THE STATE UNIVERSITY OF NEW JERSEY - PowerPoint PPT Presentation

1 / 96
About This Presentation
Title:

THE STATE UNIVERSITY OF NEW JERSEY

Description:

(Ingersent and Schiller, Georges and Kotliar, cluster expansion in real space, ... [ A. Georges and G. Kotliar, A. Schiller PRL75, 113 (1995) ... – PowerPoint PPT presentation

Number of Views:58
Avg rating:3.0/5.0
Slides: 97
Provided by: gabriel6
Category:

less

Transcript and Presenter's Notes

Title: THE STATE UNIVERSITY OF NEW JERSEY


1
Outline
  • Model Hamiltonians and qualitative
    considerations in the physics of materials. Or
    what do we want to know? An example from the
    physics of the Mott transition.
  • Merging band structure methods with many body
    theory, where to improve? A) basis set? B)
    parameter estimates of your model Hamiltonian C)
    DMFT impurity solver? D) Improvements of DMFT ?
    An intro to Cellular DMFT G. Kotliar S.
    Savrasov G. Palsson and G. Biroli PRL87, 186401
    2001

2
(No Transcript)
3
Extensions of DMFT
  • Spin Orbital Ordered States
  • Longer range interactions Coulomb, interactions,
    Random Exchange (Sachdev and Ye, Parcollet and
    Georges, Kajueter and Kotliar, Si and Smith,
    Chitra and Kotliar,)
  • Short range magnetic correlations. Cluster
    Schemes. (Ingersent and Schiller, Georges and
    Kotliar, cluster expansion in real space,
    momentum space cluster DCA Jarrell et.al., C-DMFT
    Kotliar et. al ).

4
Extensions of DMFT
  • Formulation as an electronic structure method
    (Chitra and Kotliar)
  • Density vs Local Spectral Function
  • Extensions to treat strong spatial
    inhomogeneities. Anderson Localization
    (Dobrosavlevic and Kotliar),Surfaces
    (Nolting),Stripes (Fleck Lichtenstein and Oles)
  • Practical Implementation (Anisimov and Kotliar,
    Savrasov, Katsenelson and Lichtenstein)

5
Cuprates Photoemission

Transfer of Spectral Weight with a) temperature
and b) doping
6
Challenges
  • The photoemission in cuprates has a strong
    momentum dependence
  • Strong Magnetic Correlations (no orbital
    degeneracy)
  • Single Site DMFT does not capture these effects

7
Cuprates Photoemission

Transfer of Spectral Weight with a) temperature
and b) doping
8
(No Transcript)
9

Mott transition in the actinide series. B.
Johanssen 1974 Smith and Kmetko Phase Diagram
1984.
10
Schematic DMFT phase diagram one band Hubbard
model (half filling, semicircular DOS, partial
frustration) Rozenberg et.al PRL (1995)
11
Robustness of the finite T results
  • Underlying Landau Free energy which is
    responsible of all the qualitative features of
    the phase diagram. Of the frustrated Hubbard
    model in large d G. Kotliar EPJB 99
  • Around the finite temperature Mott endpoint, the
    Free energy has a simple Ising like form as in a
    liquid gas transition R. Chitra, G. Kotliar
    E.Lange M. Rozenberg
  • Changing the model (DOS, degeneracy, etc) just
    changes the coefficients of the Landau theory.

12
Robustness of the finite T results and Functional
Approach
  • Different impurity solvers, different values of
    the Landau coefficients, as long as they preserve
    the essential (non) analytic properties of the
    free energy functional.
  • The functional approach can be generalized to
    combine DFT and DMFT R. Chitra G. Kotliar , S.
    Savrasov and G. Kotliar
  • Justification for applying simple models to some
    aspects of the crossover in Ni(SeS)2And V2O3.

13
Qualitative phase diagram in the U, T , m plane
(two band Kotliar and Rozenberg (2001) cond-matt
0110625)
  • Coexistence regions between localized and
    delocalized spectral functions.

14
QMC calculationof n vs m (Murthy Rozenberg and
Kotliar 2001, 2 band, U3.0, cond-matt 0110625)
k diverges at generic Mott endpoints
15
Compressibilty divergence One band case
(Kotliar Murthy and Rozenberg 2001, cond-matt
0110625)
16
Minimum in melting curve and divergence of the
compressibility at the Mott endpoint
17
Minimum in melting curve and divergence of the
compressibility at the Mott endpoint
18
A (non comprehensive )list of extensions of DMFT
  • Two impurity method. A. Georges and G. Kotliar,
    A. Schiller PRL75, 113 (1995)
  • M. Jarrell Dynamical Cluster Approximation Phys.
    Rev. B 7475 1998
  • Continuous version periodic cluster M.
    Katsenelson and A. Lichtenstein PRB 62, 9283
    (2000).
  • Extended DMFT H. Kajueter and G. Kotliar
  • Rutgers Ph.D thesis 2001, Q. Si and J L Smith PRL
    77 (1996)3391 Coulomb interactions R . Chitra
  • Cellular DMFT PRL87, 186401 2001

19
DMFT cavity construction
Weiss field
20
Elements of the Dynamical Mean Field Construction
and Cellular DMFT, G. Kotliar S. Savrasov G.
Palsson and G. Biroli PRL 2001
  • Definition of the local degrees of freedom
  • Expression of the Weiss field in terms of the
    local variables (I.e. the self consistency
    condition)
  • Expression of the lattice self energy in terms of
    the cluster self energy.

21
Cellular DMFT Basis selection
22
Lattice action
23
Elimination of the medium variables
24
Determination of the effective medium.
25
Connection between cluster and lattice self
energy.
The estimation of the lattice self energy in
terms of the cluster energy has to be done using
additional information Ex. Translation invariance
  • C-DMFT is manifestly causal causal impurity
    solvers result in causal self energies and Green
    functions (GK S. Savrasov G. Palsson and G.
    Biroli PRL 2001)
  • In simple cases C-DMFT converges faster than
    other causal cluster schemes.

26
Improved estimators
  • Improved estimators for the lattice self energy
    are available (Biroli and Kotliar)

27
Real Space Formulation of the DCA approximation
of Jarrell et.al.
28
Affleck Marston model.
29
Convergence test in the Affleck Marston
30
Convergence of the self energy
31
Recent application to high Tc
  • A. Perali et.al. cond-mat 2001, two patch model,
    phenomenological fit of the functional form of
    the vertex function of C-DMFT to experiments in
    optimally doped and overdoped cuprates
  • Flexibility in the choice of basis seems
    important.

32
Extended DMFT electron phonon
33
Extended DMFT e.ph. Problem
34
E-DMFT classical case, soft spins
35
E-DMFT classical case Ising limit
36
E-DMFT test in the classical caseBethe Lattice,
S. Pankov 2001
37
Advantage and Difficulties of E-DMFT
  • The transition is first order at finite
    temperatures for dlt 4
  • No finite temperature transition for d less than
    2 (like spherical approximation)
  • Improved values of the critical temperature

38
Conclusion
  • For first principles work there are several
    many body tools waiting to be used, once the one
    electron aspects of the problem are clarified.
  • E-DMFT or C-DMFT for Ni, and Fe ?
  • Promising problem Qualitative aspects of the
    Mott transition within C-DMFT ?? Cuprates?

39
Realistic Theories of Correlated Materials
  • ITP, Santa-Barbara
  • July 20 December 20 (2002)
  • O.K. Andesen, A. Georges,
  • G. Kotliar, and A. Lichtenstein
  • http//www.itp.ucsb.edu/activities/future/

40
(No Transcript)
41
Functional Approach
G. Kotliar EPJB (1999)
42
Recent phase diagram of the frustrated Half
filled Hubbard model with semicircular DOS (QMC
Joo and Udovenko PRB2001).
43
Case study IPT half filled Hubbard one band
  • (Uc1)exact 2.1 (Exact diag, Rozenberg,
    Kajueter, Kotliar 1995) , (Uc1)IPT 2.4
  • (Uc2)exact 2.95 (Projective self consistent
    method, Moeller Si Rozenberg Kotliar PRL 1995 )
    (Uc2)IPT 3.3
  • (TMIT ) exact .026_ .004 (QMC Rozenberg Chitra
    and Kotliar PRL 1999), (TMIT )IPT .5
  • (UMIT )exact 2.38 - .03 (QMC Rozenberg Chitra
    and Kotliar PRL 1991), (UMIT )IPT 2.5 For
    realistic studies errors due to other sources
    (for example the value of U, are at least of the
    same order of magnitude).

44
The Mott transition as a bifurcation in effective
action

Zero mode with S0 and p0, couples generically
Divergent compressibility (R. Chitra and
G.Kotliar
45
(No Transcript)
46
Realistic implementation of the self consistency
condition
  • H and S, do not commute
  • Need to do k sum for each frequency
  • DMFT implementation of Lambin Vigneron
    tetrahedron integration (Poteryaev et.al 1987)

47
Solving the impurity
  • Multiorbital situation and several atoms per unit
    cell considerably increase the size of the space
    H (of heavy electrons).
  • QMC scales as N(N-1)/23 N dimension of H
  • Fast interpolation schemes (Slave Boson at low
    frequency, Roth method at high frequency, 1st
    mode coupling correction), match at intermediate
    frequencies. (Savrasov et.al 2001)

48
Good method to study the Mott phenomena
  • Evolution of the electronic structure between
    the atomic limit and the band limit. Basic solid
    state problem. Solved by band theory when the
    atoms have a closed shell. Motts problem Open
    shell situation.
  • The in between regime is ubiquitous central
    them in strongly correlated systems. Some
    unorthodox examples
  • Fe, Ni, Pu .

49
Two Roads for calculations of the electronic
structure of correlated materials
Crystal Structure atomic positions
Model Hamiltonian
Correlation functions Total energies etc.
50
LDA functional
Conjugate field, VKS(r)
51
Minimize LDA functional
52
LDAU functional
53
LDADMFT
  • 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 of viewed as parameters

54
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)

55
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 from the atomic limit, but 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.

56
LDADMFT functional
F Sum of local 2PI graphs with local U matrix and
local G
57
Comments on LDADMFT
  • Static limit of the LDADMFT functional , with F
    FHF reduces to LDAU
  • Removes inconsistencies of this approach,
  • 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.

58
LDADMFTConnection with atomic limit
Weiss field
59
LDADMFT Self-Consistency loop
E
U
DMFT
60
Realistic DMFT loop
61
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, funcional
    formulation for full self consistent
    implementation (2001)

62
Functional Approach
  • The functional approach offers a direct
    connection to the atomic energies. One is free to
    add terms which vanish quadratically at the
    saddle point.
  • Allows us to study states away from the saddle
    points,
  • All the qualitative features of the phase
    diagram, are simple consequences of the non
    analytic nature of the functional.
  • Mott transitions and bifurcations of the
    functional .

63
Functional Approach
G. Kotliar EPJB (1999)
64

Case study in f electrons, Mott transition in
the actinide series
65
Pu Anomalous thermal expansion (J. Smith LANL)
66
Small amounts of Ga stabilize the d phase
67
Delocalization-Localization across the actinide
series
  • f electrons in Th Pr U Np are itinerant . From
    Am on they are localized. Pu is at the
    boundary.
  • Pu has a simple cubic fcc structure,the d phase
    which is easily stabilized over a wide region in
    the T,p phase diagram.
  • The d phase is non magnetic.
  • Many LDA , GGA studies ( 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 one of the largest discrepancy ever known
    in DFT based calculations.

68
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.

69
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, when full potential LMTO (Soderlind and
    Wills). This is usually taken as an indication
    that a Pu is a weakly correlated system

70
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 and Kotliar,
    Bouchet et. Al. ) Delta Pu has U4,
  • Alpha Pu has U 0.

71
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.

72
Pu Specific Heat
73
Anomalous ResistivityJ. Smith LANL
74
MAGNETIC SUSCEPTIBILITY
75
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 electronic
    structure is about to vary.
  • This result resolves one of the basic paradoxes
    in the physics of Pu.

76
Pu DMFT total energy vs Volume
77
Lda vs Exp Spectra
78
Pu Spectra DMFT(Savrasov) EXP (Arko et. Al)
79
Earlier Studies of Magnetic Anisotropy
  • Erickson
  • Daalderop

80
Case study Fe and Ni
  • Archetypical itinerant ferromagnets
  • LSDA predicts correct low T moment
  • Band picture holds at low T
  • Main challenge, finite T properties
    (Lichtensteins talk).
  • Magnetic anisotropy puzzle. LDA predicts the
    incorrect easy axis for Nickel .
  • LDA Fermi surface has features which are not seen
    in DeHaas Van Alphen ( Lonzarich)

81
Iron and Nickel crossover to a real space
picture at high T
82
Photoemission Spectra and Spin Autocorrelation
Fe (U2, J.9ev,T/Tc.8) (Lichtenstein,
Katsenelson,GK prl 2001)
83
Photoemission and T/Tc.8 Spin Autocorrelation
Ni (U3, J.9 ev)
84
Iron and Nickelmagnetic properties
(Lichtenstein, Katsenelson,GK cond-mat 0102297)
85
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)

86
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.

87
Ni moment
88
Fe moment\
89
Magnetic anisotropy Ni
90
Magnetic anisotropy Fe
91
Magnetic anisotropy
92
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 interpolating between atoms and band,
    encouraging results for simple elements

93
DMFT Review A. Georges, G. Kotliar, W. Krauth
and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)
Weiss field
94
Outlook
  • Systematic improvements, short range
    correlations.
  • Take a cluster of sites, include the effect of
    the rest in a G0 (renormalization of the
    quadratic part of the effective action). What
    to take for G0
  • DCA (M. Jarrell et.al) , CDMFT ( Savrasov and GK
    )
  • include the effects of the electrons to
    renormalize the quartic part of the action (spin
    spin , charge charge correlations) E. DMFT
    (Kajueter and GK, Si et.al)

95
Outlook
  • Extensions of DMFT implemented on model systems,
    (e.g. Motome and GK ) carry over to more
    realistic framework. Better determination of Tcs.
  • First principles approach determination of the
    Hubbard parameters, and the double counting
    corrections long range coulomb interactions
    E-DMFT
  • Improvement in the treatement of multiplet
    effects in the impurity solvers, phonon
    entropies,

96
Functional Approach
G. Kotliar EPJB (1999)
Write a Comment
User Comments (0)
About PowerShow.com