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Gabriel Kotliar

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Title: Gabriel Kotliar


1
  • Gabriel Kotliar
  • Physics Department and
  • Center for Materials Theory
  • Rutgers University

2
The Strong Correlation Problem
  • Two limiting cases of the electronic structure of
    solids are understoodthe high density limit and
    the limit of well separated atoms.
  • High densities, the is electron be a wave, use
    band theory, k-space
  • One particle excitations quasi-particle,quasi-hol
    e bands, collective modes.
  • Density Functional Theory with approximations
    suggested by the Kohn Sham formulation, (LDA GGA)
    is a successful computational tool for the total
    energy, and a good starting point for
    perturbative calculation of spectra, GW.

3
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.) Magnetic and Orbital Ordering at low
T Quantitative calculations of Hubbard bands and
exchange constants, LDA U, Hartree Fock. Atomic
Physics.
4
Localization vs Delocalization Strong Correlation
Problem
  • A large number of compounds with electrons which
    are not close to the well understood limits
    (localized or itinerant).
  • These systems display anomalous behavior
    (departure from the standard model of solids).
  • Neither LDA or LDAU or Hartree Fock works well
  • Dynamical Mean Field Theory Simplest approach to
    the electronic structure, which interpolates
    correctly between atoms and bands

5
Outline
  • Motivation plutonium puzzles.
  • Review of Dynamical Mean Field Theory an
    Extension to realistic systems . DMFT and DFT.
  • A case study of system specific properties f
    .electrons DMFT Results for d Pu.
  • A case study of system specific properties d
    electrons in Fe and Ni.

6
Collborators and References
  • Reviews of DMFT A. Georges G. Kotliar W krauth
    and M . Rozenberg Rev Mod Pnys 68, 13 (1996).
    DMFT and LDA R. Chitra and G. Kotliar Phys. Rev.
    B 62,, 12715 (2000).
  • S. Savrasov and G. Kotliar cond-mat cond-mat
    0106308.
  • DMFT study of Plutonium. S. Savrasov, G. Kotiar
    and E. Abrahams, Nature 410, 793 (2001). S.
    Savrasov and G. Kotliar
  • DMFT study of Iron and Nickel. S. Lichtenstein M
    Katsenelson and G. Kotliar Phys. Rev. Lett 87,
    (2001).

7
Outline
  • Motivation plutonium puzzles.
  • Review of Dynamical Mean Field Theory an
    Extension to realistic systems . DMFT and DFT.
  • A case study of system specific properties f
    .electrons DMFT Results for d Pu.
  • A case study of system specific properties d
    electrons in Fe and Ni.

8

Case study in f electrons, Mott transition in
the actinide series. B. Johanssen 1974 Smith and
Kmetko Phase Diagram 1984.
9
Pu Complex Phase Diagram (J. Smith LANL)
10
Small amounts of Ga stabilize the d phase
11
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.

12
Problems with LDA
  • 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
  • LDA predicts correctly the volume of the a phase
    of Pu, using full potential LMTO (Soderlind and
    Wills). This is usually taken as an indication
    that a Pu is a weakly correlated system.

13
Other Methods
  • LDA U (Savrasov and Kotliar Phys. Rev. Lett.
    84, 3670, 2000, Bouchet et. al 2000) predicts
    correct volume of Pu with the constrained LDA
    estimate of U4 ev. However, it predicts spurious
    magnetic long range order and a spectra which is
    very different from experiments.
  • Requires U0 to treat the alpha phase, which has
    many physical properties in common with the delta
    phase.
  • Similar problems with the constrained (4 of the
    5f electrons are treated as core ) LDA approach
    of Erikson and Wills.

14
Conventional viewpoint
  • Alpha Pu is a simple metal, it can be described
    with LDA correction. In contrast delta Pu is
    strongly correlated.
  • Constrained LDA approach (Erickson, Wills,
    Balatzki, Becker). In Alpha Pu, all the 5f
    electrons are treated as band like, while in
    Delta Pu, 4 5f electrons are band-like while one
    5f electron is deloclized.
  • Same situation in LDA U (Savrasov andGK
    Bouchet et. al. ) .Delta Pu has U4,Alpha Pu
    has U 0.

15
Problems with the conventional viewpoint of Pu
  • The specific heat of delta Pu, is only twice as
    big as that of alpha Pu.
  • The susceptibility of alpha Pu is in fact larger
    than that of delta Pu.
  • The resistivity of alpha Pu is comparable to that
    of delta Pu.
  • Only the structural and elastic properties are
    completely different.

16
MAGNETIC SUSCEPTIBILITY
17
Pu Specific Heat
18
Anomalous ResistivityJ. Smith LANL
19
Outline
  • Motivation plutonium puzzles.
  • Review of Dynamical Mean Field Theory an
    Extension to realistic systems . DMFT and DFT.
  • A case study of system specific properties f
    .electrons DMFT Results for d Pu.
  • A case study of system specific properties d
    electrons in Fe and Ni.

20
DMFT Impurity cavity construction A. Georges,
G. Kotliar, PRB, (1992)
Weiss field
21
Single site DMFT, functional formulation
  • Express in terms of Weiss field (semicircularDOS)
  • The Mott transition as bifurcation point in
    functionals oGG or FD, (G. Kotliar EPJB 99)

Local self energy (Muller Hartman 89)
22
Solving the DMFT equations
  • Wide variety of computational tools (QMC,
    NRG,ED.)
  • Analytical Methods

23
DMFT
  • Construction is easily extended to states with
    broken translational spin and orbital order.
  • Large number of techniques for solving DMFT
    equations for a review see
  • A. Georges, G. Kotliar, W. Krauth and M.
    Rozenberg Rev. Mod. Phys. 68,13 (1996)

24
Schematic DMFT phase diagram one band Hubbard
(half filling, semicircular DOS, role of partial
frustration) Rozenberg et.al PRL (1995)
25
Evolution of the Spectral Function with
Temperature
Anomalous transfer of spectral weight connected
to the proximity to an Ising Mott endpoint
(Kotliar et.al.PRL 84, 5180 (2000))
26
Localization Delocalization
  • The Mott transition/crossover is driven by
    transfer of spectral weight from low to high
    energy as we approach the localized phase
  • Control parameters doping, temperature,pressure
  • Intermediate U region is NOT perturbatively
    accessible. DMFT a new starting point to access
    this regime.

27
Qualitative phase diagram in the U, T , m plane
(two band Kotliar and Rozenberg (2001))
  • Coexistence regions between localized and
    delocalized spectral functions.

28
QMC calculationof n vs m (Murthy Rozenberg and
Kotliar 2001, 2 band, U3.0)
k diverges at generic Mott endpoints
29
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, substract this out by
    shifting the heavy level (double counting term)
  • The U matrix can be estimated from first
    principles or viewed as parameters

30
Spectral Density Functional effective action
construction (Fukuda, Valiev and Fernando ,
Chitra and GK).
  • 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 sources for r(r) and G
    and performing a Legendre transformation,
    Gr(r),G(R,R)(iw)

31
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.

32
LDADMFT functnl
F Sum of local 2PI graphs with local U matrix and
local G
33
LDADMFT Self-Consistency loop
E
U
DMFT
34
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.

35
Outline
  • Motivation plutonium puzzles.
  • Review of Dynamical Mean Field Theory an
    Extension to realistic systems . DMFT and DFT.
  • A case study of system specific properties f
    .electrons DMFT Results for d Pu.
  • A case study of system specific properties d
    electrons in Fe and Ni.

36
Pu DMFT total energy vs Volume Savrasov Kotliar
Abrahams to appear in Nature
37
Lda vs Exp Spectra
38
Pu Spectra DMFT(Savrasov) EXP (Arko et.al)
39
PU ALPHA AND DELTA
40
Dynamical Mean Field View of Pu(Savrasov Kotliar
and Abrahams, Nature 2001)
  • Delta and Alpha Pu are both strongly correlated,
    the DMFT mean field free energy has a double
    well structure, for the same value of U. One
    where the f electron is a bit more localized
    (delta) than in the other (alpha).
  • Is the natural consequence of the model
    Hamiltonian phase diagram once the structure is
    about to vary.
  • This result resolves one of the basic paradoxes
    in the physics of Pu.

41
Minimum of the melting point
  • Divergence of the compressibility at the Mott
    transition endpoint.
  • Rapid variation of the density of the solid as a
    function of pressure, in the localization
    delocalization crossover region.
  • Slow variation of the volume as a function of
    pressure in the liquid phase

42
Minimum in melting curve and divergence of the
compressibility at the Mott endpoint
43
Cerium melting T vs p
44
Pu Anomalous thermal expansion (J. Smith LANL)
45
Double well structure and d Pu
  • Qualitative explanation
    of negative thermal expansion
  • Sensitivity to impurities which easily raise the
    energy of the a -like minimum.

46
Double well structure and d Pu
  • negative thermal expansion
  • Sensitivity to impurities which easily raise the
    energy of the a -like minimum.

47
Future directions
  • Including short range correlations. Less local
    physics, C-DMFT.
  • Life without U, including the effects of long
    range Coulomb interactions, E-DMFT and GW.
  • Applications are just beginning, many surprises
    ahead

48
Outline
  • Motivation plutonium puzzles.
  • Review of Dynamical Mean Field Theory an
    Extension to realistic systems . DMFT and DFT.
  • A case study of system specific properties f
    .electrons DMFT Results for d Pu.
  • A case study with d electrons in Fe and Ni.

49
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

50
Iron and Nickel crossover to a real space
picture at high T (Lichtenstein, Katsnelson and
GK)
51
Photoemission Spectra and Spin Autocorrelation
Fe (U2, J.9ev,T/Tc.8) (Lichtenstein,
Katsenelson,GK prl 2001)
52
Photoemission and T/Tc.8 Spin Autocorrelation
Ni (U3, J.9 ev)
53
Iron and Nickelmagnetic properties
(Lichtenstein, Katsenelson,GK PRL 01)
54
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)

55
Fe and Ni
  • 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.

56
However not everything in low T phase is OK as
far as LDA goes..
  • Magnetic anisotropy puzzle. LDA predicts the
    incorrect easy axis(100) for Nickel .(instead of
    the correct one (111)
  • LDA Fermi surface has features which are not seen
    in DeHaas Van Alphen ( Lonzarich)
  • Use LDA U to tackle these refined issues, (
    compare parameters with DMFT results )
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