Towards an Electronic Structure Method for Correlated Electron Systems based on Dynamical Mean Field

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Towards an Electronic Structure Method for
Correlated Electron Systems based on Dynamical
Mean Field Theory

• G. Kotliar
• Physics Department and Center for Materials
  Theory
• Rutgers

Second International Workshop 2004 Ordering
Phenomena in Transition Metal Oxides September 26
- 29, 2004, Wildbad Kreuth
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Outline

• Can we build a controlled but practical many
  body scheme, for first principles electronic
  structure calculations of correlated solids ?
• Application electronic and elastic properties
  of Pu, a strongly correlated element.
• Collaborators S. Savrasov (NJIT) and N. Zein
  (Kurchatov-Rutgers)

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Density functional and Kohn Sham reference
system

• Kohn Sham spectra, proved to be an excelent
  starting point for doing perturbatio theory in
  screened Coulomb interactions GW.

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GW approximation (Hedin )
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LDAGW semiconducting gaps
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Strongly Correlated Materials

• Can we construct a conceptual framework and
  computational tools, for studying strongly
  correlated materials, which will be as successful
  as the Fermi Liquid LDA-GW program ?
• Is a local perspective reasonable ? accurate ?
• Dynamical Mean Field Theory . Unify band theory
  and atomic physics. Use an impurity model, (local
  degrees of freedom free electron enviroment )
  to describe the local spectra of a correlated
  system.

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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.
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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)
Testing CDMFT (G.. Kotliar,S. Savrasov, G.
Palsson and G. Biroli, Phys. Rev. Lett. 87,
186401 (2001) ) with two sites in the Hubbard
model in one dimension.
U/t4.
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Comparison with other cluster methods.
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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.
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Functional formulation. Chitra and Kotliar Phys.
Rev. B 62, 12715 (2000)and Phys. Rev.B (2001) . 
Introduce Notion of Local Greens functions, Wloc,
Gloc GGlocGnonloc .
Ex. IrgtR, rgt GlocG(R r, R r) dR,R
Sum of 2PI graphs
One can also view as an approximation to an
exact Spetral Density Functional of Gloc and
Wloc.
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EDMFT loop G. Kotliar and S. Savrasov in New
Theoretical Approaches to Strongly Correlated G
Systems, A. M. Tsvelik Ed. 2001 Kluwer Academic
Publishers. 259-301 . cond-mat/0208241 S. Y.
Savrasov, G. Kotliar, Phys. Rev. B 69, 245101
(2004)

• Full implementation in the context of a a one
  orbital model. P Sun and G. Kotliar Phys. Rev. B
  66, 85120 (2002).
• After finishing the loop treat the graphs
  involving Gnonloc Wnonloc in perturbation theory.
  P.Sun and GK PRL (2004). Related work, Biermann
  Aersetiwan and Georges PRL 90,086402 (2003) .
•  

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Further Approximations.

• 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) .
• Truncate the W operator act on the H sector
  only. i.e.
• Replace W(w) by a static U. This quantity can
  be estimated by a constrained LDA calculation or
  by a GW calculation with light electrons only.
  e.g.
• M.Springer and F.Aryasetiawan,Phys.Rev.B57,
  4364(1998) T.Kotani,J.PhysCondens.Matter12,2413(2
  000). FAryasetiawan M Imada A Georges G Kotliar
  S Biermann and A Lichtenstein cond-matt (2004)

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• or the U matrix can be adjusted
  empirically.
• At this point, the approximation can be derived
  from a functional (Savrasov
  and Kotliar 2001)
• FURTHER APPROXIMATION, ignore charge self
  consistency, namely set
• LDADMFT V. Anisimov, A. Poteryaev, M. Korotin,
  A. Anokhin and G. Kotliar, J. Phys. Cond. Mat.
  35, 7359 (1997) See also . A Lichtenstein and M.
  Katsnelson PRB 57, 6884 (1988).
• ReviewsHeld, K., I. A. Nekrasov, G. Keller, V.
  Eyert, N. Blumer, A. K. McMahan, R. T.
  Scalettar, T. Pruschke, V. I. Anisimov, and D.
  Vollhardt, 2003, Psi-k Newsletter 56, 65.
• Lichtenstein, A. I., M. I. Katsnelson, and G.
  Kotliar, in Electron Correlations and Materials
  Properties 2, edited by A. Gonis, N. Kioussis,
  and M. Ciftan (Kluwer Academic, Plenum
  Publishers, New York), p. 428.
• Georges, A., 2004, Electronic Archive, .lanl.gov,
  condmat/ 0403123 .

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Further approximations, use approximate impuirity
solvers rational Interpolative Perturbative
Theory. Savrasov Udovenko Villani Haule and
Kotliar . Cond-matt 0401539
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Pu in the periodic table
actinides
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Small amounts of Ga stabilize the d phase (A.
Lawson LANL)
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Delta phase of Plutonium Problems with LDA

• Many implementations.(Freeman, Koelling 1972)APW
  , 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. Negative shear modulus (Bouchet et.
  al.).
• 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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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)
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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.

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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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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)
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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 G. Ackland et. al. PRB.65, 184104
  (2002) (and neglecting electronic entropy).
  TC 600 K.

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

• Can we build a controlled but practical many
  body scheme, for first principles electronic
  structure calculations of correlated solids ?
• Application electronic and elastic properties
  of Pu, a strongly correlated element.

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Conclusions/Questions/Discussion

• Is self consistency important ? Or can we
  derive accurate model Hamiltonians, how do we
  separate scales
• what is currently limiting the accuracy of a
  given calculation. Is the neglect of a
  frequency dependent in U (W) ? Do we need to
  improve in the treatment of the self energy of
  the light electrons ? What is the minimal scale
  that is needed to take into account for a given
  material. How large of a cluster to use ? How
  about the impurity solver ? And the basis set ?

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Outline

• Can we build a controlled but practical many
  body scheme, for first principles electronic
  structure calculations of correlated solids ?
• Application electronic and elastic properties
  of Pu, a strongly correlated element.

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Conclusion

• Spectral Density Functional. Connection between
  spectra and bonding. Microscopic theory of Pu,
  connecting its anomalies to the vicinity of a
  Mott point.
• Combining theory and experiment we can more than
  the sum of the parts. Next step in Pu, much
  better defined problem, discrepancy in (111 )
  zone boundary, may be due to either the
  contribution of QP resonance, or the inclusion of
  nearest neighbor correlations. Both can be
  individually studied.
• Also needed, more experiment. Recent Neutron
  scattering .

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Further approximations r ational Interpolative
Perturbative Theory. Savrasov Udovenko Villani
Haule and Kotliar . Cond-matt 0401539
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CDMFT G.. Kotliar,S. Savrasov, G. Palsson and G.
Biroli, Phys. Rev. Lett. 87, 186401 (2001
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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)
t(K) hopping expressed in the superlattice
notations.

• Other cluster extensions (DCA Jarrell
  Krishnamurthy, Katsnelson and Lichtenstein
  periodized scheme, Nested Cluster Schemes ,
  causality issues, O. Parcollet, G. Biroli and GK
  cond-matt 0307587 .

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LDADMFT Self-Consistency loop. S. Savrasov and
G. Kotliar (2001) and cond-matt 0308053
E
U
DMFT
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LDADMFT Formalism V. Anisimov, A. Poteryaev,
M. Korotin, A. Anokhin and G. Kotliar, J. Phys.
Cond. Mat. 35, 7359-7367 (1997). S. Y. Savrasov
and G. Kotliar, Phys. Rev. B 69, 245101 (2004).
V. Udovenko S. Savrasov K. Haule and G. Kotliar
Cond-mat 0209336
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Comments on LDADMFT

• Gives the local spectra and the total energy
  simultaneously, treating QP and H bands on the
  same footing.
• Gives an approximate starting point, for
  perturbation theory in the non local part of the
  Coulomb interactions. See for example, P.
  SunPhys. Rev. Lett. 92, 196402 (2004)
• Good approximate starting point for optics.

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• Topics for discussion. Test model Hamiltonian
  approaches. Could it be that
• Adjusting one parameter we get just one
• Quantity.

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K. Haule , Pu- photoemission with DMFT using
vertex corrected NCA.
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Benchmarking SUNCA, V. Udovenko and K. Haule
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Interpolative scheme with slave bosons.
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Specific Heat Resistivity susceptibility.
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Functional formulation. Chitra and Kotliar Phys.
Rev. B 62, 12715 (2000)and Phys. Rev.B (2001) . 
IrgtR, rgt
Sum of all 2 PI graphs.
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• Experimentally delta Pu stable. It has a
  negative coefficient of thermal expansion.
• Delta Pu has the largest shear anisotropy of all
  elements , in spite of its cubic fcc
  structure,fcc Al, c44/c1.2, in Pu C44/C 6.

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Functional formulation. Chitra and Kotliar Phys.
Rev. B 62, 12715 (2000)and Phys. Rev.B (2001) . 
IrgtR, rgt
Sum of all 2 PI graphs.
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• Introduce localized basis set (e.g. LMTOs)
• Make local (i.e. cluster approximation)
• Treat non local pieces in perturbation theory.

DMFT eqs for Gloc and Wloc
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Functional formulation. Chitra and Kotliar Phys
Rev B62, 12715 (2000) and Phys. Rev. B Phys.
Rev. B 63, 115110 (2001).
IrgtR, rgt
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Inelastic X Ray. Phonon energy 10 mev, photon
energy 10 Kev.
E Ei - Ef Q ki - kf
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Experiments . Joe Wong, Michael Krisch, Daniel L.
Farber, Florent Occelli, Adam J. Schwartz, Tai-C.
Chiang, Mark Wall Carl Boro Ruqing Xu,
Science, Vol 301, Issue 5636, 1078-1080 , 22
August 2003.
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Alpha and delta Pu
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LDADMFT functional
F Sum of local 2PI graphs with local U matrix and
local G
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LDADMFT Self-Consistency loop
E
U
DMFT
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Electronic Structure of Materials

• Issues of principle.
• How do we think bout the electronic excitations
  of a given material ?
• Is it possible to obtain practical computational
  schemes of the free energies and excitations of
  a material ? What quantities do we have a
  reasonable expectations to be able to compute ?
  with which methods ?
• Issues of implementation.