THE STATE UNIVERSITY OF NEW JERSEY

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Excitation spectra
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Comments on realistic calculations using DMFGT

• Gabriel Kotliar
• Rutgers University
• Trieste 2002

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Spectral Evolution at T0 half filling full
frustration
X.Zhang M. Rozenberg G. Kotliar (PRL 1993) Joo
and Udovenko (20010)
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Summary

• Basis set LMTO (Savrasov)
• Materials Information and Design Lab. (Savrasovs
  MINDLAB)
• Computations of U (Anisimov)
• Derivation of model hamiltonian
• Solution via DMFT mapping onto degenerate
  Anderson model in a self consistent bath.
• Solution of the multiorbital anderson model
• Using QMC (Rozenber and Lichtenstein).

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Basis set, bands , DOS
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Computation of Us
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Comments

• U is a basis dependent concept.
• Dynamical mean field theory is a basis dependent
  technique.

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Unitary transformation
K dependent!
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Two Roads for calculations of the electronic
structure of correlated materials
Crystal Structure atomic positions
Model Hamiltonian
Correlation functions Total energies etc.
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LDA functional
Conjugate field, VKS(r)
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Minimize LDA functional
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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)

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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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LDADMFT functional
F Sum of local 2PI graphs with local U matrix and
local G
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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.

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LDADMFT Self-Consistency loop
E
U
DMFT
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Realistic DMFT loop
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(No Transcript)
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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 (1998).
• S. Savrasov and G.Kotliar, funcional
  formulation for full self consistent
  implementation Nature (2001)

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Applications

• Look for situations which
• Are in between atomic and band behavior.
• Many Many Many Compounds Oxides.
• BUT ALSO SOME ELEMENTS!

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Mott transition in the actinide series. B.
Johanssen 1974 Smith and Kmetko Phase Diagram
1984.
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Pu DMFT total energy vs Volume(S. Savrasov 2001)
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Lda vs Exp Spectra
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Pu Spectra DMFT(Savrasov) EXP (Arko et. Al)
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Iron and Nickel crossover to a real space
picture at high T(Lichtenstein,Katsnelson andGK)
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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 many systems

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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
• Cluster DMFT, periodic clusters (Lichtenstein and
  Katsnelson)DCA (M. Jarrell et.al) , CDMFT ( 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)

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C-DMFT test in one dimension. (Bolech, Kancharla
and Gk2002)
Gap vs U, Exact solution Lieb and Wu, Ovshinikov
PRL 20,1445 (1968)
Nc2 CDMFT vs Nc1
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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 GK Savrasov Palsson and Biroli
  PRL87, 186401 2001

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DMFT cavity construction
Weiss field
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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.

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Cellular DMFT Basis selection
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Lattice action
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Elimination of the medium variables
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Determination of the effective medium.
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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.

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Improved estimators

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

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Real Space Formulation of the DCA approximation
of Jarrell et.al.
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Affleck Marston model.
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Convergence test in the Affleck Marston
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Convergence of the self energy
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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.

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Extended DMFT electron phonon
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Extended DMFT e.ph. Problem
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E-DMFT classical case, soft spins
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E-DMFT classical case Ising limit
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E-DMFT test in the classical caseBethe Lattice,
S. Pankov 2001
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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

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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?

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Realistic Theories of Correlated Materials

• ITP, Santa-Barbara
• July 27 December 13 (2002)
• Conference November15-19 (2002)
• O.K. Andesen, A. Georges,
• G. Kotliar, and A. Lichtenstein
• http//www.itp.ucsb.edu/activities/future/

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Recent phase diagram of the frustrated Half
filled Hubbard model with semicircular DOS (QMC
Joo and Udovenko PRB2001).
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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)

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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 .

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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 .

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Functional Approach
G. Kotliar EPJB (1999)
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Case study in f electrons, Mott transition in
the actinide series
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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

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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.
• 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

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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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DMFTConnection with atomic limit
Weiss field