Correlation Effects in Itinerant Magnets : Towards a realistic Dynamical Mean Field Approach

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Correlation Effects in Itinerant Magnets
Towards a realistic Dynamical Mean Field Approach

• Gabriel Kotliar
• Physics Department
• Rutgers University

In Electronic Structure and Computational
Magnetism July 15-17 (2002)
2
Outline

• Dynamical Mean Field Theory a tool for treating
  correlations in model Hamiltonians.
• Towards Realistic implementations of DMFT.
• Applications to Fe and Ni.
• Conclusions and outlook.

3
Acknowledgements

• Collaborators and References
• A. Lichtenstein M. Katsnelson and G. Kotliar
  Phys. Rev Lett. 87, 067205 (2001).
• I Yang S. Savrasov and G. Kotliar Phys.
  Rev. Lett. 87, 216405 (2001).
• Useful Discussions K. Hathaway and G. Lonzarich
• Support NSF and ONR

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Strong Correlation Problem

• Two limiting cases of the electronic structure
  problem are well understood. The high density
  limit ( spectrum of one particle excitations
  forms bands) and the low density limit (spectrum
  of atomic like excitations, Hubbard bands).
• Correlated compounds electrons in partially
  filled shells. Not close to the well understood
  limits . Non perturbative regime.
• Standard approaches (LDA, HF ) do not work well.

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Motivations for going beyond density functional
theory.

• DFT is a theory for ground state properties.
  Its Kohn Sham spectra can be taken a starting
  point for perturbative (eg. GW ) calculations of
  the excitation spectra and transport.
• This does not work for strongly correlated
  systems, eg oxides containing 3d, 4f, 5f
  elements. Character of the spectra (QP bands
  Hubbard bands ) is not captured by LDA.
• LDA GGA is less accurate in determining some
  ground state properties in correlated materials.

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DMFT

• DMFT simplest many body technique which describes
  correctly the open shell atomic limit and the
  band limit . Exact in the limit of large lattice
  coordination.
• Band physics (i.e. kinetic energy) survive in the
  atomic limit (superexchange). Some aspects of
  atomic physics survive even in itinerant systems
  (J, U, Hubbard bands, satellites, L)
• Computations of one electron spectra, transport
  properties
• Spectral density functional. Connects the one
  electron spectral function and the total energy.

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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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DMFT Impurity cavity construction
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C-DMFT
CDMFT The lattice self energy is inferred from
the cluster self energy.

Alternative approaches DCA (Jarrell et.al.)
Periodic clusters (Lichtenstein and Katsnelson)
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C-DMFT test in one dimension. (Bolech, Kancharla
GK cond-mat 2002)
Gap vs U, Exact solution Lieb and Wu, Ovshinikov
Nc2 CDMFT vs Nc1
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Solving the DMFT equations

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

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From model hamiltonians to realistic calculations.

• DMFT as a method to be incorporated in electronic
  structure calculations.
• Important in regimes where local moments are
  present (e.g. NiO above its Neel temperature)
• Incorporate realistic structure and orbital
  degeneracy information in many body studies.
• Combination of electronic structure(LDA,GGA,GW)
  and many body methods (DMFT)

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Interface with electronic structure.

• Derive model hamiltonians, solve by DMFT
• (or cluster extensions). Total energy?
• Full many body aproach, treat light electrons byt
  GW or screend HF, heavy electrons by DMFT GK and
  Chitra, GK 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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Combining LDA and DMFT

• The light, SP electrons well described by LDA
• The heavier D electrons treat by model DMFT.
• LDA already contains an average interaction of
  the heavy electrons, subtract this out by
  shifting the heavy level (double counting term,
  Lichtenstein et.al.)
• Atomic physics parameters . UF0 cost of double
  occupancy irrespectively of spin, JF2F4, Hunds
  energy favoring spin polarization .F2/F4.6
• Calculations of U, Edc, study as a function of
  these parameters.

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Combine Dynamical Mean Field Theory with
Electronic structure methods.

• Single site DMFT made correct qualitative
  predictions.
• Make realistic by
• Incorporating all the electrons.
• Add realistic orbital structure. U, J..
• Add realistic crystal structure.
• Allow the atoms to move.

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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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Realistic Calculationsof the Electronic
Structure of Correlated materials

• Combinining DMFT with state of the art electronic
  structure methods to construct a first principles
  framework to describe complex materials.
• Anisimov Poteryaev Korotin Anhokin and Kotliar J.
  Phys. Cond. Mat. 35, 7359 (1997).
• Lichtenstein and Katsenelson PRB (1998)
• Savrasov Kotliar and Abrahams Nature 410, 793
  (2001)) Kotliar, Savrasov, in New Theoretical
  approaches to strongly correlated systems, Edited
  by A. Tsvelik, Kluwer Publishers, 2001)

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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, subtract this out by
  shifting the heavy level (double counting term)
• The U matrix can be estimated from first
  principles (Gunnarson and Anisimov, Mc Mahan et.
  Al. Hybertsen et.al) or viewed as parameters

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Density functional theory and Dynamical Mean
Field Theory

• DFT Static mean field, electrons in an effective
  potential.
• Functional of the density.
• DMFT Promote the local (or a few quasilocal
  Greens functions or observables) to the basic
  quantities of the theory.
• Express the free energy as a functional of those
  quasilocal quantities.

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Spectral Density Functional effective action
construction (Fukuda, Valiev and Fernando ,
Chitra and Kotliar).

• 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 Gr(r),G(R,R)(iw)
• A useful approximation to the exact functional
  can be constructed, the DMFT LDA functional.
  Savrasov Kotliar and Abrahams Nature 410, 793
  (2001))
• Full self consistent implementation.

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LDADMFT-outer loop relax
Edc
U
DMFT
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Realistic DMFT loop
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LDADMFT functional
F Sum of local 2PI graphs with local U matrix and
local G
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Very Partial list of application of realistic
DMFT to materials

• QP bands in ruthenides A. Liebsch et al (PRL
  2000)
• N phase of Pu S. Savrasov et al (Nature 2001)
• MIT in V2O3 K. Held et al (PRL 2001)
• Magnetism of Fe, Ni A. Lichtenstein et al PRL
  (2001)
• J-G transition in Ce K. Held et al (PRL 2000)
  M. Zolfl et al PRL (2000).
• 3d doped Mott insulator La1-xSrxTiO3 (Anisimov
  et.al 1997, Nekrasov et.al. 1999, Udovenko et.al
  2002)
• ..

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DMFT

• Developed initially to treat correlation effects
  in model Hamiltonians.
• Review A. Georges, G. Kotliar, W. Krauth and M.
  Rozenberg Rev. Mod. Phys. 68,13 (1996)
• Extension to realistic setting V. Anisimov, A.
  Poteryaev, M. Korotin, Anokhin and G. Kotliar,
  J. Phys. Cond. Mat 9, 7359 (1997). S. Savrasov,
  G. Kotliar and E. Abrahams, Nature 410, 793
  (2001). Lichtenstein and Katsnelson
  Phys.Rev. B 57, 6884(1998)
• Unlike DFT, DMFT computes both free energies
  and one electron (photoemission ) spectra and
  many other physical quantities at finite
  temperatures.

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LDADMFT Spectral Density Functional (Fukuda,
Valiev and Fernando , Chitra and GK, Savrasov
and GK).

• DFT, consider the exact free energy as a
  functional of an external potential. Express the
  free energy as a functional of the local density
  by Legendre transformation.
• 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 double Legendre transformton

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Spectral Density Functional

• Formal construction of a functional of the d
  spectral density
• 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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LDADMFT functional
FAtom Sum of all local 2PI graphs build with
local Coulomb interaction matrix, parametrized
by Slater integrals F0, F2 and F4 and local
G.Express F in terms of AIM model.
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Outer loop relax
Edc
G0
Impurity Solver
G,S
Hartree-Fock
U
SCC
DMFT
LDAU
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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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LDADMFT Self-Consistency loop
E
U
DMFT
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LDADMFT and LDAU

• Static limit of the LDADMFT functional ,
• with F FHF reduces to the LDAU functional
• of Anisimov et.al.
• Crude approximation. Reasonable in ordered
  situations.

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DMFT

• If the self energy matrix is weakly k dependent
  is the physical self
  energy.
• Since is a matrix, DMFT changes the
  shape of the Fermi surface
• DMFT is absolutely necessary in the high
  temperature local momentregime. LDAU with an
  effective U is OK at low energy.
• DMFT is needed to describe spectra with QP and
  Hubbard bands or satellites.

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Applications of LDADMFT

• Organics
• Alpha-Gamma Cerium
• V2O3
• Volume collapse in Pu
• Photoemission of ruthenates
• Doping driven Mott transition in LaSrTiO3
• Itinerant Ferromagnetism
• Bucky Balls

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Applications Itinerant Ferromagnetism, Ni Fe

• Compromise in the resources used for the solution
  of the one electron problem, and the many body
  problem.
• Goal obtain an overall approximate but
  consistent picture of how correlations affect
  physical properties. Estimate sensitivity on
  parameters.
• Tc, spectra, susceptibility, QMC- impurity
  solver ASA, relatively small number of k
  points
• Magnetic anisotropy HF-impurity solverfull
  potential LMTO, large number of k points, non
  collinear magnetization

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Case study Fe and Ni

• Archetypical itinerant ferromagnets
• LSDA predicts correct low T moment
• Band picture holds at low T

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Iron and Nickel crossover to a real space
picture at high T
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Other aspects that require to treate correlations
beyond LDA

• Magnetic anisotropy. L.S effect. LDA predicts
  the incorrect easy axis(100) for Nickel
  .(instead of the correct one (111) )
• LDA Fermi surface in Nickel has features which
  are not seen in DeHaas Van Alphen ( G. Lonzarich)
  
• Photoemission spectra of Ni 6 ev satellite 30
  band narrowing, reduction of exchange splitting.

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DMFT-QMC Numerical Details

• 256 k points
• 105 - 106 QMC sweeps
• Analytic continuation via maximum entropy.
• Tight binding LMTO-ASA

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Photoemission Spectra and Spin Autocorrelation
Fe (U2, J.9ev,T/Tc.8) (Lichtenstein,
Katsenelson,GK prl 2001)
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Photoemission and T/Tc.8 Spin Autocorrelation
Ni (U3, J.9 ev)
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Iron and Nickelmagnetic properties
(Lichtenstein, Katsenelson,GK PRL 01)
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Ni and Fe theory vs exp

• 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)

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Magnetic anisotropy

• MAE is small (1 meV/Atom)
• Ni 2.8 meV/Atom easy axis 111
• Fe 1.4 meV/Atom easy axis 100
• Long standing problem Early papers
• Van Vleck (PR 1937)
• Brooks (PR 1940)

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LDA calculations

• Trygg et.al (1995) SCF Total energy with large
  of k-points Wrong easy axis for Ni.
• Other related works
• Halilov et al. (1998)
• G. Schneider et al. (1997)
• Wang et al. (1993)
• Beiden et.al. (1998)

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Method

Full-potential multiple kappa LMTO method. Pauli
treatment of relativistic effects. Non-collinear
intraatomic magnetism included. Explore
different Edc. (Details I Yang Ph.D
thesis) Generalized relativistic LDAU with
occupancies nab,ss
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Numerical Considerations

Work of Trygg et.al. proves equivalence of
special points and tetrahedra. Confirmed.
(broadening 0.15 mRy.) Convergent Etot needs
15000 ks. We use 28000ks. Convergency checked
to 100000 ks. SUN E10K with 64 processors
used. LDA results of Trygg et.al. reproduced Ni
0.5 meV 001, exp. 2.8 meV 111, Fe 0.5 meV 001,
exp. 1.4 meV 001.
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LDAU Results

Studies of MAE as function of U and J. Both U
and J influence magnetic moment which is OK in
LDA 0.6 mB for Ni and 2.2 mB for Fe. How to fix
moment in LDAU Find M(U,J) and trace path for
which moment does not change.
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N i - M(U,J)

Magnetic moment as function of U and J for Ni
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Fe - M(U,J)

Magnetic moment as function of U and J for Fe
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MAE as a function of U(J)
U1.9 eV, J1.2 eV
Ni
U1.2 eV, J0.8 eV
Fd
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LDA vs LDAU for Ni


eg forming X2 pocket
eg
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Ni U2,J.1 PT (Katsenelson and
Lichtenstein)cond-matt 2002
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Conclusions

• Satellite in majority band at 6 ev, 30
  reduction of bandwidth, exchange splitting
  reduction from band theory value (.6ev) to .3 ev
• Spin wave stiffness controls the effects of
  spatial flucuations, it is about twice as large
  in Ni and in Fe. Single site should work for Ni,
  and overestimate Tc for 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.

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Conclusions

• Overall consistent picture of the
• effects of correlations on itinerant magnets
  using
• DMFT.
• Can reproduce correct easy axis and MAE
• of Fe and Ni.
• Can correct the Fermi surface of Ni.

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Work in progress

• With existing techniques, derive practical
  formulae for the magnetic anisotropy of systems
  containing partially localized and itinerant
  electrons.
• Further tests of DMFT on interesting materials.
• Incorporate extensions of DMFT to incorporate
  frequency dependent interations (GWDMFT) and to
  larger clusters.

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LDA and LDAU bands for Fe

No changes of Fermi surface found
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E(k) for Ni


LDA electronic structure for Ni
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Fermi Surface for Ni

Calculated Fermi surface for Ni using LDAU. No
artificial X2 pocket
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G0
Impurity Solver
G,S
Hartree-Fock
SCC
DMFT
LDAU
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GWDMFT functional.

• S. Savrasov and GK. P. Sun and GK. (cond matt).

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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/

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Solving the DMFT equations

• Wide variety of computational tools (QMC,
  NRG,ED.)
• Analytical Methods

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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)

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Minimize LDA functional
Kohn Sham eigenvalues, auxiliary quantities.
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LDA functional
Conjugate field, VKS(r)
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Double counting term (Lichtenstein et.al)
subtracts average correlation
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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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DMFT Impurity cavity construction A. Georges,
G. Kotliar, PRB, (1992)
Weiss field
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Single site DMFT, functional of local Greens
function G.

• Express in terms of Weiss field (semicircularDOS)
  

Local self energy (Muller Hartman 89)
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LDADMFT functional
F Sum of local 2PI graphs with local Coulomb
interaction matrix and local G
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Calculated MAE for Ni and Fe using LDAU method
S. Y. Savrasov New Jersey Institute of Technology
In collaboration with Imseok Yang (Ph.D Thesis,
RU) Gabriel Kotliar (RU)
Sponsored by Office of Naval Research Grant No
ONR 4-2650
Phys. Rev. Lett. 87, 216405 (2001)
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Calculations

• Total Energy DFT job with
• huge k-point summation problem.
• Eckard et.al (1987) Right order Wrong easy
  axis for Fe.
• Daalderlop et.al (1990) Force theorem Wrong
  easy axis for Ni.
• Varying position of Fermi level, artificial X2
  pocket influences easy axis.

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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 5 ( theory) 1043(expt)
• Ni 700 (theory) 631 (expt)

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Correlations
Many-body Hubbard interactions are important (not
captured by LDA) DMFT onsite correlations are
treated exactly, both atomic and band limit are
OK. Static limit of DMFT LDAU
method Self-energy S(w)-gtS(static) Solution of
impurity model collapses to determination of
nab Problem can be solved now.
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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.
• Many materials have electron states that are in
  between these two limiting situations and require
  the development of new electronic structure
  methods to predict some of its properties
  (spectra, energy, transport,.)
• DMFT simplest many body technique which treats
  simultaneously the open shell atomic limit and
  the band limit .

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Mean-Field Classical vs Quantum
Classical case
Quantum case
A. Georges, G. Kotliar (1992)
Phys. Rev. B 45, 6497