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Title: Strongly Correlated Electron Systems a Dynamical Mean Field Perspective


1
Strongly Correlated Electron Systems a
Dynamical Mean Field Perspective
  • G. Kotliar
  • Physics Department and Center for Materials
    Theory
  • Rutgers

5th International Conference on Inelastic X-Ray
Scattering. Argonne National Labs Chicago
September 20 2004
2
Outline
  • Introduction to the concepts of the dynamical
    mean field method.
  • Application The temperature driven Mott
    transition. Theoretical predictions, and
    experiments. IXS ?
  • Application elemental Pu. DMFT predictions, and
    a key IXS experiment.

3
Electronic states in weakly and strongly
correlated materials
  • Simple metals, semiconductors. Fermi Liquid
    Description Quasiparticles and quasiholes, (and
    their bound states ). Computational tool
    Density functional theory perturbation theory
    in W, GW method.
  • Correlated electrons. Atomic states. Hubbard
    bands. Narrow bands. Many anomalies.
  • Need tool that treats Hubbard bands, and
    quasiparticle bands, real and momentum space on
    the same footing. DMFT!

4
Strongly Correlated Electron Systems Display
remarkable phenomena, that cannot be understood
within the standard model of solids.
Resistivities that rise without sign of
saturation beyond the Mott limit, (e.g. H.
Takagis work on Vanadates), temperature
dependence of the integrated optical weight up
to high frequency (e.g. Vandermarels work on
Silicides).
THE WHY
Correlated electrons do big things, large
volume collapses, colossal magnetoresitance, high
temperature superconductivity . Properties are
very sensitive to structure chemistry and
stoichiometry, and control parameters large non
linear susceptibilites,etc.
5
C. Urano et. al. PRL 85, 1052 (2000)
6
THE HOW
Need non perturbative tool.
How to think about their electronic states ? How
to compute their properties ? Mapping onto
connecting their properties, a simpler reference
system. A self consistent impurity model living
on SITES, LINKS and PLAQUETTES......
  • DYNAMICAL MEAN FIELD THEORY.
  • "Optimal Gaussian Medium " " Local Quantum
    Degrees of Freedom " "their interaction "
  • is a good reference frame for understanding,
    and predicting physical properties
  • of correlated materials. Focus on local
    quantities, construct functionals of those
    quantities, similarities with DFT.

7
Two paths for ab-initio calculation of electronic
structure of strongly correlated materials
Crystal structure Atomic positions
Model Hamiltonian
Correlation Functions Total Energies etc.
DMFT ideas can be used in both cases.
8

Dynamical Mean Field Theory (DMFT) Cavity
Construction A. Georges and G. Kotliar PRB 45,
6479 (1992). C.DMFT G. Kotliar et. al. Phys. Rev.
Lett 87,186401 (2001).
9
One dimensional Hubbard model 2 site (LINK)
CDMFT compare with Bethe Anzats, V. Kancharla
C. Bolech and GK PRB 67, 075110
(2003)M.CaponeM.Civelli V Kancharla
C.Castellani and GK P. R B 69,195105 (2004)
A rapidly convergent algorithm ?
U/t4.
10
Functional formulation. Chitra and Kotliar
(2001), Savrasov and Kotliarcond- matt0308053
(2003).
IrgtR, rgt
Double loop in Gloc and Wloc
11
Impurity model representability of spectral
density functional.
12
LDADMFT V. Anisimov, A. Poteryaev, M. Korotin,
A. Anokhin and G. Kotliar, J. Phys. Cond. Mat.
35, 7359 (1997). A Lichtenstein and M. Katsnelson
PRB 57, 6884 (1988).
  • The light, SP (or SPD) electrons are extended,
    well described by LDA .The heavy, D (or F)
    electrons are localized treat by DMFT.
  • LDA Kohn Sham Hamiltonian already contains an
    average interaction of the heavy electrons,
    subtract this out by shifting the heavy level
    (double counting term)
  • Kinetic energy is provided by the Kohn Sham
    Hamiltonian (sometimes after downfolding ). The U
    matrix can be estimated from first principles of
    viewed as parameters. Solve resulting model
    using DMFT.

13
What did we learn ? Schematic DMFT phase diagram
and DOS of a partially frustrated integer filled
Hubbard model and pressure driven Mott transition.
M. Rozenberg G. Kotliar H. Kajueter G Thomas D.
Rapkine J Honig and P Metcalf Phys. Rev. Lett.
75, 105 (1995)
14
T
15
How do we know there is some truth in this
picture ? Qualitative Predictions Verified
  • Two different features in spectra. Quasiparticles
    bands and Hubbard bands.
  • Transfer of spectral weight which is non local in
    frequency. Optics and Photoemission.
  • Two crossovers, associated with gap closure and
    loss of coherence. Transport.
  • Mott transition endpoint, is Ising like, couples
    to all electronic properties.
  • Recently numerical approaches in two dimensions
    found the first order line(M. Imada), C-DMFT 4
    site studies (Parcollet et. al.).

16
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)
17
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)

18
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19
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).
M. Rozenberg G. Kotliar H. Kajueter G Tahomas D.
Rapkikne J Honig and P Metcalf Phys. Rev. Lett.
75, 105 (1995)
20
Anomalous Resistivity and Mott transition Ni
Se2-x Sx
Crossover from Fermi liquid to bad metal to
semiconductor to paramagnetic insulator.
21
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22
Anomalous transfer of optical spectral weight,
NiSeS. Miyasaka and Takagi 2000
23
Ising critical endpoint found! In V2O3 P.
Limelette et.al. (Science 2003)
24
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25
Ising critical endpoint found! In V2O3 P.
Limelette et.al. (Science 2003)
26
Why does it work Energy Landscape of a
Correlated Material and a top to bottom approach
to correlated materials.
Single site DMFT. High temperature universality
vs low temperature sensitivity to detail for
materials near a temperature-pressure driven
Mott transition
Energy
T
Configurational Coordinate in the space of
Hamiltonians
27
What did we gain?
  • Conceptual understanding of how the electronic
    structure evolves when the electron goes from
    localized to itinerant.
  • Uc1 Uc2, transfer of spectral weight, .
  • A general methodology which was extended to
    clusters (non trivial!) and integrated into an
    electronic structure method, which allows us to
    incorporate structure and chemistry. Both are
    needed away from the high temperature universal
    region.

28
  • Mott transition across the 5fs, a very
    interesting playground for studying correlated
    electron phenomena.
  • DMFT ideas have been extended into a framework
    capable of making first principles first
    principles studies of correlated materials. Pu
    Phonons. Combining theory and experiments to
    separate the contributions of different energy
    scales, and length scales to the bonding
  • In single site DMFT , superconductivity is an
    unavoidable consequence when we try to go move
    from a metallic state to a Mott insulator
    where the atoms have a closed shell (no entropy).
    Realization in Am under pressure ?

29
DMFT Phonons in fcc d-Pu connect bonding to
energy and length scales.
( Dai, Savrasov, Kotliar,Ledbetter, Migliori,
Abrahams, Science, 9 May 2003)
(experiments from Wong et.al, Science, 22 August
2003)
30
Where do we go now ?
  • One can study a large number of experimentally
    relevant problems within the single site
    framework.
  • Continue the methodological development, we need
    tools!
  • Solve the CDMFT Mott transition problem on the
    plaquette problem, hard, but it is a significant
    improvement, the early mean field theories while
    keeping its physical appeal.
  • Study material trends, make contact with
    phenomenological approaches, doped semiconductors
    (Bhatt and Sachdev), heavy fermions ,
    115s(Nakatsuji, Pines and Fisk )

31
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)
32
Pu in the periodic table
actinides
33

Mott transition in the actinide series (Smith
Kmetko phase diagram)
34
Electronic Physics of Pu
35
Small amounts of Ga stabilize the d phase (A.
Lawson LANL)
36
Elastic Deformations
Uniform compressionDp-B DV/V
Volume conserving deformations
F/Ac44 Dx/L
F/Ac Dx/L
In most cubic materials the shear does not depend
strongly on crystal orientation,fcc Al,
c44/c1.2, in Pu C44/C 6 largest shear
anisotropy of any element.
37
Anomalous Resistivity
Maximum metallic resistivity
38
Specific heat and susceptibility.
39
Delta phase of Plutonium Problems with LDA
  • Many studies and implementations.(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).all give
    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 (Solovyev
    et.al.) Experimentally d Pu is not magnetic.
  • If one treats the f electrons as part of the core
    LDA overestimates the volume by 30

40
DFT Studies of a Pu
  • DFT in GGA predicts correctly the volume of the a
    phase of Pu, when full potential LMTO (Soderlind
    Eriksson and Wills) is used. This is usually
    taken as an indication that a Pu is a weakly
    correlated system. Experimentally, there are
    clear signs of electron correlation in a Pu .
  • .

41
Pu DMFT total energy vs Volume (Savrasov
Kotliar and Abrahams 2001)
42
Total Energy as a function of volume for Pu W
(ev) vs (a.u. 27.2 ev)
(Savrasov, Kotliar, Abrahams, Nature ( 2001) Non
magnetic correlated state of fcc Pu.
Zein Savrasov and Kotliar (2004)
43
Lda vs Exp Spectra
44
Alpha and delta Pu
45
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.

46
Phonon freq (THz) vs q in delta Pu X. Dai et. al.
Science vol 300, 953, 2003
47
Inelastic X Ray. Phonon energy 10 mev, photon
energy 10 Kev.
E Ei - Ef Q ki - kf
48
Expt. Wong et. al.
49
Expts Wong et. al.
50
DMFT Phonons in fcc d-Pu
( Dai, Savrasov, Kotliar,Ledbetter, Migliori,
Abrahams, Science, 9 May 2003)
(experiments from Wong et.al, Science, 22 August
2003)
51
Conclusion
  • DMFT. Electronic Structure Method in Development.
    a) quantitative results b) qualitative
    understanding by linking real materials to
    impurity models. Concepts to think about
    correlate materials.
  • System specific. Many materials to be studied,
    realistic matrix elements for each spectroscopy.
    Optics. IXS.
  • Interplay of theory and experiment. DMFT can
    enhance joint theoretical- experimental advances
    in the field of correlated electron materials.

52
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53
Epsilon Plutonium.
54
Phonon entropy drives the epsilon delta phase
transition
  • Epsilon is slightly more delocalized than delta,
    has SMALLER volume and lies at HIGHER energy than
    delta at T0. But it has a much larger phonon
    entropy than delta.
  • At the phase transition the volume shrinks but
    the phonon entropy increases.
  • Estimates of the phase transition following
    Drumont and G. Ackland et. al. PRB.65, 184104
    (2002) (and neglecting electronic entropy).
    TC 600 K.

55
Transverse Phonon along (0,1,1) in epsilon Pu in
self consistent Born approximation.
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