Dynamical Mean Field Approach to Strongly Correlated Electrons

1
Dynamical Mean Field Approach to Strongly
Correlated Electrons

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

Field Theory and Statistical Mechanics Rome
10-15 June (2002)
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Outline

• Correlated Electrons and the Mott transition
  problem.
• Dynamical Mean Field Theory. Cavity construction.
  Effective action construction.
• G Jona-Lasinio, Nuovo Cimento 34, (1964), De
  Dominicis and Martin, Fukuda
• Model Hamiltonian Studies of the Mott transition
  in frustrated systems. Universal aspects.
• Application to itinerant ferromagnets Fe,Ni.
• Outlook

3
The electron in a solid wave picture
Standard model of solid (Sommerfeld) (Bloch
)Periodic potential, waves form bands , k in
Brillouin zone . (Landau) Interactions
renormalize away. Justification perturbative RG
(Benfatto Gallavotti)

Consequences Maximum metallic resistivity 200
mohm cm
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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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Evolution of the spectra from localized to
itinerant

• Low densities. Electron as particle bound to
  atom.
• High densities. Electrons are waves spread thru
  the crystal.
• Mott transition problem evolution between the
  two limits, in the open shell case.
• Non perturbative problem.
• Key to understanding many interesting solids.

6
Mott transition in V2O3 under pressure or
chemical substitution on V-site
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Failure of the Standard Model NiSe2-xSx
Miyasaka and Takagi (2000)
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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
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Missing in this limit

• Short Range Magnetic Correlations without
  magnetic order. Long wavelength modes.
• Trust more in frustrated situations and at high
  temperatures.

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DMFT cavity construction A. Georges G. Kotliar
92
Weiss field
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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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Solving the DMFT equations

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

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Different Extensions

• Take larger clusters in the cavity construction,
  e.g. cellular DMFT.Kotliar Savrasov Palsson and
  Biroli, DCAJarrell and Krishnamurthy
• Take into account approximately the
  renormalization of the quartic coupling, e.g.
  extended DMFT. Sachdev and Ye, Kajueter Kotliar,
  Si and Smith

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

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

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C-DMFT functional formulation. Construct a
functional of the restriction of the Greens
function to the cluster and its supercell
translations.
Sigma and G are non zero on the selected cluster
and its supercell translations and are non zero
otherwise. Lattice quantities are inferred or
projected out from the local quantities.
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C-DMFT test in one dimension. (Bolech, Kancharla
and Kotliar 2002)
Gap vs U, Exact solution Lieb and Wu, Ovshinikov
Nc2 CDMFT vs Nc1
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Results Schematic DMFT phase diagram Hubbard
model (partial frustration)
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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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Kuwamoto Honig and Appell PRB (1980)M. Rozenberg
G. Kotliar H. Kajueter G Thomas D. Rapkikne J
Honig and P Metcalf Phys. Rev. Lett. 75, 105
(1995)
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Qualitative phase diagram in the U, T , m
plane,full frustration ( GK Murthy and
Rozenberg 2002)

• Shaded regions the DMFT equations have a
  metallic-like and an insulating-like solution).
  

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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). Foreshadowed by Castellani Di Castro
Feinberg Ranninger (1979).
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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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ARPES measurements on NiS2-xSexMatsuura et. al
Phys. Rev B 58 (1998) 3690. Doniach and Watanabe
Phys. Rev. B 57, 3829 (1998)
.
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Anomalous Spectral Weight Transfer Optics
Below energy
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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
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Anomalous Resistivity and Mott transition 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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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.
• Anisimov Poteryaev Korotin Anhokin and Kotliar J.
  Phys. Cond. Mat. 35, 7359 (1997)
• Savrasov Kotliar and Abrahams Nature 410, 793
  (2001))

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Spectral Density Functional effective action
construction ( Chitra and GK PRB 2001).

• DFT, exact free energy as a functional of an
  external potential. Legendre transform to obtain
  a functional of the density GDFTr(r).
  Hohenberg and Kohn, Lieb, Fukuda
• 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.

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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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LDADMFT Self-Consistency loop
E
EdcU
DMFT
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Case study Fe and Ni

• Band picture holds at low T. LSDA predicts
  correct low T moment
• At high temperatures c has a Curie Weiss law with
  a (fluctuating) moment larger than the T0
  ordered moment.
• Localization delocalization crossover as a
  function of T.

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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, Katsnelson,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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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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Fe and Ni

• Consistent picture of Fe (more localized) and Ni
• (more itinerant but 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, twice as large in Ni and in
  Fe
• Cluster methods.

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Outlook

• Many open problems!
• Strategy advancing our understanding scale by
  scale.
• New local physics in plaquettes.
• Cluster methods to capture longer range magnetic
  correlations. New structures in k space. Cellular
  DMFT
• Many applications to real materials.

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LDADMFT functional
F Sum of local 2PI graphs with local U matrix and
local G
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Mott transition in layered organic conductors
S Lefebvre et al. cond-mat/0004455, Phys. Rev.
Lett. 85, 5420 (2000)