• Theory (DMFT) and experiments (optical
  conductivity) complement each other extraordinary
  well.
• Interpretation.
• Predictions.
• Access to regimes that cannot be easily reached
  in real materials.

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Standard Model . Kohn Sham reference system
Excellent starting point for computation of
spectra in perturbation theory in screened
Coulomb interaction GW. Bethe Salpeter equation
for optics.
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Standard Model fails when Correlations localize
the electron
Hubbard bands. One particle excitations
corresponding to adding or removing electrons. In
solids they broaden by their incoherent motion
(eg. Mott insulators NiO, CoO MnO.) H H H
H H H motion of H forms the
lower Hubbard band H H H H- H H
motion of H_ forms the upper Hubbard band
Optical conductivity, start from atomic physics
and broaden the atomic transitions (on site
processes). Transitions to neighboring atomic
states (transitions between the Hubbard bands ).
One needs a tool that treats quasiparticle bands
and Hubbard bands on the same footing to contain
the band and atomic limit. DMFT!
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Strong correlation anomalies

• Metals with resistivities which exceed the Mott
  Ioffe Reggel limit.
• Gigantic linear and non linear responses.
• Dramatic failure of DFT based approximations
  (say DFT-GW) in predicting physical properties.
• Breakdown of the rigid band picture.

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Transfer of optical spectral weight non local in
frequency Schlesinger et.al (FeSi) PRL 71 ,1748 ,
(1993) B Bucher et.al. Ce2Bi4Pt3 PRL 72, 522
(1994),
Neff depends on T
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Two roads for ab-initio calculation of electronic
structure of strongly correlated materials
Crystal structure Atomic positions
Model Hamiltonian
Correlation Functions Total Energies etc.
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RESTRICTED SUM RULES
Below energy
ApreciableT dependence found.
M. Rozenberg G. Kotliar and H. Kajueter PRB 54,
8452, (1996).
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RESTRICTED SUM RULES
Below energy
ApreciableT dependence found.
M. Rozenberg G. Kotliar and H. Kajueter PRB 54,
8452, (1996).
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DMFT Cavity Construction. A. Georges and G.
Kotliar PRB 45, 6479 (1992). First happy marriage
of a technique from atomic physics and a
technique band theory.
Reviews A. Georges G. Kotliar W. Krauth and M.
Rozenberg RMP68 , 13, 1996 Gabriel Kotliar and
Dieter Vollhardt Physics Today 57,(2004)
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Self-Consistency loop. S. Savrasov and G.
Kotliar (2001) and cond-matt 0308053
E
U
DMFT
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Spectral Density Functional effective action
construction G. Kotliar, and S. Savrasov, in New
Theoretical approaches to strongly correlated
systems, edited by A.M. Tsvelik, Kluwer Academic
Publishers, 259 (2001) S. Y. Savrasov and G.
Kotliar, Phys. Rev. B 69, 245101 (2004).)

• DFT, consider the exact free energy as a
  functional of an external potential. Express the
  free energy as a functional of the density by
  Legendre transformation. GDFTr(r)
• Introduce local orbitals, caR(r-R)orbitals, and
  local GF
• G(R,R)(i w)
• The exact free energy can be expressed as a
  functional of the local Greens function and of
  the density by introducing sources for r(r) and G
  and performing a Legendre transformation,
  Gr(r),G(R,R)(iw)

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LDADMFT References

• V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin
  and G. Kotliar, J. Phys. Cond. Mat. 35,
  7359-7367 (1997).
• A Lichtenstein and M. Katsenelson Phys. Rev. B
  57, 6884 (1988).
• G. Kotliar, and S. Savrasov, in New Theoretical
  ap-
• proaches to strongly correlated systems, edited
  by A.
• M. Tsvelik, Kluwer 259 (2001) S. Y. Savrasov
  and G. Kotliar, Phys. Rev. B 69, 245101 (2004).

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LDADMFT Formalism.
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Optics formula
double pole
One divergence integrated out!
single pole
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Comments on LDADMFT

• Gives the local spectra and the total energy
  simultaneously, treating QP and H bands on the
  same footing.
• Gives an approximate starting point, for
  perturbation theory in the non local part of the
  Coulomb interactions. See for example, P. Sun
  and G. Kotliar PRL .
• Good approximate starting point for optics.

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Outline

• Correlated Electrons and the Dynamical Mean Field
  Theory (DMFT) framework.
• Restricted Sum Rules and Transfer of Optical
  Spectral Weight.
• Optics near the temperature driven Mott
  transition.
• The Cerium alpha-gamma transition, Mott
  transition or Kondo collapse ? A perspective from
  optics.
• Doping driven Mott transition in La1-x SrxTiO3.
• A perspective from the optical conductivity.

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Pressure Driven Mott transition
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Insights from DMFT

• Low temperature Ordered phases . Stability
  depends on chemistry and crystal structure
• High temperature behavior around Mott endpoint,
  more universal regime, captured by simple models
  treated within DMFT. Role of magnetic frustration.

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Schematic DMFT phase diagram of a partially
frustrated integered filled Hubbard model. M. J.
Rozenberg, G. Kotliar, H. Kajueter, G. A. Thomas,
D. H. Rapkine, J. M. Honig, and P. Metcalf, Phys.
Rev. Lett. 75, 105, 1995
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Spectral Evolution at T0 half filling full
frustrationX.Zhang M. Rozenberg G. Kotliar (PRL
70,16661993)

• Spectra of the strongly correlated metallic
  regime contains both quasiparticle-like and
  Hubbard band-like features.
• Mott transition is driven by transfer of spectral
  weight.

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Evolution of the Spectral Function with
Temperature
Anomalous transfer of spectral weight connected
to the proximity to the Ising Mott endpoint
(Kotliar Lange nd Rozenberg Phys. Rev. Lett. 84,
5180 (2000)
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Consequences for the optical conductivity
Evidence for QP peak in V2O3 from optics.
M. Rozenberg G. Kotliar H. Kajueter G Thomas D.
Rapkine J Honig and P Metcalf Phys. Rev. Lett.
75, 105 (1995)
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Anomalous transfer of spectral weight
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Anomalous transfer of optical spectral weight V2O3

• M Rozenberg G. Kotliar and H. Kajuter Phys. Rev.
  B 54, 8452 (1996).
• M. Rozenberg G. Kotliar H. Kajueter G Tahomas D.
  Rapkikne J Honig and P Metcalf Phys. Rev. Lett.
  75, 105 (1995)

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Anomalous transfer of optical spectral weight,
NiSeS. Miyasaka and Takagi 2000
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Optical transfer of spectral weight , kappa
organics. Eldridge, J., Kornelsen, K.,Wang,
H.,Williams, J., Crouch, A., and Watkins, D.,
Sol. State. Comm., 79, 583 (1991).
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Epilogue, the search for a quasiparticle peak and
its demise, photoemission, transport.
Confirmation of the DMFT predictions

• ARPES measurements on NiS2-xSexMatsuura et. Al
  Phys. Rev B 58 (1998) 3690. Doniaach and Watanabe
  Phys. Rev. B 57, 3829 (1998)
• S.-K. Mo et al., Phys Rev. Lett. 90, 186403
  (2003).
• Limelette et. al. Science G. Kotliar
  Science.

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Case study Cerium.

• Study the alpha to gamma transition.
• Test the approach, in a well studied setting.
• Differentiate between the Kondo volume collapse
  picture and the Mott transition picture.

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Overview
volumes exp. LDA LDAU
a 28Å3 24.7Å3
g 34.4Å3 35.2Å3

• ? ? -phase (localized)
• High T phase
• Curie-Weiss law (localized magnetic moment),
• Large lattice constant
• Tk around 60-80K

• ? ?-phase (delocalizedKondo-physics)
• Low T phase
• Loss of Magnetism (Fermi liquid Pauli
  susceptibility) - completely screened magnetic
  moment
• smaller lattice constant
• Tk around 1000-2000K

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Qualitative Ideas.

• Johanssen, Mott transition of the f electrons as
  a function of pressure. Ce alpha gamma
  transition. spd electrons are spectators.
• Mathematical implementation, metallic phase
  treat spdf electrons by LDA, insulating phase
  put f electron in the core.
• Allen and Martin. Kondo volume collapse picture.
  The dominant effect is the spd-f hybridization.

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Qualitative Ideas

• screened moment alpha phase Kondo effect
  between spd and f takes place. unscreend moment
  gamma phase no Kondo effect (low Kondo
  temperature).
• Mathematical implementation, Anderson impurity
  model in the Kondo limit suplemented with elastic
  terms. (precursor of DMFT ideas, but without self
  consistency condition).

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Photoemissionexperiment

• A. Mc Mahan K Held and R. Scalettar (2002)
• K. Haule V. Udovenko and GK. (2003)

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Unfortunately photoemission cannot decide between
the Kondo collapse picture and the Mott
transition picture.Evolution of the spectra as a
function of U , half filling full frustration,
Hubbard model!!!!
X.Zhang M. Rozenberg G. Kotliar (PRL 1993) A.
Georges, G. Kotliar (1992) Phys. Rev. B 45, 6497
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Resolution Turn to Optics!

• Qualitative idea. The spd electrons have much
  larger velocities, so optics will be much more
  senstive to their behavior.
• See if they are simple spectators (Mott
  transition picture ) or wether a Kondo binding
  unbinding takes pace (Kondo collapse picture).

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LDA and LDADMFT studies.K.Haule et. al.
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Optical Conductivity Temperature dependence.
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Origin of the features.
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Conclusion

• The anomalous temperature dependence
• and the formation of a pseudogap, suggests
  that the Kondo collapse picture is closer to the
  truth for Cerium.
• Possible experimental verification in Ce(ThLa)
  alloys.
• Qualitative agreement with experiments,
  quantitative discrepancies. (see however J.Y.
  Rhee, X. Wang, B.N. Harmon, and D.W. Lynch, Phys.
  Rev. B 51, 17390 (1995) ).

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Conclusion

• Dynamical mean field theory, a first principles
  approach to the computation of physical
  properties of correlated materials.
• Tool under construction! Many improvements are
  possible.
• Already giving interesting results.
• Violations of the restricted sum rule near the
  temperature driven Mott transition of the order
  or 5 -10 . Prediction of DMFT. Verified in
  experiments.

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Conclusion

• Complementary tool to photoemission/inverse
  photoemission.
• Experimental advantages. Ex. V2O3, Cerium.
• Future work, investigate vertex corrections.
• Future work Where does the spectral weight go ?
• Future work, study more materials.

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La1-xSrx O3

• Adding holes to a Mott insulator in three
  dimensions.
• For very small doping,(xlt.07) interesting spin
  and orbital order takes place, non universal
  physics and lattice distortions are important.
  Small energy scales, larger dopings more robust
  universal behavior.
• Magnetic frustration. Good system to applyDMFT.

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Optical Conductivity
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Optical conductivity
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Realistic Computation of Optical Properties
La1-xSrxTiO3
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Conclusion

• Reasonable agreement, between theory and
  experiments at both low and high energy.
• The dependence of Neff on doping is due to the
  changes in the effective mass.

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(Tokura et. Al. 1993)A doped Mott
insulatorLaxSr1-xO3
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DMFT calculation U near the Mott transition,
Rozenberg et.al 94
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Hall Coefficient, electron like.
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La1-xSrxTiO3 photoemission
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Evolution of spectra with doping U4
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Haule et. al.
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