Dynamical Mean Field Theory and Electronic Structure Calculations

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Dynamical Mean Field Theory and Electronic
Structure Calculations

• Gabriel Kotliar
• Center for Materials Theory
• Rutgers University

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Outline
Physics Today Vol 57, 53 (2004) Gabriel Kotliar
and Dieter Vollhardt

• Incorporating electronic structure methods in
  DMFT. C-DMFT. M. Capone, M. Civelli
• Why do we need k-sum to do optics. Cerium
  puzzles. K. Haule V. Udovenko Why do we need
  functionals to do total energies. Phonons and
  plutonium puzzles.
• X. Dai S. Savrasov

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Two roads 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.
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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,
  substract this out by shifting the heavy level
  (double counting term)
• The U matrix can be estimated from first
  principles of viewed as parameters. Solve
  resulting model using DMFT.

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Single site DMFT Impurity cavity construction
A. Georges, G. Kotliar, PRB 45, 6497 (1992)
Weiss field
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EDMFT H. Kajueter Rutgers Ph.D Thesis 1995
Si and Smith PRL77, 3391(1996) R. Chitra and G.
Kotliar PRL84,3678 (2000)
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Realistic DMFT loop matrix inversion-tetrahedron
method
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Site? Cell. Cellular DMFT. C-DMFT. G.
Kotliar,S.. Savrasov, G. Palsson and G. Biroli,
Phys. Rev. Lett. 87, 186401 (2001)
tˆ(K) is the hopping expressed in the
superlattice notations.

• Other cluster extensions (DCA, Katsnelson and
  Lichtenstein periodized scheme, nested cluster
  schemes, PCMDFT ), causality issues, O.
  Parcollet, G. Biroli and GK cond-matt 0307587
  (2003)

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N vs mu in one dimensional Hubbard model
.Compare 2 site cluster (in exact diag with
Nb8) vs exact Bethe Anzats, M. Capone M.Civelli
C. Castellani V Kancharla and GK 2004
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Two roads for ab-initio calculation of electronic
structure of strongly correlated materials
Crystal structure Atomic positions
Model Hamiltonian
Correlation Functions Total Energies etc.
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Spectral Density Functional Effective action
construction R. Chitra G. Kotliar PRB 62,12715.
Kotliar Savrasov in New Theoretical Approaches
to Strongly Correlated Systems, A. M. Tsvelik ed.
(2001) Kluwer Academic Publishers. 259-301
 cond-mat/0208241. S Savrasov G Kotliar
cond-mat0308053.

• 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)
• Allows computation of total energy, phonons!!!!

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LDADMFT Self-Consistency loop. See also S.
Savrasov and G. Kotliar cond-matt 0308053
E
U
DMFT
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Impurity Solvers.

• Hubbard I.
• Quantum Montecarlo.
• Rational Approximations to the self energy,
  constructed with slave bosons. cond-mat/0401539
  V. Oudovenko, K. Haule, S. Savrasov D. Villani
  and G. Kotliar.
• Extensions of NCA. Th. Pruschke and N. Grewe, Z.
  Phys. B Condens. Matter 74, 439, 1989. SUNCA K.
  Haule, S. Kirchner, J. Kroha, and P. Wolfle,
  Phys. Rev. B 64, 155111, (2001).

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Application to Materials

• Cerium Alpha to Gamma Transition.
• Plutonium Alpha-Delta-Epsilon.

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Overview
volumes exp. LDA LDAU
a 28Å3 24.7Å3
g 34.4Å3 35.2Å3

• ? ? -phase (localized)
• High T phase
• Curie-Weiss law (localized magnetic moment),
• Large lattice constant
• Tk around 60-80K

• ? ?-phase (delocalizedKondo-physics)
• Low T phase
• Loss of Magnetism (Fermi liquid Pauli
  susceptibility) - completely screened magnetic
  moment
• smaller lattice constant
• Tk around 1000-2000K

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Qualitative Ideas.

• B. Johansson, Philos. Mag. 30, 469 (1974). Mott
  transition of the f electrons as a function of
  pressure. Ce alpha gamma transition. spd
  electrons are spectators.
• Mathematical implementation, metallic phase
  treat spdf electrons by LDA, insulating phase
  put f electron in the core.
• J.W. Allen and R.M. Martin, Phys. Rev. Lett. 49,
  1106 (1982) Kondo volume collapse picture. The
  dominant effect is the spd-f hybridization.

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Qualitative Ideas

• alpha phase Kondo effect between spd and f takes
  place. insulating phase no Kondo effect (low
  Kondo temperature).
• Mathematical implementation, Anderson impurity
  model in the suplemented with elastic terms.
  (precursor of realistic DMFT ideas, but without
  self consistency condition). J.W. Allen and L.Z.
  Liu, Phys. Rev. B 46, 5047 (1992).

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LDADMFTCe spectra
M.B.Zolfl,I.A.NekrasovTh.Pruschke,V.I.Anisimov
J. Keller,Phys.Rev. Lett 87, 276403 (2001). K.
Held, A.K. McMahan, and R.T. Scalettar, Phys.
Rev.Lett. 87, 276404 (2001) A.K.McMahan,K.Held,and
R.T.Scalettar,Phys Rev. B 67, 075108 (2003).
Successful calculations of thermodynamics.
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Unfortunately photoemission cannot decide between
the Kondo collapse picture and the Mott
transition picture.Evolution of the spectra as a
function of U , half filling full frustration,
Hubbard model!!!!
X.Zhang M. Rozenberg G. Kotliar (PRL 70,
1666(1993)).
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The schematic phase diagram of cannot distinguish
between the two scenarios.

• J.W. Allen and L.Z. Liu, Phys. Rev. B 46, 5047
  (1992). Kondo impurity model elastic terms.
• DMFT phase diagram of a Hubbard model at integer
  filling, has a region between Uc1(T) and Uc2(T)
  where two solutions coexist. A. Georges G.
  Kotliar W. Krauth and M Rozenberg RMP
  68,13,(1996).
• Coupling the two solutions to the lattice gives a
  phase diagram akin to alpha gamma cerium.
  Majumdar and Krishnamurthy PRL 73 (1994).

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Photoemissionexperiment

• A. Mc Mahan K Held and R. Scalettar (2002)
• Zoffl et. al (2002)
• K. Haule V. Udovenko S. Savrasov and GK. (2004)

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To resolve the conflict between the Mott
transition and the volume collapse picture
Turn to Optics! Haule et.al.

• Qualitative idea. The spd electrons have much
  larger velocities, so optics will be much more
  senstive to their behavior.
• See if they are simple spectators (Mott
  transition picture ) or wether a Kondo binding
  unbinding takes pace (Kondo collapse picture).
• General method, bulk probe.

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Optics formula
double pole
One divergence integrated out!
single pole
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Temperature dependence of the optical
conductivity.
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Theory Haule et. al. cond-matt 04Expt J.W.
vanderEb PRL 886,3407 (2001)
The volume of alpha is 28.06A and the
temperature 580K. The volume of the gamma phase
is 34.37A and T 1160K. Experiments alpha
at 5 K and gamma phase at 300 K.
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Optical conductivity of Ce (expt. Van Der Eb
et.al. theory Haule et.al)
experiment
LDADMFT

• K. Haule et.al.

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Origin of the features.
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Conclusion Cerium

• Qualitatively good agreement with existing
  experiment.
• Some quantitative disagreement, see however .
• Experiments should study the temperature
  dependence of the optics.
• Optics Theory can provide a simple resolution
  of the Mott vs K-Collapse conundrum.

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Phases of Pu (A. Lawson LANL) Los Alamos Science
26, (2000)
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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

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Pu DMFT total energy vs Volume (Savrasov
Kotliar and Abrahams 2001)
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DMFT studies of Pu.

• Savrasov, S. Y., and G. Kotliar, 2003, Phys. Rev.
  Lett. 90(5), 056401/1.
• Savrasov, S. Y., and G. Kotliar, 2003,
  cond-mat/0308053 .
• Savrasov, S. Y., G. Kotliar, and E. Abrahams,
  2001, Nature 410, 793
• Dai X. Savrasov S.Y. Kotliar G. Migliori A.
  Letbetter H, Abrahams A. Science 300, 953, (2003)

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DFT Studies of 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
• The shear moduli in the delta phase were
  calculated within LDA and GGA by Bouchet et. al.
  (2000) and c is negative!
• .

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Evolution of the spectra as a function of U ,
half filling full frustration.
X.Zhang M. Rozenberg G. Kotliar (PRL 70,
1666(1993)).
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Alpha and delta Pu Expt. Arko et.al. PRB 62,
1773 (2000). DMFT Savrasov and Kotliar
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Phonon freq (THz) vs q in delta Pu X. Dai et. al.
Science vol 300, 953, 2003
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Expts Wong et. al. Science 301. 1078 (2003)
Theory Dai et. al. Science 300, 953, (2003)
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The delta epsilon transition

• The high temperature phase, (epsilon) is body
  centered cubic, and has a smaller volume than the
  (fcc) delta phase.
• What drives this phase transition?
• Having a functional, that computes total energies
  opens the way to the computation of phonon
  frequencies in correlated materials (S. Savrasov
  and G. Kotliar 2002)

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Phases of Pu (A. Lawson LANL) Los Alamos Science
26, (2000)
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Epsilon Plutonium.
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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 Ackland et. al. PRB.65, 184104
  (2002) (and neglecting electronic entropy).
  TC 600 K.

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Phonons epsilon
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Summary

• Incorporating electronic structure methods in
  DMFT. C-DMFT. M. Capone, M. Civelli
• Why do we need k-sum to do optics. Cerium
  puzzles. K. Haule V. Udovenko Why do we need
  functionals to do total energies. Phonons and
  plutonium puzzles.
• X. Dai S. Savrasov

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Why is optics calculation not completely trivial?
Analytic tetrahedron method
Integral is analytic and simple (combination of
logarithms)

• Energies linearly interpolated

no simple analytic expression

• Product of two energies linearly interpolated

ATM applicable
but numerically very unstable because of
quadratic pole
1D example
Parabola has 2 zeros (2poles) Line has no zeros
(no poles)
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LDADMFT functional
F Sum of local 2PI graphs with local U matrix and
local G
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Schematic DMFT phase diagram one band Hubbard
model. Rozenberg et. al. 1996. Introduce
coupling to the lattice will cause a volume jump
across the first order transition. (Majumdar and
Krishnamurthy ).
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Shear anisotropy. Expt. vs Theory

• C(C11-C12)/2 4.78 GPa C3.9 GPa
• C44 33.59 GPa C4433.0 GPa
• C44/C 7 Largest shear anisotropy in any
  element!
• C44/C 8.4

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Benchmarking SUNCA, V. Udovenko and K. Haule