Optical Properties of Strongly Correlated
Electrons: A Dynamical Mean Field Approach
G. Kotliar
Physics Department and Center for Materials
Theory Rutgers University
Outline
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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

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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
- Ñ / 2 + VKS (r ) y kj = ekj y kj
2
r (r ) =
å
f (ekj ) | y kj (r ) |2
kj
VKS (r )[r (r )] = Vext (r ) +
ò
dExcLDA
r (r ')
dr '+
[r ]
| r- r'|
dr (r )
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
H
H-
H H
H H
motion of H+ forms the lower Hubbard band
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
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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
H hamiltonian, J electric current , P polarization


0
 ( )d 
Below energy


0
 ( )d 

iV
  P, J  
H eff , J eff , Peff
 ne 2
m


iV
  Peff , J eff  
ApreciableT dependence
found.

 nk
k
2
k
k
2
M. Rozenberg G. Kotliar and H. Kajueter PRB 54, 8452, (1996).
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RESTRICTED SUM RULES
H hamiltonian, J electric current , P polarization


0
 ( )d 
Below energy


0
 ( )d 

iV
  P, J  
H eff , J eff , Peff
 ne 2
m


iV
  Peff , J eff  
ApreciableT dependence
found.

 nk
k
2
k
k
2
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
c ka | - Ñ 2 + Vxc (r ) | c ka = H LMTO ( k )
G
0
Impurity
Solver
E
G
U

S.C.C.
DMFT
r (r) = T
å
G( r, r, iw)e
iw0+
nHH = T
å
iw
iw
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+
GHH ( r , r , iw)eiw 0
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).)
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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. GDFTr(r)]
Introduce local orbitals, caR(r-R)orbitals, and local GF
G(R,R)(i ) =
 dr ' dr c R(r ) *G(r, r ')(i ) ca R(r ')
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, Gr(r),G(R,R)(i)]
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LDA+DMFT References
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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 approaches 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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LDA+DMFT Formalism.
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Optics formula
double pole
One divergence integrated out!
single pole
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Comments on LDA+DMFT
•
•
•
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

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
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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 T=0 half filling full
frustrationX.Zhang M. Rozenberg G. Kotliar (PRL
70,16661993)
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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
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ARPES measurements on NiS2-xSex
Matsuura 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
 Various phases :
isostructural phase transition (T=298K, P=0.7GPa)
 (fcc) phase
[ magnetic moment
(Curie-Wiess law) ]
 a (fcc) phase
[ loss of magnetic
moment (Pauli-para) ]
with large
volume collapse
v/v  15
( -phase a  5.16 Å
a-phase a  4.8 Å)
volumes exp.
28Å3
a
34.4Å3

  -phase
LDA
24.7Å3
LDA+U
35.2Å3
(localized):
High T phase
Curie-Weiss law (localized
magnetic moment),
Large lattice constant
Tk around 60-80K
 a-phase (delocalized:Kondo-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.
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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
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“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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Photoemission&experiment
•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!
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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 LDA+DMFT
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
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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
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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
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Adding holes to a Mott insulator in three
dimensions.
For very small doping,(x<.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
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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
insulator:LaxSr1-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 U=4
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Haule et. al.
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