Strongly Correlated Electron Systems: a DMFT Perspective

1
Strongly Correlated Electron Systems a DMFT
Perspective

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

Statistical Mechanics Conference
Rutgers December 2003
2
Outline

• Introduction to strongly correlated electrons.
• Introduction to Dynamical Mean Field Theory
  (DMFT)
• The Mott transition problem some theoretical
  insights from DMFT studies of simple models.
  Some recent Experiments.
• Towards a DMFT based electronic structure
  method. Some highlight of recent results.

3
Outline

• Introduction to strongly correlated electrons.
• Essentials of Dynamical Mean Field Theory (DMFT)
• The Mott transition problem some theoretical
  insights from DMFT studies of simple models.
  Some recent Experiments.
• Towards a DMFT based electronic structure
  method. Some highlight of recent results.

4
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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The electron in a solid particle picture.

• NiO, MnO, solid as a collection of atoms.
• e_ e_ e_
  e_

• High T local moments,
• Low T spin-orbital order

• Superexchange

• One particle excitations Hubbard bands . Excited
  atomic states adding or removing electrons,
  broaden into bands .

• 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

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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 (large
  metallic resistivities, optical responses that
  cannot be interpreted in terms of rigid bands,
  ..).
• None of the standard electronic structure tools
  (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.

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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.
• Huge Volume Collapses in Lanthanides (Ce, Pr)
  and Actinides ( Pu, Am)
• ..
• Unusual behavior, large resistivity, non-rigid
  bands,
• failures of the standard model.

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The Mott transition

• Pressure driven MIT.
• Forces to face directly a central issue of the
  strongly correlated electron systems.
• Localization delocalization problem.
• Relevant to many materials, eg V2O3,organics
• Techniques applicable to a very broad
• range or problems.

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Pressure Driven Mott transition
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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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Limit of large lattice coordination
Metzner Vollhardt, 89
Muller-Hartmann 89
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Single site DMFT Impurity cavity construction
A. Georges, G. Kotliar, PRB, (1992)
Weiss field
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Extension to clusters. Cellular DMFT. C-DMFT. G.
Kotliar,S.Y. Savrasov, G. Palsson and G. Biroli,
Phys. Rev. Lett. 87, 186401 (2001)
t(K) is the hopping expressed in the
superlattice notations.

• Other cluster extensions (DCA, nested cluster
  schemes, PCMDFT ), causality issues, O.
  Parcollet, G. Biroli and GK cond-matt 0307587
  (2003)

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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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Comments on DMFT.

• Review of DMFT, technical tools for solving DMFT
  eqs. A. Georges, G. Kotliar, W. Krauth and M.
  Rozenberg Rev. Mod. Phys. 68,13 (1996)
• CDMFT , instead of studying finite systems with
  open or periodic boundary conditions, study a
  system in a medium. Connection with DMRG, infer
  the density matrix by using a Gaussian anzats,
  and the periodicity of the system.

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How good is the LOCAL approximation?
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C-DMFT test in one dimension. (Bolech, Kancharla
GK PRB 2003)
Gap vs U, Exact solution Lieb and Wu, Ovshinikov
Nc2 CDMFT vs Nc1
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N vs mu in one dimensional Hubbard model
.Compare 2 site cluster (in exact diag with
Nb8) vs exact Bethe Anzats, M. Capone C.
Castellani M.Civelli and GK (2003)
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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. A look at recent
  experiments.
• Towards an electronic structure method
  applications to materials
• Outlook

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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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DMFT results. Hubbard model on a frustrated
lattice.

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

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Insight, in the strongly correlated region the
one particle density of states Density of
states for adding and removing one particle.
Measureable in photoemission and inverse
photoemissionhas a three peak structurelow
energy quasiparticle peak plus Hubbard bands.
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Density of states for adding and removing one
particle. Measureable in photoemission and
inverse photoemission.

• 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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Anomalous metallic resistivities

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

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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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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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Recent study of V2O3 under pressure Limlette
et.al. Science 2003. Ising behavior at the
endpoint finally found!
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Ising critical endpoint! In V2O3 P. Limelette
et.al.
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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 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 studies of the Mott transition

• Single site DMFT study of the Mott transition,
  based on a study of the Hubbard model on
  frustrated lattices made several interesting
  qualitative predictions. Confirmed by CDMFT
  study on 2 by2 cluster with intersting
  modifications. O. Parcollet G. Biroli and G.
  Kotliar cond-matt 0308577
• New experiments and of 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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Outline

• Introduction to strongly correlated electrons.
• Introduction to Dynamical Mean Field Theory
  (DMFT)
• The Mott transition problem some theoretical
  insights from DMFT studies of simple models.
  Some recent Experiments.
• Towards a DMFT based electronic structure
  method. Some highlight of recent results.

42
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 cond-matt 0308053.
  (2003.)
• Reviews Held et.al. , Psi-k Newsletter 56
  (April 2003), p. 65
• A. Lichtenstein M. Katsnelson and G. Kotliar
  cond-mat/0211076

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Overview

• ? ? -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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Photoemissionexperiment

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

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Optical conductivity of Ce (expt. Van der Marel
theory Haule et.al)
experiment
LDADMFT

• K. Haule V. Udovenko and G Kotliar (2003)

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Mott transition in the actinide series (Smith
Kmetko phase diagram)
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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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Pu DMFT total energy vs Volume (Savrasov
Kotliar and Abrahams 2001)
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Alpha and delta Pu photoemission and dos (S.
Savrasov and G. Kotliar)
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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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Outline

• Introduction to strongly correlated electrons.
• Introduction to Dynamical Mean Field Theory
  (DMFT)
• The Mott transition problem some theoretical
  insights from DMFT studies of simple models.
  Some recent Experiments.
• Towards a DMFT based electronic structure
  method. Some highlight of recent results.

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Evolution of the Spectral FunctionU/D2, U/D2.25
(Parcollet Biroli and Kotliar 2003.)
Uc2.35-.05, Tc/D1/44
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Conjecture

• Formation of hot regions is a more general
  phenomena due to the proximity to the Mott point.

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Deviations from single site DMFT
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Lattice and cluster self energies
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Mechanism for hot spot formation
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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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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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V2O3 resistivity
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Photoemissionexperiment
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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.

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Mott transition in layered organic conductors
S Lefebvre et al. cond-mat/0004455
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Mott transition in V2O3 under pressure or
chemical substitution on V-site
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Single site DMFT, functional formulation

Local self energy (Muller Hartman 89)
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DMFT functional constructions.

• Construction of approximations in the cluster
  case requires care to maintain causality.
• Cellular DMFT a) take a supercell of the desired
  range,b)

• c) obtain estimate of the lattice self energy by
  restoring translational symmetry.
• Many other cluster approximations (eg. DCA, the
  use of lattice self energy in self consistency
  condition, restrictions of BK functional, etc.
  exist). Causality and classical limit of these
  methods has recently been clarified G Biroli O
  Parcollet and GK

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N vs mu in one dimensional Hubbard model
.Compare 2 site cluster (in exact diag with
Nb8) vs exact Bethe Anzats, M. Capone C.
Castellani M.Civelli and GK (2003)
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Failure of the Standard Model NiSe2-xSx
Miyasaka and Takagi (2000)
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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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Start with the TOE
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Rewrite the TOE as an electron boson problem.
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Build effective action for the local greens
functions of the fermion and Bose field

• rRr
• R unit cell vector
• r position within the unit cell. IrgtR, rgt
• Couple sources to

84

• Legendre transfor the sources, eliminating the
  field f,
• Build exact functional of the correlation
  functionsW(r R,r R)
• and G (r R,r R)

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Kohn Sham decomposition.
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(E)DMFT pproximation to
Sum over all LOCAL 2PI graphs (integrations are
restricted over the unit cell ) built with W and
G Map into impurity model to generate G and W Go
beyond this approximation by returning to many
body theory and adding the first non local
correction.