From%20LDA U%20to%20LDA DMFT

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From LDAU to LDADMFT
S. Y. Savrasov, Department of Physics, University
of California, Davis
Collaborators Q. Yin, X. Wan, A. Gordienko (UC
Davis) G. Kotliar, K Haule (Rutgers)
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Content

• From LDAU to LDADMFT
• Extension I LDAHubbard 1 Approximation
• Extension II LDACluster Exact Diagonalization
• Applications to Magnons Spectra

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LDAU
Idea of LDAU is borrowed from the Hubbard
Hamiltonian
Spectrum of Antiferromagnet at half filling
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LDAU Orbital Dependent Potential
The Schroedingers equation for the electron is
solved
with orbital dependent potential
When forming Hamiltonian matrix
The correction acts on the correlated orbitals
only
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Main problem with LDAU
LDAU is capable to recover insulating behavior
in magnetically ordered state. However, systems
like NiO are insulators both above and below the
Neel temperature. Magnetically disordered state
is not described by LDAU Another example is 4f
materials which show atomic multiplet structure
reflecting atomic character of 4f states. Late
actinides (5fs) show similar behavior. Atomic
limit cannot be recovered by LDAU because LDAU
correction is the Hartree-Fock approximation for
atomic self-energy, not the actual self-energy of
the electron!
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NiO Comparison with Photoemission
Electronic Configuration Ni 2O2- d8p6
(T2g6Eg2p6)
LSDA
LDAU
Paramagnetic LDA
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LDADMFT as natural extension of LDAU
In LDAU correction to the potential
is just the Hartree-Fock value of the exact
atomic self energy.
Why dont use exact atomic self-energy itself
instead of its Hartree-Fock value? This is so
called Hubbard I approximation to the electronic
self-energy.
LDA
LDA
LDAU
Next step use self-energy from atom allowing to
hybridize with conduction bath, i.e. finding it
from the Anderson impurity problem.
LDA
LDA
Impose self-consistency for the bath full
dynamical mean field theory is recovered.
LDADMFT
LDA
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Localized electrons LDADMFT
Electronic structure is composed from LDA
Hamiltonian for sp(d) electrons and dynamical
self-energy for (d)f-electrons extracted from
solving Anderson impurity model
Poles of the Green function
have information about atomic multiplets, Kondo,
Zhang-Rice singlets, etc.
N(w)
dn-gtdn1
dn-gtdn-1
w
0
Better description compared to LDA or LDAU is
obtained
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Exact Diagonalization Methods
For capturing physics of localized electrons
combination of LDA and exact diagonalization
methods can be utilized
The cluster Hamiltonian is exact diagonalized
The Green function is calculated
Self-energy is extracted
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Exact Diagonalization Methods
In the limit of small hybridization Vkd0 this is
reduced to calculating atomic d(f)-shell
self-energy Hubbard I approximation (Hubbard,
1961). If Hatree Fock estimate is used here,
LDAU method is recovered. Corrections due to
finite hybridization can be alternatively
evaluated using QMC, or NCA, OCA, SUNCA
approximations (K. Haule, 2003)
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Excitations in Atoms
Electron Removal Spectrum
Electron Addition Spectrum
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Atomic Self-Energies have singularities
Ground state energies for configurations dn,
dn1, dn-1 give rise to electron removal En-En-1
and electron addition En-En1 spectra. Atoms are
always insulators!
Electron removal
Electron addition
or two poles in one-electron Green function
Coulomb gap
Self-energy with a pole is required
This is missing in DFT effective potential or
LDAU orbital dependent potential
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Mott Insulators as Systems near Atomic Limit
Classical systems MnO (d5), FeO (d6), CoO (d7),
NiO (d8). Neel temperatures 100-500K. Remain
insulating both below and above TN
LDA/LDAU, other static mean field theories,
cannot access paramagnetic insulating state.
Frequency dependence in self-energy is required
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Electronic Structure calculation with LDAHub1
LDAHubbard 1 Hamiltonian is diagonalized
Green Function is calculated
Density of states can be visualized
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LDAHub1 Densities of States for NiO and MnO
Results of LDAHubbard 1 calculation
paramagnetic insulating state is recovered
LHB
UHB
U
Dielectric Gap
LHB
UHB
U
Dielectric Gap
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NiO Comparison with Photoemission
LHB
UHB
U
Dielectric Gap
Insulator is recovered, satellite is recovered as
lower Hubbard band Low energy feature due to d
electrons is not recovered!
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Americium Puzzle
Experimental Equation of State
(after Heathman et.al, PRL 2000)
Mott Transition?
Soft
Hard

• Density functional based electronic structure
  calculations
• Non magnetic LDA/GGA predicts volume 50 off.
• Magnetic GGA corrects most of error in volume but
  gives m6mB
• (Soderlind et.al., PRB 2000).
• Experimentally, Am has non magnetic f6 ground
  state with J0 (7F0)

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Photoemission in Am, Pu, Sm
Atomic multiplet structure emerges from measured
photoemission spectra in Am (5f6), Sm(4f6) -
Signature for f electrons localization.
after J. R. Naegele, Phys. Rev. Lett. (1984).
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Am Equation of State LDAHub1 Predictions
Self-consistent evaluations of total energies
with LDADMFT using matrix Hubbard I method.
Accounting for full atomic multiplet structure
using Slater integrals F(0)4.5 eV, F(2)8 eV,
F(4)5.4 eV, F(6)4 eV
New algorithms allow studies of complex
structures.
Theoretical P(V) using LDAHub1
Predictions for Am I

• LDAHub1 predictions
• Non magnetic f6 ground state with J0 (7F0)
• Equilibrium Volume
• Vtheory/Vexp0.93
• Bulk Modulus Btheory47 GPa
• Experimentally B40-45 GPa

Predictions for Am II
Predictions for Am III
Predictions for Am IV
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Atomic Multiplets in Americium
LDAHub1 Density of States
Matrix Hubbard I Method
F(0)4.5 eV F(2)8.0 eV F(4)5.4 eV F(6)4.0 eV
Exact Diag. for atomic shell F(0)4.5 eV
F(2)8.0 eV F(4)5.4 eV F(6)4.0 eV
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Many Body Electronic Structure for 7F0 Americium
Insights from LDADMFT Under pressure energies
of f6 and f7 states become degenerate which
drives Americium into mixed valence
regime. Explains anomalous growth in resistivity,
confirms ideas pushed forward recently by
Griveau, Rebizant, Lander, Kotliar, (2005)
Experimental Photoemission Spectrum after J.
Naegele et.al, PRL 1984
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Bringing k-resolution to atomic multiplets
Effective (DFT-like) single particle
spectrum always consists of delta like peaks
Real excitational spectrum can be quite different
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Many Body Calculations with speed of LDA
(Savrasov, Haule, Kotliar, PRL2006)
Non-linear over energy Dyson equation
with pole representation for self energy
is exactly reduced to linear set of equations in
the extended space
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Many Body Electronic Structure Method
Green function G(r,r,w)
The proof lies in mathematical identity
S(w)
Physical part of the electron is described by the
first component of the vector
Electronic Green function is non interacting like
but with more poles
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Many Body Electronic Structure for 7F0 Americium
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Cluster Exact Diagonalization
Cluster Hamiltonian
The cluster Hamiltonian is exact diagonalized
The Green function is calculated
Self-energy is extracted
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Electronic Structure calculation with LDACED
LDAS Hamiltonian is diagonalized
Green Function is calculated
Density of states can be visualized
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Illustration Many Body Bands for NiO
LDAcluster exact diagonaization for NiO above
TN
(atomic like UHB)
O-hole coupled to local d moment (Zhang-Rice like)
Egs
(atomic like LHB)
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Generalized Zhang-Rice Physics
JAF
CuO2(d9)
Zhang-Rice Singlet (Stot0)
NiO(d8)
CoO(d7)
FeO(d6)
MnO(d5)
Doublet (Stot1/2)
Triplet (Stot1)
Quartet (Stot3/2)
Quintet (Stot2)
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NiO LDACED compared with ARPES
Dispersion of doublet
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CoO LDACED compared with ARPES
Dispersion of triplet
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LDACED for HTSCs Dispersion of Zhang-Rice
singlet
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Magnons, Exchange Interactions, Tcs

• Realistic treatment of magnetic exchange
  interactions
• in strongly-correlated systems
• Spin waves, magnetic ordering temperatures
• Necessary input to Heisenberg, Kondo
  Hamiltonians
• Spin-phonon interactions, incommensurability,
• magnetoferroelectricity

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Magnetic Force Theorem
Exchange Constants via Linear Response
(Lichtenstein et. al, 1987)
Spin Wave spectra, Curie temperatures, Spin
Dynamics (Antropov et.al, 1995)
Fe
After Halilov, et. al, 1998
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Exchange Constants, Spin Waves, Neel Tcs
Magnetic force theorem for DMFT has been recently
discussed (Katsnelson, Lichtenstein, PRB 2000)
Using rational representation for self-energy,
magnetic force theorem can be simplified (X. Wan,
Q. Yin, SS, PRL 2006) Expression for exchange
constants looks similar to DFT
However, eigenstates which describe Hubbard
bands, quasiparticle bands, multiplet
transitions, etc. appear here.
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Spin Wave Spectrum in NiO
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Calculated Neel Temperatures
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Conclusion

• There are natural extensions of LDAU method
• LDAHubbard 1 is a method where full frequency
  dependent atomic self-energy is used
• LDAED is a method where self-energy is
  extracted from cluster calculations
• LDADMFT is a general method where correlated
  orbitals are treated with
• full frequency resolution
• Many new phenomena (atomic multiplets, mixed
  valence, Kondo effect) can
• be studied with the electronic structure
  calculations