university of california daviskashiwa, july 27, 2007 from lda+u to lda+dmft s. y. savrasov,...
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University of California DavisKashiwa, July 27, 2007
From LDA+U to LDA+DMFTFrom LDA+U to LDA+DMFT
S. Y. Savrasov,Department of Physics, University of California, Davis
Collaborators:
Q. Yin, X. Wan, A. Gordienko (UC Davis)G. Kotliar, K Haule (Rutgers)
University of California DavisKashiwa, July 27, 2007
ContentContent
From LDA+U to LDA+DMFT
Extension I: LDA+Hubbard 1 Approximation
Extension II: LDA+Cluster Exact Diagonalization
Applications to Magnons Spectra
University of California DavisKashiwa, July 27, 2007
T U U
T
( )N E( )N E
U
k kt U
k kt
1ˆ ˆ ˆ ˆ ˆ2ij i j i i
ij i
H t c c Un n
Idea of LDA+U is borrowed from
the Hubbard Hamiltonian:
LDA+ULDA+U
T
T UU
Spectrum of Antiferromagnet at half fillingSpectrum of Antiferromagnet at half filling
FE
University of California DavisKashiwa, July 27, 2007
LDA+U Orbital Dependent PotentialLDA+U Orbital Dependent Potential
ˆ ˆ ˆ( ) [ ( 1/ 2)]LDA U LDA dm m dmV V r P U n P
The correction acts on the correlated orbitals only:
ˆ ˆ[ ( 1/ 2)]dm m dmP U n P | |
' ' '' '
ˆ| ( ) ( )l l ml m
r Y r
{ }ˆ ˆ( ) ( ) ( ) ( )dm d dm dmSP r r Y r r dV
The Schroedinger’s equation for the electron is solved
2 ˆ( )LDA U kj kj kjV
with orbital dependent potential
( ) ( )kj r A r
When forming Hamiltonian matrix 2 ˆ| |LDA UH V
EF
( )N E( )N E
Up
d
d
University of California DavisKashiwa, July 27, 2007
Main problem with LDA+UMain problem with LDA+U
LDA+U 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 LDA+U
Another example is 4f materials which show atomic multiplet structure reflecting atomic character of 4f states. Late actinides (5f’s)show similar behavior.
Atomic limit cannot be recovered by LDA+U because LDA+U correction is the Hartree-Fock approximation for atomic self-energy, not the actual self-energy of the electron!
University of California DavisKashiwa, July 27, 2007
NiO: Comparison with PhotoemissionNiO: Comparison with Photoemission
LSDALDA+UParamagnetic LDA
Electronic Configuration: Ni 2+O2- d8p6 (T2g6Eg
2p6)
University of California DavisKashiwa, July 27, 2007
LDA+DMFT as natural extension of LDA+ULDA+DMFT as natural extension of LDA+U
In LDA+U correction to the potential
,( )LDA U atomic DC atomic HF DCV V V
is just the Hartree-Fock value of the exact atomic self energy.
Why don’t use exact atomic self-energy itself instead of its Hartree-Fock value? This is so called Hubbard I approximation to the electronic self-energy.
Next step: use self-energy from atom allowing to hybridize with conduction bath, i.e. finding it from the Anderson impurity problem.
Impose self-consistency for the bath: full dynamical mean field theory is recovered.
LDA+U LDA+ ,atomic HF LDA+ ( )atomic
LDA+ ( )atomic LDA+ ( )inpurity
LDA+ ( )inpurity LDA+DMFT
University of California DavisKashiwa, July 27, 2007
Localized electrons: LDA+DMFTLocalized electrons: LDA+DMFT
2 ˆ ˆ[ ( ) ( ( ) ) ] ( ) ( )LDA d imp DC d kj kj kjV r P V P r r
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 1( , )
kj
G k
have information about atomic multiplets, Kondo, Zhang-Rice singlets, etc.
N(N())
ddnn->d->dn+1n+1
Better description compared to LDA or LDA+U is obtained
ddnn->d->dn-1n-1
21ˆˆ ( ) [ ( )] kd
imp d impk k
VG
University of California DavisKashiwa, July 27, 2007
Exact Diagonalization MethodsExact Diagonalization Methods
[ . .]imp d k k k kd kk k
H d d Ud d dd c c V c d c c
For capturing physics of localized electrons combination of LDA and exact diagonalization methods can be utilized:
| |impH E
kd
kdV
The cluster Hamiltonian is exact diagonalized
' '
0
0 | | | | 0( )
( )m m
imp
d dG
E E
The Green function is calculated:
Self-energy is extracted:2
1ˆˆ ( ) [ ( )] kdimp d imp
k k
VG
University of California DavisKashiwa, July 27, 2007
Exact Diagonalization MethodsExact Diagonalization Methods
In the limit of small hybridization Vkd=0 this is reduced to calculating atomic d(f)-shell self-energy: Hubbard I approximation (Hubbard, 1961). IfHatree Fock estimate is used here, LDA+U method is recovered.
Corrections due to finite hybridization can be alternatively evaluated using QMC, or NCA, OCA, SUNCA approximations (K. Haule, 2003)
21ˆˆ ( ) [ ( )] kd
imp d impk k
VG
University of California DavisKashiwa, July 27, 2007
Excitations in AtomsExcitations in Atoms
nd
1( 1)
2n dE n Un n
1nd
1
1( 1) ( 1)( 2)
2n dE n U n n
1nd
1
1( 1) ( 1)
2n dE n U n n
Electron Removal Spectrum
1 / 2n n dE E U Electron Addition Spectrum
1 / 2n n dE E U
/ 2d U / 2d U
Im ( )G
University of California DavisKashiwa, July 27, 2007
Atomic Self-Energies have singularitiesAtomic Self-Energies have singularities
2
1 1( )
( )4( )
dd
d
GU
Self-energy with a pole is required:
Ground state energies for configurations dn, dn+1, dn-1 give rise to electron removal En-En-1 and electron addition En-En+1 spectra. Atoms are always insulators!
1 / 2n n dE E U 1 / 2n n dE E U
1/ 2 1/ 2( )
/ 2 / 2atomicd d
GU U
or two poles in one-electron Green function
Electron removalElectron addition
Coulomb gap
This is missing in DFT effective potential or LDA+U orbital dependent potential
University of California DavisKashiwa, July 27, 2007
Mott Insulators as Systems near Atomic LimitMott 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
1( )LDA
d LDA
GV
1 2
1 1( )
( ) ( )( )
4( )
1/ 2 1/ 2
( ) / 2 ( ) / 2
Hubbardd
d
G kUk
k
k U k U
Frequency dependence in self-energy is required:
LDA/LDA+U, other static mean field theories, cannot access paramagnetic insulating state.
University of California DavisKashiwa, July 27, 2007
Electronic Structure calculation with LDA+Hub1Electronic Structure calculation with LDA+Hub1
1( )
( )
ss ps ds
LDA Hub sp pp dp
sd pd dd Atomic DC
H H H
H H H H
H H H V
LDA+Hubbard 1 Hamiltonian is diagonalized
Green Function is calculated2
1
| ( ) |( )
( )
kj
LDA Hubj kj
AG k
Density of states can be visualized
1
1( ) Im ( )LDA Hub
k
N G k
University of California DavisKashiwa, July 27, 2007
LDA+Hub1 Densities of States for NiO and MnOLDA+Hub1 Densities of States for NiO and MnO
Results of LDA+Hubbard 1 calculation: paramagnetic insulating state is recovered
LHB UHB
LHB UHB
U
U
Dielectric Gap
Dielectric Gap
University of California DavisKashiwa, July 27, 2007
NiO: Comparison with Photoemission NiO: Comparison with Photoemission
Insulator is recovered, satellite is recovered as lower Hubbard bandLow energy feature due to d electrons is not recovered!
LHB UHBU
Dielectric Gap
University of California DavisKashiwa, July 27, 2007
Americium PuzzleAmericium Puzzle
Density functional based electronic structure calculations: Non magnetic LDA/GGA predicts volume 50% off. Magnetic GGA corrects most of error in volume but gives m~6B
(Soderlind et.al., PRB 2000). Experimentally, Am has non magnetic f6 ground state with J=0 (7F0)
Experimental Equation of State (after Heathman et.al, PRL 2000)
Mott Transition?“Soft”
“Hard”
University of California DavisKashiwa, July 27, 2007
Photoemission in Am, Pu, SmPhotoemission in Am, Pu, Sm
after J. R. Naegele, Phys. Rev. Lett. (1984).
Atomic multiplet structure emerges from measured photoemission spectra in Am (5f6), Sm(4f6) - Signature for f electrons localization.
University of California DavisKashiwa, July 27, 2007
Am Equation of State: LDA+Hub1 PredictionsAm Equation of State: LDA+Hub1 Predictions
LDA+Hub1 predictions: Non magnetic f6 ground
state with J=0 (7F0) Equilibrium Volume: Vtheory/Vexp=0.93
Bulk Modulus: Btheory=47 GPa
Experimentally B=40-45 GPa
Theoretical P(V) using LDA+Hub1
Self-consistent evaluations of total energies with LDA+DMFT 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 eVNew algorithms allow studies of complex structures.
Predictions for Am II
Predictions for Am IV
Predictions for Am III
Predictions for Am I
University of California DavisKashiwa, July 27, 2007
Atomic Multiplets in AmericiumAtomic Multiplets in AmericiumLDA+Hub1 Density of States
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
Matrix Hubbard I Method
F(0)=4.5 eV F(2)=8.0 eVF(4)=5.4 eV F(6)=4.0 eV
University of California DavisKashiwa, July 27, 2007
Many Body Electronic Structure for Many Body Electronic Structure for 77FF00 Americium Americium
Experimental Photoemission Spectrum after J. Naegele et.al, PRL 1984
Insights from LDA+DMFT: 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)
University of California DavisKashiwa, July 27, 2007
Effective (DFT-like) single particle spectrumalways consists of delta like peaks
Real excitational spectrumcan be quite different
Bringing k-resolution to atomic multipletsBringing k-resolution to atomic multiplets
0[ ( ) ( )] ( , ) 1H k G k
University of California DavisKashiwa, July 27, 2007
Many Body Calculations with speed of LDA Many Body Calculations with speed of LDA
( ) ( ) i
i i
W
P
(Savrasov, Haule, Kotliar, PRL2006)
0[ ( )] 0kjH
Non-linear over energy Dyson equation
with pole representation for self energy
(1)0 1 2
(2)1 1
(3)
2 2
( )
00
0
kj
kj
kj
H W W
W P
W P
is exactly reduced to linear set of equations in the extended space
University of California DavisKashiwa, July 27, 2007
Many Body Electronic Structure MethodMany Body Electronic Structure Method
11
0 1 2 0
1 1
2 2
( ) [ ( ) ]
0
0
i
i i
WH W W H
PW P
W P
The proof lies in mathematical identityGreen function G(r,r’,)
Physical part of the electron is described by the first component of the vector(1)
(2)
(3)
kj
kj
kj
2| |( )
kj
j kj
AG k
Electronic Green function is non interacting like but with more poles:
University of California DavisKashiwa, July 27, 2007
Many Body Electronic Structure for Many Body Electronic Structure for 77FF00 Americium Americium
University of California DavisKashiwa, July 27, 2007
Cluster Exact DiagonalizationCluster Exact Diagonalization
[ . .]imp d k k k kd kk k
H d d Ud d dd c c V c d c c
Cluster Hamiltonian
| |impH E
kd
kdV
The cluster Hamiltonian is exact diagonalized
' '
0
0 | | | | 0( )
( )m m
imp
d dG
E E
The Green function is calculated:
Self-energy is extracted:2
1ˆˆ ( ) [ ( )] kdimp d imp
k k
VG
University of California DavisKashiwa, July 27, 2007
Electronic Structure calculation with LDA+CEDElectronic Structure calculation with LDA+CED
( )
( )
ss ps ds
LDA sp pp dp
sd pd dd d DC
H H H
H H H H
H H H V
LDA+ Hamiltonian is diagonalized
Green Function is calculated2| ( ) |
( )( )
kj
LDAj kj
AG k
Density of states can be visualized
1( ) Im ( )LDA
k
N G k
University of California DavisKashiwa, July 27, 2007
Eg’s
Illustration: Many Body Bands for NiOIllustration: Many Body Bands for NiO
LDA+cluster exact diagonaization for NiO above TN: 8 6 2 6
2 2[ ]g g pd t e o
8 7d d
8 9d d
6 5o o
(atomic like LHB)
(atomic like UHB)
O-hole coupledto local d moment(Zhang-Rice like)
University of California DavisKashiwa, July 27, 2007
Generalized Zhang-Rice PhysicsGeneralized Zhang-Rice Physics
NiO(d8) CoO(d7) FeO(d6) MnO(d5)
SNi=1
SO=1/2
SO=1/2 SCu=1/2
CuO2(d9)
Zhang-Rice Singlet (Stot=0)
Doublet (Stot=1/2) Triplet (Stot=1) Quartet (Stot=3/2) Quintet (Stot=2)
SO=1/2 SMn=5/2SO=1/2 SFe=2SO=1/2 SCo=3/2
JAF
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NiO: LDA+CED compared with ARPESNiO: LDA+CED compared with ARPES
Dispersion of doublet
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CoO: LDA+CED compared with ARPESCoO: LDA+CED compared with ARPES
Dispersion of triplet
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LDA+CED for HTSCs: Dispersion of Zhang-Rice singletLDA+CED for HTSCs: Dispersion of Zhang-Rice singlet
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Magnons, Exchange Interactions, Tc’sMagnons, Exchange Interactions, Tc’s
Realistic treatment of magnetic exchange interactionsin strongly-correlated systems:
• Spin waves, magnetic ordering temperatures
• Necessary input to Heisenberg, Kondo Hamiltonians
• Spin-phonon interactions, incommensurability, magnetoferroelectricity
University of California DavisKashiwa, July 27, 2007
Magnetic Force TheoremMagnetic Force Theorem
0'
'
( , ')RR LDAR R
B BJ r r
Exchange Constants via Linear Response (Lichtenstein et. al, 1987)
Spin Wave spectra, Curie temperatures, Spin Dynamics (Antropov et.al, 1995)
After Halilov, et. al, 1998
Fe
University of California DavisKashiwa, July 27, 2007
Exchange Constants, Spin Waves, Neel Tc’sExchange Constants, Spin Waves, Neel Tc’s
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.
[( ) ( )] [ ]DMFTJ Tr GG Tr BGGB
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Spin Wave Spectrum in NiOSpin Wave Spectrum in NiO
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Calculated Neel Temperatures Calculated Neel Temperatures
University of California DavisKashiwa, July 27, 2007
ConclusionConclusion
There are natural extensions of LDA+U method:
LDA+Hubbard 1 is a method where full frequency dependent atomic self-energy is used
LDA+ED is a method where self-energy is extracted from cluster calculations
LDA+DMFT is a general method where correlated orbitals are treated with full frequency resolution
Many new phenomena (atomic multiplets, mixed valence, Kondo effect) canbe studied with the electronic structure calculations