Electronic%20Structure%20of%20Strongly%20Correlated%20Materials%20:%20a%20DMFT%20Perspective

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Electronic Structure of Strongly Correlated
Materials a DMFT Perspective

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

Boston March 2002
2
Outline

• Introduction to strongly correlated electrons
• Dynamical Mean Field Theory
• Model Hamiltonian Studies. Universal aspects
  insights from DMFT
• System specific studies LDADMFT
• Outlook

3
Strongly Correlated Materials

• 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.

4
Strongly Correlated Materials.

• High Temperature Superconductivity in doped
  filled Bucky Balls (J. H. Schon et.al Science
  Express 1064773 (2001)) CHBr3 C60 Tc117K .
• 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) )

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

Maximum metallic resistivity 200 mohm cm
Standard model of solids (Bloch, Landau)
Periodic potential, waves form bands , k in
Brillouin zone . Interactions renormalize away.
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Standard Model of Solids

• Qualitative predictions low temperature
  dependence of thermodynamics and transport.
• Optical response, transition between the bands.
• Qualitative predictions filled bands give rise
  to insulting behavior. Compounds with odd number
  of electrons are metals.
• Quantitative tools Density Functional Theory
  with approximations suggested by the Kohn Sham
  formulation, (LDA GGA) is a successful
  computational tool for the total energy. Good
  starting point for perturbative calculation of
  spectra,eg. GW. Kinetic equations yield transport
  coefficients.

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The electron in a solid particle picture.

• Array of hydrogen 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, 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 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 b
  and Hubbard bands.

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Failure of the standard model Mott transition in
V2O3 under pressure or chemical substitution on
V-site
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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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Failure of the Standard Model NiSe2-xSx
Miyasaka and Takagi (2000)
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Failure of the standard model Anomalous
ResistivityLiV2O4
Takagi et.al. PRL 2000
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Failure of the StandardModel Anomalous Spectral
Weight Transfer
Optical Conductivity o of FeSi for T,20,20,250
200 and 250 K from Schlesinger et.al (1993)
Neff depends on T
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Strong Correlation Problem

• Large number of compounds (d,f,p.).Qualitative
  and quantitive failures of the standard model.
• Treat the itinerant and localized aspect of the
  electron
• The Mott transition, head on confrontation with
  this issue
• Dynamical Mean Field Theory simplest approach
  interpolating between bands and atoms with open
  shells.

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Hubbard model

• U/t
• Doping d or chemical potential
• Frustration (t/t)
• T temperature

Mott transition as a function of doping, pressure
temperature etc.
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Limit of large lattice coordination
Metzner Vollhardt, 89
Muller-Hartmann 89
18
Dynamical Mean Field Theory, cavity construction
A. Georges G. Kotliar
Phys. Rev. B 45, 6497,1992
19
Mean-Field Classical vs Quantum
A. Georges, G. Kotliar (1992)
Phys. Rev. B 45, 6497
Quantum case
Classical case
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Solving the DMFT equations

• Wide variety of computational tools
  (QMC,NRG,ED.)Analytical Methods
• Extension to ordered states.
• Review A. Georges, G. Kotliar, W. Krauth and
  M. Rozenberg Rev. Mod. Phys. 68,13 (1996)

21
Single site DMFT, functional formulation.
Construct a functional of the local Greens
function

• Expressed in terms of Weiss field
  (semicircularDOS) G. Kotliar EBJB 99

22
Insights from DMFT

• Low temperatures several competing phases .
  Their relative stability depends on chemistry
  and crystal structure
• High temperature behavior around Mott endpoint,
  more universal regime, captured by simple models
  treated within DMFT

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Schematic DMFT phase diagram Hubbard model
(partial frustration)
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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Kuwamoto Honig and AppellPRB (1980)
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Phase Diag Ni Se2-x Sx
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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 and Rozenberg Phys. Rev. Lett. 84,
5180 (2000)
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Anomalous transfer of optical spectral weight,
NiSeS. Miyasaka and Takagi
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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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ARPES measurements on NiS2-xSexMatsuura et. Al
Phys. Rev B 58 (1998) 3690
.
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Insights from DMFT think in term of spectral
functions (branch cuts) instead of well defined
QP (poles )
Resistivity near the metal insulator endpoint (
Rozenberg et.al 1995) exceeds the Mott limit
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Anomalous Resistivity and Mott transition Ni
Se2-x Sx
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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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Insights from DMFT

• Mott transition as a bifurcation of an effective
  action
• Important role of the incoherent part of the
  spectral function at finite temperature
• Physics is governed by the transfer of spectral
  weight from the coherent to the incoherent part
  of the spectra. Real and momentum space.

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Realistic Calculationsof the Electronic
Structure of Correlated materials

• Combinining DMFT with state of the art electronic
  structure methods to construct a first principles
  framework to describe complex materials. Beyond
  LDAU approach (Anisimov, Andersen and Zaanen)
• Anisimov Poteryaev Korotin Anhokin and Kotliar J.
  Phys. Cond. Mat. 35, 7359 (1997)

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Combining LDA and DMFT

• The light, SP (or SPD) electrons are extended,
  well described by LDA
• The heavy, D (or F) electrons are localized,treat
  by DMFT.
• LDA already contains an average interaction of
  the heavy electrons, subtract this out by
  shifting the heavy level (double counting term)
• The U matrix can be estimated from first
  principles or viewed as parameters

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Spectral Density Functional effective action
construction (Fukuda, Valiev and Fernando ,
Chitra and Kotliar).

• 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 Gr(r),G(R,R)(iw)
• A useful approximation to the exact functional
  can be constructed, the DMFT LDA functional.
  Savrasov Kotliar and Abrahams Nature 410, 793
  (2001))

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LDADMFT Self-Consistency loop
E
U
DMFT
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Case study in f electrons, Mott transition in
the actinide series. B. Johanssen 1974 Smith and
Kmetko Phase Diagram 1984.
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Total energy vs Volume (Savrasov Kotliar and
Abrahams Nature 410, 793 (2001))
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Small amounts of Ga stabilize the d phase
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Problems with density functional treatements of d
Pu

• DFT in the LDA or GGA is a well established tool
  for the calculation of ground state properties.
• Many studies (APW Freeman, Koelling 1972, ASA and
  FP-LMTO, Soderlind et. al 1990, Kollar et.al
  1997, Boettger et.al 1998, Wills et.al. 1999)
  show
• an equilibrium volume of the d phase Is 35
  lower than experiment
• This is the largest discrepancy ever known in DFT
  based calculations.
• LSDA predicts magnetic long range order which is
  not observed experimentally (Solovyev et.al.)
• If one treats the f electrons as part of the core
  LDA overestimates the volume by 30
• Weak correlation picture for alpha phase.

43
Pu DMFT total energy vs Volume (Savrasov Kotliar
and Abrahams Nature 410, 793 (2001)
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Lda vs Exp Spectra (Joyce et.al.)
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Pu Spectra DMFT(Savrasov) EXP (Joyce , Arko et.al)
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Case study Fe and Ni

• Archetypical itinerant ferromagnets
• LSDA predicts correct low T moment
• Band picture holds at low T
• Main puzzle at high temperatures c has a Curie
  Weiss law with a moment much larger than the
  ordered moment.
• Magnetic anisotropy

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Iron and Nickel crossover to a real space
picture at high T (Lichtenstein, Katsnelson and
Kotliar Phys Rev. Lett 87, 67205 , 2001)
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Iron and Nickelmagnetic properties
(Lichtenstein, Katsenelson,GK PRL 01)
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Ni and Fe theory vs exp

• m/ mB ordered moment
• Fe 2.5 ( theory) 2.2(expt)
• Ni .6 (theory) .6(expt)
• meff / mB high T moment
• Fe 3.1 (theory) 3.12 (expt)
• Ni 1.5 (theory) 1.62 (expt)
• Curie Temperature Tc
• Fe 1900 ( theory) 1043(expt)
• Ni 700 (theory) 631 (expt)

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Fe and Ni

• Consistent picture of Fe (more localized) and Ni
• (more correlated)
• Satellite in minority band at 6 ev, 30
  reduction of bandwidth, exchange splitting
  reduction .3 ev
• Spin wave stiffness controls the effects of
  spatial flucuations, it is about twice as large
  in Ni and in Fe
• Mean field calculations using measured exchange
  constants(Kudrnovski Drachl PRB 2001) right Tc
  for Ni but overestimates Fe , RPA corrections
  reduce Tc of Ni by 10 and Tc of Fe by 50.

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Photoemission Spectra and Spin Autocorrelation
Fe (U2, J.9ev,T/Tc.8) (Lichtenstein,
Katsenelson,Kotliar Phys Rev. Lett 87, 67205 ,
2001)
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Photoemission and T/Tc.8 Spin Autocorrelation
Ni (U3, J.9 ev)
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Summary

• Introduction to strongly correlated electrons
• Dynamical Mean Field Theory
• Model Hamiltonian Studies. Universal aspects
  insights from DMFT
• System specific studies LDADMFT
• Outlook

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Outlook

• The Strong Correlation ProblemHow to deal with
  a multiplicity of competing low temperature
  phases and infrared trajectories which diverge in
  the IR
• Strategy advancing our understanding scale by
  scale
• Generalized cluster methods to capture longer
  range magnetic correlations
• New structures in k space. Cellular DMFT

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Challenges

• Short Range Magnetic Correlations without
  magnetic order.
• Single Site DMFT does not capture these effects

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Outlook

• Extensions to take into account longer range
  correlations and interactions Cellular DMFT G.
  Kotliar S. Savrasov G. Palsson and G. Biroli
  Phys. Rev. Lett. 87, 186401, 2001
• Mott transition magnetic correlations and
  momentum space differentiation. RVB, multipatch
  models of transport A. Perali M. Sindel and G.
  Kotliar Eur. Phys. J. B 24, 487 (2001).
• Exploration of materials.

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Acknowledgements
Collaborators V. Anisimov, R. Chitra, V.
Dobrosavlevic, D. Fisher, A. Georges, H.
Kajueter, W.Krauth, E. Lange, G. Moeller, Y.
Motome, G. Palsson, M. Rozenberg, S.
Savrasov, Q. Si, V. Udovenko, X.Y. Zhang
Support National Science Foundation. Work on Fe
and Ni Office of Naval Research Work on Pu
Departament of Energy and LANL.
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LDADMFT functional
F Sum of local 2PI graphs with local U matrix and
local G
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Comments on LDADMFT

• Static limit of the LDADMFT functional , with F
  FHF reduces to LDAU
• Removes inconsistencies and shortcomings of this
  approach. DMFT retain correlations effects in
  the absence of orbital ordering.
• Only in the orbitally ordered Hartree Fock limit,
  the Greens function of the heavy electrons is
  fully coherent
• Gives the local spectra and the total energy
  simultaneously, treating QP and H bands on the
  same footing.

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Anomalous Resistivities DopedHubbard Model (QMC)
Prushke and Jarrell 1993
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Anomalous ResistivitiesDoped Hubbard ModelG.
Palsson 1998
NCA
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DMFT Methods of Solution
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LDA functional
Conjugate field, VKS(r)
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Minimize LDA functional
Kohn Sham eigenvalues, auxiliary quantities.
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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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LDADMFT Self-Consistency loop
E
U
DMFT
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DMFT Impurity cavity construction A. Georges,
G. Kotliar, PRB, (1992)
Weiss field
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Spectral Evolution at T0 half filling full
frustration
X.Zhang M. Rozenberg G. Kotliar (PRL 1993)
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Parallel development Fujimori et.al
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Landau Functional
G. Kotliar EPJB (1999)
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Ni and Fe theory vs exp

• m( T.9 Tc)/ mB ordered moment
• Fe 1.5 ( theory) 1.55 (expt)
• Ni .3 (theory) .35 (expt)
• meff / mB high T moment
• Fe 3.1 (theory) 3.12 (expt)
• Ni 1.5 (theory) 1.62 (expt)
• Curie Temperature Tc
• Fe 1900 ( theory) 1043(expt)
• Ni 700 (theory) 631 (expt)

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Problems with LDA

• DFT in the LDA or GGA is a well established tool
  for the calculation of ground state properties.
• Many studies (Freeman, Koelling 1972)APW methods
• ASA and FP-LMTO Soderlind et. Al 1990, Kollar
  et.al 1997, Boettger et.al 1998, Wills et.al.
  1999) give
• an equilibrium volume of the d phase Is 35
  lower than experiment
• This is the largest discrepancy ever known in DFT
  based calculations.

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Spectral Density Functional

• The exact functional can be built in perturbation
  theory in the interaction (well defined
  diagrammatic rules )The functional can also be
  constructed expanding around the the atomic
  limit. No explicit expression exists.
• DFT is useful because good approximations to the
  exact density functional GDFTr(r) exist, e.g.
  LDA, GGA
• A useful approximation to the exact functional
  can be constructed, the DMFT LDA functional.
  Savrasov Kotliar and Abrahams Nature 410, 793
  (2001))

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LDA functional
Conjugate field, VKS(r)