Elemental Plutonium: a strongly correlated metal

1
Elemental Plutonium a strongly correlated metal

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

Collaborators S. Savrasov (NJIT) X. Dai(
Rutgers )

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Physics of Pu
The Problem This? Or this?
3
For me the problem is THIS. The Mott
Phenomena

• Evolution of the electronic structure between
  the atomic limit and the band limit in an open
  shell situation.
• The in between regime is ubiquitous central
  them in strongly correlated systems, gives rise
  to interesting physics. Example Mott
  transition across the actinide series B.
  Johansson Phil Mag. 30,469 (1974)
• Revisit the problem using a new insights and new
  techniques from the solution of the Mott
  transition problem within dynamical mean field
  theory in the model Hamiltonian context.
• Use the ideas and concepts that resulted from
  this development to give physical qualitative
  insights into real materials.
• Turn the technology developed to solve simple
  models into a practical quantitative electronic
  structure method .

4
Outline

• Introduction some Pu puzzles.
• Results Minimum of the melting curve,
• Delta Pu Most probable valence, size of the
  local moment
• Equilibrium Volume.
• Photoemission Spectral.
• Stabilization of Epsilon Pu
• Conclusions

5

Mott transition in the actinide series (Smith
Kmetko phase diagram)
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Small amounts of Ga stabilize the d phase (A.
Lawson LANL)
7
Shear anisotropy.

• C(C11-C12)/2 4.78
• C44 33.59 19.70
• C44/C 8 Largest shear anisotropy in any
  element!
• LDA Calculations (Bouchet) C -48

8
Plutonium Puzzles

• 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.

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DFT Studies

• 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
• 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
• Alterantive approach Wills et. al. (5f)4 core
  1f(5f)in conduction band.

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Pu Specific Heat
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Anomalous Resistivity
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Pu is NOT MAGNETIC
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Specific heat and susceptibility.
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Problems with the conventional viewpoint of a
Pu

• U/W is not so different in alpha and delta
• 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.

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Outline

• Introduction some Pu puzzles.
• DMFT , qualitative aspects of the Mott
  transition from model Hamiltonians
• DMFT as an electronic structure method.
• DMFT results for delta Pu, and some qualitative
  insights.
• Conclusions

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What do we want from materials theory?

• New concepts , qualitative ideas
• Understanding, explanation of existent
  experiments, and predictions of new ones.
• Quantitative capabilities with predictive
• power.
• Notoriously difficult to achieve in strongly
  correlated materials.
• We have solved the hydrogen atom problem of
  strongly correlated electron systems.

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Evolution of the Spectral Function with
Temperature
Anomalous transfer of spectral weight connected
to the proximity to the Ising Mott endpoint
(Kotliar Lange and Rozenberg Phys. Rev. Lett. 84,
5180 (2000)
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Generalized phase diagram
T
U/W
Structure, bands, orbitals
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Qualitative phase diagram in the U, T , m plane
(two band Kotliar Murthy Rozenberg PRL (2002).

• Coexistence regions between localized and
  delocalized spectral functions.
• k diverges at generic Mott endpoints

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Mott transition in layered organic conductors
S Lefebvre et al. Ito et.al, Kanodas talk
Bourbonnais talk
Magnetic Frustration
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Ultrasound study of
Fournier et. al. (2002)
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Minimum in melting curve and divergence of the
compressibility at the Mott endpoint
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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
• Elastic anomalies, more pronounced with orbital
  degeneracy.

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Minimum in melting curve and divergence of the
compressibility at the Mott endpoint
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Cerium
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Outline

• Introduction some Pu puzzles.
• DMFT , qualitative aspects of the Mott
  transition in model Hamiltonians.
• DMFT as an electronic structure method.
• DMFT results for delta Pu, and some qualitative
  insights.
• Conclusions

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Solving the DMFT equations

• Wide variety of computational tools
  (QMC,ED.)Analytical Methods
• Extension to ordered states.
• Review A. Georges, G. Kotliar, W. Krauth and
  M. Rozenberg Rev. Mod. Phys. 68,13 (1996)

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Realistic DMFT loop
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LDADMFT-outer loop relax
Edc
U
DMFT
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Outer loop relax
Edc
G0
Impurity Solver
G,S
Imp. Solver Hartree-Fock
U
SCC
DMFT
LDAU
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Outline

• Introduction some Pu puzzles.
• DMFT , qualitative aspects of the Mott
  transition in model Hamiltonians.
• DMFT as an electronic structure method.
• Realistic DMFT and Plutonium
• Conclusions

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What is the dominant atomic configuration? Local
moment?

• Snapshots of the f electron
• Dominant configuration(5f)5
• Naïve view Lz-3,-2,-1,0,1
• ML-5 mB
• S5/2 Ms5 mB
• Mtot0

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LDAU bands. (Savrasov GK ,PRL 2000).
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Magnetic moment

• L5, S5/2, J5/2, MtotMsmB gJ .7 mB
• Crystal fields G7 G8
• GGAU estimate (Savrasov and Kotliar 2000)
  ML-3.9 Mtot1.1
• This bit is quenched by Kondo effect of spd
  electrons DMFT treatment
• Experimental consequence neutrons large
  magnetic field induced form factor (G. Lander).

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

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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.

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Comments on the HF static limit

• Describes only the Hubbard bands.
• No QP states.
• Single well structure in the E vs V curve.
• (Savrasov and Kotliar PRL)

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Lda vs Exp Spectra
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Spectral Evolution at T0 half filling full
frustration
X.Zhang M. Rozenberg G. Kotliar (PRL 1993)
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Pu Spectra DMFT(Savrasov) EXP (Arko Joyce Morales
Wills Jashley PRB 62, 1773 (2000)
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Comparaison with LDAU
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Summary
Spectra
Method
E vs V
LDA
LDAU
DMFT
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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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Energy vs Volume
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Energy vs Volume
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Success story Density Functional Linear Response
Tremendous progress in ab initio modelling of
lattice dynamics electron-phonon interactions
has been achieved (Review Baroni et.al, Rev.
Mod. Phys, 73, 515, 2001)
(Savrasov, PRB 1996)
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Results for NiO Phonons
Solid circles theory, open circles exp. (Roy
et.al, 1976)
DMFT Savrasov and GK PRL 2003
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DMFT for Mott insulators
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Phonon freq (THz) vs q in delta Pu (Dai et. al. )
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Shear anisotropy. Expt. vs Theory

• C(C11-C12)/2 4.78 GPa C3.37GPa
• C44 33.59 GPa C4419.7 GPa
• C44/C 8 Largest shear anisotropy in any
  element!
• C44/C 6

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Phonon frequency (Thz ) vs q in epsilon Pu.
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Temperature stabilizes a very anharmonic phonon
mode
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Phonons epsilon
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Phonon entropy drives the epsilon delta phase
transition

• Epsilon is slightly more metallic than delta, 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 neglecting the
• Electronic entropy TC 600 K.

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Outline

• Introduction some Pu puzzles.
• DMFT , qualitative aspects of the Mott
  transition in model Hamiltonians.
• DMFT as an electronic structure method.
• DMFT results for delta Pu, and some qualitative
  insights.
• Conclusions

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Conclusions

• DMFT produces non magnetic state, around a
  fluctuating (5f)5 configuraton with correct
  volume the qualitative features of the
  photoemission spectra, and a double minima
  structure in the E vs V curve.
• Correlated view of the alpha and delta phases of
  Pu. Interplay of correlations and electron
  phonon interactions (delta-epsilon).
• Calculations can be refined.

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Conclusions

• Outsanding question electronic entropy, lattice
  dynamics.
• In the making, new generation of DMFT programs,
  QMC with multiplets, full potential DMFT,
  frequency dependent Us, multiplet effects ,
  combination of DMFT with GW

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Acknowledgements Development of DMFT
Collaborators V. Anisimov, R. Chitra, V.
Dobrosavlevic, X. Dai, D. Fisher, A. Georges,
H. Kajueter, W.Krauth, E. Lange, A.
Lichtenstein, G. Moeller, Y. Motome, G.
Palsson, M. Rozenberg, S. Savrasov, Q. Si, V.
Udovenko, I. Yang, X.Y. Zhang
Support NSF DMR 0096462 Support
Instrumentation. NSF DMR-0116068 Work on Fe
and Ni ONR4-2650 Work on Pu DOE
DE-FG02-99ER45761 and LANL subcontract No.
03737-001-02
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DMFT MODELS.
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Mean-Field Classical vs Quantum
Classical case
Quantum case
A. Georges, G. Kotliar (1992)
Phys. Rev. B 45, 6497
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Example Single site DMFT, functional formulation

• Express in terms of Weiss field (G. Kotliar EPJB
  99)

Local self energy (Muller Hartman 89)
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DMFT Impurity cavity construction
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DMFT Review A. Georges, G. Kotliar, W. Krauth
and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)
Weiss field
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Case study IPT half filled Hubbard one band

• (Uc1)exact 2.2_.2 (Exact diag, Rozenberg,
  Kajueter, Kotliar PRB 1996) , confirmed by
  Noack and Gebhardt (1999) (Uc1)IPT 2.6
• (Uc2)exact 2.97_.05(Projective self consistent
  method, Moeller Si Rozenberg Kotliar Fisher PRL
  1995 ), (Confirmed by R. Bulla 1999) (Uc2)IPT
  3.3
• (TMIT ) exact .026_ .004 (QMC Rozenberg Chitra
  and Kotliar PRL 1999), (TMIT )IPT .045
• (UMIT )exact 2.38 - .03 (QMC Rozenberg Chitra
  and Kotliar PRL 1999), (UMIT )IPT 2.5
  (Confirmed by Bulla 2001)
• For realistic studies errors due to other
  sources (for example the value of U, are at
  least of the same order of magnitude).

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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.

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Interfacing DMFT in calculations of the
electronic structure of correlated materials
Crystal Structure atomic positions
Model Hamiltonian
Correlation functions Total energies etc.
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LDADMFT functional
F Sum of local 2PI graphs with local U matrix and
local G
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LDADMFT and LDAU

• Static limit of the LDADMFT functional ,
• with Fatom FHF reduces to the LDAU
  functional
• of Anisimov Andersen and Zaanen.
• Crude approximation. Reasonable in ordered Mott
  insulators. Short time picture of the systems.
• Total energy in DMFT can be approximated by
  LDAU with an effective U . Extra screening
  processes in DMFT produce smaller Ueff.
• ULDAU lt UDMFT

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E-DMFT GW P. Sun and G. Kotliar Phys. Rev. B 2002
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LDADMFT and LDAU

• Static limit of the LDADMFT functional ,
• with Fatom FHF reduces to the LDAU
  functional
• of Anisimov Andersen and Zaanen.
• Crude approximation. Reasonable in ordered Mott
  insulators. Short time picture of the systems.
• Total energy in DMFT can be approximated by
  LDAU with an effective U .

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LDADMFT References

• Anisimov Poteryaev Korotin Anhokin and Kotliar J.
  Phys. Cond. Mat. 35, 7359 (1997).
• Lichtenstein and Katsenelson PRB (1998).
• Reviews Kotliar, Savrasov, in New Theoretical
  approaches to strongly correlated systems, Edited
  by A. Tsvelik, Kluwer Publishers, (2001).
• Held Nekrasov Blumer Anisimov and Vollhardt
  et.al. Int. Jour. of Mod PhysB15, 2611 (2001).
• A. Lichtenstein M. Katsnelson and G. Kotliar
  (2002)

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Comments on LDADMFT

• Static limit of the LDADMFT functional , with F
  FHF reduces to LDAU
• Gives the local spectra and the total energy
  simultaneously, treating QP and H bands on the
  same footing.
• Luttinger theorem is obeyed.
• Functional formulation is essential for
  computations of total energies, opens the way to
  phonon calculations.

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References

• LDADMFT
• 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 G.Kotliar funcional formulation
  for full self consistent implementation of a
  spectral density functional.
• Application to Pu S. Savrasov G. Kotliar and
  E. Abrahams (Nature 2001).

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DMFT Effective Action point of view.R. Chitra
and G. Kotliar Phys Rev. B.(2000), (2001).

• Identify observable, A. Construct an exact
  functional of ltAgta, G a which is stationary at
  the physical value of a.
• Example, density in DFT theory. (Fukuda et. al.)
• When a is local, it gives an exact mapping onto a
  local problem, defines a Weiss field.
• The method is useful when practical and accurate
  approximations to the exact functional exist.
  Example LDA, GGA, in DFT.
• It is useful to introduce a Lagrange multiplier
  l conjugate to a, G a, l .
• It gives as a byproduct a additional lattice
  information.

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Interface DMFT with electronic structure.

• Derive model Hamiltonians, solve by DMFT
• (or cluster extensions). Total energy?
• Full many body aproach, treat light electrons by
  GW or screened HF, heavy electrons by DMFT
  E-DMFT frequency dependent interactionsGK and S.
  Savrasov, P.Sun and GK cond-matt 0205522
• Treat correlated electrons with DMFT and light
  electrons with DFT (LDA, GGA DMFT)

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Spectral Density Functional effective action
construction

• Introduce local orbitals, caR(r-R), 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)

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LDADMFT approximate functional

• 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 (Gunnarson and Anisimov, McMahan
  et.al. Hybertsen et.al) of viewed as parameters

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References

• Long range Coulomb interactios, E-DMFT. R. Chitra
  and G. Kotliar
• Combining E-DMFT and GW, GW-U , G. Kotliar and S.
  Savrasov
• Implementation of E-DMFT , GW at the model level.
  P Sun and G. Kotliar.
• Also S. Biermann et. al.

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Energy difference between epsilon and delta
87

• Connection between local spectra and cohesive
  energy using Anderson impurity models
  foreshadowed by J. Allen and R. Martin PRL 49,
  1106 (1982) in the context of KVC for cerium.
• Identificaton of Kondo resonance n Ce , PRB 28,
  5347 (1983).

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E-DMFTGW effective action
G D
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Dynamical Mean Field Theory(DMFT)Review A.
Georges G. Kotliar W. Krauth M. Rozenberg. Rev
Mod Phys 68,1 (1996)

• Local approximation (Treglia and Ducastelle PRB
  21,3729), local self energy, as in CPA.
• Exact the limit defined by Metzner and Vollhardt
  prl 62,324(1989) inifinite.
• Mean field approach to many body systems, maps
  lattice model onto a quantum impurity model
  (e.g. Anderson impurity model )in a self
  consistent medium for which powerful theoretical
  methods exist. (A. Georges and G. Kotliar
  prb45,6479 (1992).

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Technical details

• 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)

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Technical details

• Atomic sphere approximation.
• Ignore crystal field splittings in the self
  energies.
• Fully relativistic non perturbative treatment of
  the spin orbit interactions.