Strongly%20Correlated%20Electron%20Systems:%20a%20DMFT%20Perspective

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Strongly Correlated Electron Systems a DMFT
Perspective

• Gabriel Kotliar
• Physics Department and
• Center for Materials Theory
• Rutgers University

Colloquium UBC September (2004)
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Outline

• Introduction to the strong correlation problem.
• Essentials of DMFT
• The Mott transition problem some insights from
  studies of models.
• Towards an electronic structure method
  applications to materials Ce, Pu
• Outlook

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The electron in a solid wave picture
Momentum Space (Sommerfeld)

Maximum metallic resistivity 200 mohm cm
Standard model of solids Periodic potential,
waves form bands , k in Brillouin zone
Landau Interactions renormalize away
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Standard Model of Solids
RIGID BAND PICTURE. Optical response, transitions
between bands.
Quantitative tools DFT, LDA, GGA, total
energies,good starting point for spectra, GW,and
transport

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The electron in a solid particle picture.

• NiO, MnO, Array of atoms is insulating if
  agtgtaB. Mott correlations localize the electron
• e_ e_ e_
  e_

• Superexchange

• Think in real space , solid collection of atoms
• High T local moments, Low T spin-orbital order

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Mott Correlations localize the electron

• Low densities, electron behaves as a particle,use
  atomic physics, work in real space.

• One particle excitations Hubbard Atoms sharp
  excitation lines corresponding to adding or
  removing electrons. In solids they broaden by
  their incoherent motion, Hubbard bands (eg.
  bandsNiO, 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

• Quantitative calculations of Hubbard bands and
  exchange constants, LDA U, Hartree Fock. Atomic
  Physics.

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Localization vs Delocalization Strong Correlation
Problem

• A large number of compounds with electrons in
  partially filled shells, are not close to the
  well understood limits (localized or itinerant).
  Non perturbative problem.

• These systems display anomalous behavior
  (departure from the standard model of solids).
• Neither LDA GW or LDAU or Hartree Fock work
  well.

• Dynamical Mean Field Theory Simplest approach to
  electronic structure, which interpolates
  correctly between atoms and bands. Treats QP
  bands and Hubbard bands. New reference point, to
  replace the Kohn Sham system.

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DFTGW program has been less succesful in
correlated situations.

• Strong interactions localize the particles. Atoms
  with open shells are not easily connected to band
  theory.
• The spectrum in this case, contain Hubbard bands
  which are NOT simply perturbatively connected to
  the Kohn Sham orbitals.
• Need an alternative reference point for doing
  perturbation theory! Situation is worse in
  between the atomic and the localized limit
• DMFT!

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Correlated Materials do big things

• Mott transition.Huge resistivity changes V2O3.
• Copper Oxides. .(La2-x Bax) CuO4 High Temperature
  Superconductivity.150 K in the Ca2Ba2Cu3HgO8 .
• Uranium and Cerium Based Compounds. Heavy
  Fermion Systems,CeCu6,m/m1000
• (La1-xSrx)MnO3 Colossal Magneto-resistance.

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Strongly Correlated Materials.

• Large thermoelectric response in CeFe4 P12 (H.
  Sato et al. cond-mat 0010017). Ando et.al.
  NaCo2-xCuxO4 Phys. Rev. B 60, 10580 (1999).
• Large and ultrafast optical nonlinearities
  Sr2CuO3 (T Ogasawara et.a Phys. Rev. Lett. 85,
  2204 (2000) )
• Huge volume collapses, Ce, Pu

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Breakdown of standard model

• LDAGW program fails badly.
• Large metallic resistivities exceeding the Mott
  limit. Anderson, Emery and Kivelson
• Breakdown of the rigid band picture. Need new
  ways to think about the excitations.
• Anomalous transfer of spectral weight in
  photoemission and optics. G. Sawatzki

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Strongly correlated systems are usually treated
with model Hamiltonians

• In practice other methods (eg constrained LDA are
  used)

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Strongly correlated systems are usually treated
with model Hamiltonians

• They are hard to derive and hard to solve.
• In practice other methods (eg. constrained LDA
  are used)

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Outline

• Introduction to the strong correlation problem
  and to the Mott transition.
• DMFT ideas
• Applications to the Mott transition problem some
  insights from studies of models.
• Towards an electronic structure method
  applications to materials Pu.
• Outlook

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Mean-Field Classical vs Quantum
Classical case
Quantum case
A. Georges, G. Kotliar (1992)
Phys. Rev. B 45, 6497
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• Insert transparency from nijmeigen
• About infinite dimensions, and about
• Greens functions.

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DMFT Effective Action point of view.

• Identify observable, A. Construct an exact
  functional of ltAgta, G a which is stationary at
  the physical value of a.
• Example, density in DFT theory. (Fukuda et. al.)
• When a is local, it gives an exact mapping onto a
  local problem, defines a Weiss field.
• The method is useful when practical and accurate
  approximations to the exact functional exist.
  Example LDA, GGA, in DFT.
• DMFT, build functionals of the LOCAL spectral
  function.
• Density of states for adding or removing and
  electron
• Exact functionals exist. We also have good
  approximations!
• Extension to an ab initio method.

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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).
• S. Savrasov and G.Kotliar and Abrahams
  funcional formulation for full self consistent
  Nature \bf 410, 793(2001).
• Reviews Held et.al. , Psi-k Newsletter \\bf
  56 (April 2003), p. 65 Lichtenstein Katsnelson
  and and Kotliar cond-mat/0211076

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How good is the LOCAL approximation?
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C-DMFT test in one dimension. (Bolech, Kancharla
GK cond-mat 2002)
Gap vs U, Exact solution Lieb and Wu, Ovshinikov
Nc2 CDMFT vs Nc1
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N vs mu in one dimension.Compare 28 vs exact
Bethe Anzats, M. Capone and M.Civelli
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Outline

• Introduction to the strong correlation problem.
• Essentials of DMFT
• Applications to the Mott transition problem some
  insights from studies of models.
• Towards an electronic structure method
  applications to materials
• Outlook

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The Mott transition

• Electronically driven MIT.
• Forces to face directly the localization
  delocalization problem.
• Relevant to many systems, eg V2O3
• Techniques applicable to a very broad
• range or problems.

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Mott transition in V2O3 under pressure or
chemical substitution on V-site
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• Resistivity.
• Limelette et. al.

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How good is the local approximation ?

• Single site DMFT study of the Mott transition,
  based on a study of the Hubbard model on
  frustrated lattices made several interesting
  qualitative predictions.
• New experiments and reexamination of old ones
  give credence to that the local picture is quite
  good.
• DMFT is a new reference frame to approach
  strongly correlated phenomena, and describes
  naturally , NON RIGID BAND picture, highly
  resistive states, etc.

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Insight

• Phase diagram in the T, U plane of a frustrated
  ((the magnetic order is supressed)) correlated
  system at integer filling.
• At high temperatures, the phase diagram is
  generic, insensitive to microscopic details.
• At low temperatures, details matters.

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Schematic DMFT phase diagram one band Hubbard
model (half filling, semicircular DOS, partial
frustration) Rozenberg et.al PRL (1995)
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Mott transition in layered organic conductors
S Lefebvre et al. cond-mat/0004455, Phys. Rev.
Lett. 85, 5420 (2000)
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Insight, in the strongly correlated region the
one particle density of states has a three peak
structurelow energy quasiparticle peak plus
Hubbard bands.
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DMFT has bridged the gap between band theory and
atomic physics.

• Delocalized picture, it should resemble the
  density of states, (perhaps with some additional
  shifts and satellites).
• Localized picture. Two peaks at the ionization
• and affinity energy of the atom.

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One electron spectra near the Mott transition,
three peak structure.
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ARPES measurements on NiS2-xSexMatsuura et. Al
Phys. Rev B 58 (1998) 3690. Doniaach and Watanabe
Phys. Rev. B 57, 3829 (1998)
.
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QP in V2O3 was recently found Mo et.al
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Insights from DMFT

• The Mott transition is driven by transfer of
  spectral weight from low to high energy as we
  approach the localized phase
• Control parameters doping, temperature,pressure

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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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ARPES measurements on NiS2-xSexMatsuura et. Al
Phys. Rev B 58 (1998) 3690. Doniaach and Watanabe
Phys. Rev. B 57, 3829 (1998)
.
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Anomalous metallic resistivities

• In the in between region anomalous
• resistivities are the rule rather than the
  exception.

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Failure of the Standard Model NiSe2-xSx
Miyasaka and Takagi (2000)
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Anomalous Resistivity and Mott transition
(Rozenberg et. Al. ) Ni Se2-x Sx
Insights from DMFT think in term of spectral
functions (branch cuts) instead of well defined
QP (poles )
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More recent work, organics, Limelette et.
al.(PRL 2003)
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Anomalous Resistivities when wave picture does
not apply. Doped Hubbard model
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Qualitative single site DMFT predictions Optics

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

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Anomalous transfer of spectral weight in v2O3
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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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Anomalous transfer of spectral weight heavy
fermions Rozenberg Kajueter Kotliar (1996)
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Anomalous transfer of optical weight A.
Damascelli D. Van der Marel
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Anomalous Spectral Weight Transfer Optics
Below energy
ApreciableT dependence found.
Schlesinger et.al (FeSi) PRL 71 ,1748 , (1993) B
Bucher et.al. Ce2Bi4Pt3PRL 72, 522 (1994),
Rozenberg et.al. PRB 54, 8452, (1996).
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DMFT and the strong correlation anomalies
crossover from momentum space to real space
picture

• Metals with resistivities which exceed the Mott
  Ioffe Reggel limit.
• Three peak structure of DOS
• Transfer of spectral weight which is non local in
  frequency.
• Dramatic failure of DFT based approximations in
  predicting physical properties.

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Outline

• Introduction to the strong correlation problem.
• Essentials of DMFT
• Applications to the Mott transition problem some
  insights from studies of models.
• Towards an electronic structure method
  applications to materials Pu, Fe, Ni, Ce,
  LaSrTiO3, NiO,MnO,CrO2,K3C60,2d and quasi-1d
  organics, magnetic semiconductors,SrRuO4,V2O3.
  
• Outlook

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Generalized phase diagram
T
U/W
Relax Structure, bands, orbitals
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Pu in the periodic table
actinides
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Electronic Physics of Pu
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DFT studies.

• Underestimates the volume by 35
• Predicts Pu to be magnetic.
• Largest quantitative failure of DFT-LDA-GA
• Fail to predict a stable delta phase.

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Phonon Spectra

• Electrons are the glue that hold the atoms
  together. Vibration spectra (phonons) probe the
  electronic structure.
• Phonon spectra reveals instablities, via soft
  modes.
• Phonon spectrum of Pu had not been measured until
  recently.

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Phonon freq (THz) vs q in delta Pu X. Dai et. al.
Science vol 300, 953, 2003
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Inelastic X ray scattering. Wong et. al.
Science 301, 1078 (2003).
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Pu DMFT total energy vs Volume (Savrasov
Kotliar and Abrahams 2001)
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Alpha and delta Pu
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Outline

• Introduction to the strong correlation problem.
• Essentials of DMFT
• The Mott transition problem some insights from
  studies of models.
• Towards an electronic structure method
  applications to materials Pu
• Outlook

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What do we want from materials theory?

• New concepts , qualitative ideas
• Understanding, explanation of existent
  experiments, and predictions of new ones.
• Quantitative capabilities with predictive
• power.
• Notoriously difficult to achieve in strongly
  correlated materials. DMFT is delivering on both
  fronts.

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Outlook

• Local approach to strongly correlated electrons.
• Many extensions, make the approach suitable for
  getting insights and quantitative results in
  correlated materials.

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Conclusion

• The character of the localization delocalization
  in simple( Hubbard) models within DMFT is now
  fully understood, nice qualitative insights.
• This has lead to extensions to more realistic
  models, and a beginning of a first principles
  approach to the electronic structure of
  correlated materials.

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Outlook

• Systematic improvements, short range
  correlations, cluster methods, improved mean
  fields.
• Improved interfaces with electronic structure.
• Exploration of complex strongly correlated
  materials. Correlation effects on surfaces,
• large molecules, systems out of equilibrium,
  illumination, finite currents, aeging.

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Acknowledgements Development of DMFT
Collaborators V. Anisimov,G. Biroli, R.
Chitra, V. Dobrosavlevic, X. Dai, D. Fisher,
A. Georges, H. Kajueter, K. Haujle, W.Krauth, E.
Lange, A. Lichtenstein, G. Moeller, Y. Motome,
O. Parcollet , G. Palsson, M. Rozenberg, S.
Savrasov, Q. Si, V. Udovenko, I. Yang, X.Y. Zhang
Support NSF DMR 0096462 Support
Instrumentation. NSF DMR-0116068 Work on Fe
and Ni ONR4-2650 Work on Pu DOE
DE-FG02-99ER45761 and LANL subcontract No.
03737-001-02
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High Performance Computing http//beowulf.rutgers
.edu
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Shear anisotropy fcc Pu (GPa)

• C(C11-C12)/2 4.78
• C44 33.59
• C44/C 8 Largest shear anisotropy in any
  element!
• LDA Calculations (Bouchet et. al.) C -48

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Dai et. al.
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Epsilon Plutonium.
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Anomalous transfer of spectral weight heavy
fermions
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Anomalous transfer of spectral weight
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Anomalous transfer of spectral weigth heavy
fermions
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V2O3 resistivity
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Example DMFT for lattice model (e.g. single band
Hubbard).Muller Hartman 89, Chitra and Kotliar 99.

• Observable Local Greens function Gii (w).
• Exact functional G Gii (w) .
• DMFT Approximation to the functional.

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Spectral Density Functional effective action
construction (Chitra and GK).

• 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)
• Approximate functional using DMFT insights.