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Prospects of LDA+DMFT
Silke BiermannCentre de Physique Theorique
Ecole Polytechnique, Palaiseau, France
(*) LDA = the local density approximation (LDA) of density
functional theory
LDA+DMFT = the combination of dynamical mean field theory
(DMFT) with the LDA. – p.1/69
The Mott transition within DMFTSpectral function ofthe Hubbard model
2
0
2
0
2
0
2
0
2
0
−Im
G
ω−4 −2 0 2 4
U/D=1
U/D=2
U/D=2.5
U/D=3
U/D=4
/D
Fig.30
. – p.2/69
Spectral function – survival kitAdd/remove an electron – at which energy?Two “easy” limiting cases:
1. Non-interacting limit:state of N electrons = Slater determinant(N+1)th electron can jump into any (unoccupied) band
probe unoccupied density of states
0
0.2
0.4
0.6
−2 −1 0 1 2E−E
Fermi
DOS
. – p.3/69
Spectral function – survival kit2. “Atomic limit” (complete localization):probe local Coulomb interaction!
0
0.1
0.2
0.3
0.4
0.5
−10 −5 0 5 10E−E
Fermi
U
In the general, interacting case:Spectral function
describes the possibility ofadding an electron with energy (includingrelaxation effects)
. – p.4/69
The Mott transition within DMFTSpectral function ofthe Hubbard model
2
0
2
0
2
0
2
0
2
0
−Im
G
ω−4 −2 0 2 4
U/D=1
U/D=2
U/D=2.5
U/D=3
U/D=4
/D
Fig.30
. – p.5/69
Mott insulator and correl. metal:YTiO and SrVO
Inte
nsity
(ar
b. u
nits
)
2.0 1.0 0.0Binding Energy (eV)
Sr0.5Ca0.5VO3
SrVO3 (x = 0)
CaVO3 (x = 1)
hν = 900 eV hν = 275 eV hν = 40.8 eV hν = 21.2 eV
2
0
2
0
2
0
2
0
2
0−
ImG
ω−4 −2 0 2 4
U/D=1
U/D=2
U/D=2.5
U/D=3
U/D=4
/D
Fig.30
. – p.6/69
Outline
Reminder: Dynamical Mean Field Theory(DMFT)
The “LDA+DMFT” method
:Reminder: an “effective atom” point of view
A functional point of view
Examples
What can we calculate ?The current status
Beyond LDA+DMFT: the GW+DMFT
scheme
LDA = The local density approximation to Density Functional Theory
LDA+DMFT = The combination of LDA and dynamical mean field theory (DMFT)
(**) GW+DMFT = The combination of Hedin’s GW approximation with DMFT. – p.7/69
The local Green’s function ...... is the central object of DMFT
Definition of Green’s function:
Relation to local spectral function:
. – p.8/69
Green’s function – survival kitQuasi-particles are poles of
All correlations are hidden in the self-energy:
. – p.9/69
Dynamical mean field theory ...... maps a lattice problem
onto a single-site (Anderson impurity) problem
with a self-consistency condition
. – p.10/69
Effective dynamics ...... for single-site problem
with the dynamical mean field
. – p.11/69
DMFT (contd.)Green’s function:
Self-energy (k-independent):
DMFT assumption :
Self-consistency condition for
. – p.12/69
The DMFT self-consistency cycleAnderson impurity model solver
Self-consistency condition:
. – p.13/69
Realistic Approach to CorrelationsCombine DMFT with band structure calculations
(Anisimov et al. 1997, Lichtenstein et al. 1998)
effective one-particle Hamiltonian within LDArepresent in localized basis
add Hubbard interaction term for correlated orbitals
solve within Dynamical Mean Field Theory
. – p.14/69
Hamiltonian formulation
(correl. orb.)
(correl. orb.)
in localized basis set( lecture by F. Lechermann)
all valence electrons
Hubbard terms for “correlated orbitals”. – p.15/69
Dynamical mean field theory ...... maps a solid
onto an “effective atom” problem
with a self-consistency condition
. – p.16/69
The DMFT self-consistency cycleAnderson impurity model solver
Self-consistency condition:
. – p.17/69
What do we mean by this?Represent the Green’s function in localized basis,e.g. LMTO’s:
,
where
double counting correctionsis a matrix in orbital space
is a matrix in the space of the correlated orbitals
lecture by F. Lechermann
. – p.18/69
Hubbard interaction termsScreened Coulomb interaction, onsite terms only:
with
Simplification: keep “diagonal density” terms only:
. – p.19/69
Parametrization of interaction:
with the Slater integrals
, e.g. for d-electrons:
. – p.20/69
Double counting corrections
of Hartree type (cf. “LDA+U”, Anisimov et al.)
Lichtenstein et al., 2001:
Tr
( no zero frequency contribution,cf. LDA Fermi surface Luttinger’s theorem)
. – p.21/69
A functional point of view ?Reminder about Density Functional Theory:Energy is a functional of the density:E= E[n(r)]
Kohn-Sham potential can be viewed as a Lagrangemultiplier to fix the density of the non-interactingreference system
. – p.22/69
A functional point of view ...... of LDA+DMFT
“Spectral Density Functional Theory”:Free energy a functional of(1) the density n(r)(2) the local Green’s function of the correlated orbitals
Construction by introduction of source termslecture by M. Katsnelson
Kotliar, Savrasov, PRB 2004
. – p.23/69
LDA+DMFT as a ...... spectral density functional theoryFree energy functional:
tr
tr
Kotliar, Savrasov, 2004 . – p.24/69
LDA+DMFT – the full scheme
DMFT loop
DMFT preludeDFT part
update
PSfrag replacements
VKS = Vext + VH + Vxc
[
−∇2
2 + VKS
]
|ψkν〉 = εkν |ψkν〉
from charge density ρ(r) constructupdate
|χRm
〉
GKS =[
iωn + µ+ ∇2
2 − VKS
]−1
G0
build GKS =[
iωn + µ+ ∇2
2 − VKS
]−1
construct initial G0
impurity solver
Gimpmm′(τ − τ ′) = −〈T dmσ(τ)d†
m′σ′(τ ′)〉Simp
self-consistency condition: construct Gloc
G−10 = G−1
loc + Σimp
Gloc = P(C)R
[
G−1KS −
(
Σimp − Σdc
)]−1
P(C)R
Σimp = G−10 − G−1
imp
ρ
compute new chemical potential µ
ρ(r) = ρKS(r) + ∆ρ(r)
(Appendix A)
(from F. Lechermann, A. Georges, A. Poteryaev, S. B., M. Posternak, A. Yamasaki, O. K. Ander-
sen, Phys. Rev. B 74 125120 (2006))
. – p.25/69
Some more examples
. – p.26/69
Cerium sesquioxide Ce O Mott insulator,
paramagnetic above 10 K
. – p.27/69
Cerium sesquioxide Ce O
Metallic in LDA:
-6
-4
-2
0
2
4
6
8
10
G K M G A L H A
Ene
rgy
(eV
)Ce2O3 bands (Ce 6p in basis)
Note the narrow f-bands!. – p.28/69
Cerium sesquioxide Ce O
LDA+DMFT splits f-states into upper and lowerHubbard bands:
(L. Pourovskii et al., arXiv:0705.2161). – p.29/69
Spectral functionCe O
25
50
0
25
50
Den
sity
of
stat
es (
1/eV
)
LDA
DMFT-nonSC
DMFT-SC
-6 -4 -2 0 2 4Energy (eV)
0
25
50
(L. Pourovskii et al., arXiv:0705.2161)
. – p.30/69
Total energyCe O
3.6 3.7 3.8 3.9 4
a (Ao
)
0
25
50
75
100
E-E
min
(m
Ry)
LDA DMFT (without charge SC)DMFT (with charge SC)
aexp
=3.89 Ao
(L. Pourovskii et al., arXiv:0705.2161)
. – p.31/69
Vanadium dioxide: VO
High temperature rutile phase: metallic
Low temperature monoclinic phase: insulating
. – p.32/69
Metal-insulator Transition
. – p.33/69
VO in LDA and LDA+DMFT
0
0.5
1.0
−2 0 2 4
Spe
ctra
lFun
ctio
n(a
rb.u
nits
)
ω (eV)
VO2
rutile phase
VO2
monoclinic phase
LDA: dashed lineLDA+DMFT: solid line
S.B., A.Poteryaev, A. Georges, A.Lichtenstein, PRL 2005 . – p.34/69
LDA+DMFT simulations ...for the monoclinic phase have to be done with care:
Inter-site (intra-pair)fluctuations ?
"Cluster-DMFT" :embed V pair in bath non-local self-energy !
. – p.35/69
VO recent photoemission
Koethe et al.,PRL 2006
lecture byL.-H. Tjeng
. – p.36/69
What can we calculate ?
. – p.37/69
What can we calculate ?
total energy
Examples: Ce O Pu
3.6 3.7 3.8 3.9 4
a (Ao
)
0
25
50
75
100
E-E
min
(m
Ry)
LDA DMFT (without charge SC)DMFT (with charge SC)
aexp
=3.89 Ao
(Pourovskii et al., arXiv:0705.2161) (Savrasov et al., Nature 2001)
(see also Held et al., PRL 2001, and Amadon et al., PRL 2006, for total energy calculations for
Cerium)
. – p.38/69
Total energy functional
with:
= crystal energy
= Hartree energy
= exchange correlation energysecond line = energy terms stemming from manybody Hamiltonian
(from B. Amadon et al., PRL 2006)
. – p.39/69
What can we calculate ?
local spectral functions
Examples:VO2-M1 and Ce O
25
50
0
25
50
Den
sity
of
stat
es (
1/eV
)
LDA
DMFT-nonSC
DMFT-SC
-6 -4 -2 0 2 4Energy (eV)
0
25
50
0
0.5
1.0
−2 0 2 4
Spe
ctra
lFun
ctio
n(a
rb.u
nits
)
ω (eV)
VO2
rutile phase
VO2
monoclinic phase
. – p.40/69
What can we calculate ?
total energy
local spectral functions
self-energies
-2
-1
0
1
-3 -2 -1 0 1 2 3
Σ(ω
)
ω [eV]
lecture by D. Vollhardt
. – p.41/69
Self-energy and spectral function
0
0.1
0.2
0.3
0.4
0.5
0.6
-3 -2 -1 0 1 2 3
A(ω
)
ω [eV]
-2
-1
0
1
-3 -2 -1 0 1 2 3
Σ(ω
)
ω [eV]
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
-4 -2 0 2 4
A(ω
)
ω [eV]
-5
-4
-3
-2
-1
0
1
2
3
-4 -2 0 2 4
Σ(ω
)ω [eV]
(from J.Tomczak, PhD thesis, 2007) . – p.42/69
What can we calculate ?
self-energies
Example: VO insulating phase
-3
-2
-1
0
1
2
3
4
-3 -2 -1 0 1 2 3 4
ω [eV]
Re
Σ (ω
) [e
V]
a1g
a1g-a1g
egπ1
egπ2
-3
-2
-1
0
1
2
3
4
-3 -2 -1 0 1 2 3 4
ω [eV]Im
Σ (
ω)
[eV
]
. – p.43/69
What can we calculate ?
... and k-resolved spectral functions
Example: VO insulating and metallic phases:
0.1
1
ω [e
V]
ΓZCYΓ-2
-1
0
1
2
3
0.1
1
ω [e
V]
ΓZCYΓ-2
-1
0
1
2
3
effective bandstructures (and absence thereof!)
J.Tomczak, F. Aryasetiawan, SB, arXiv:0704.0902 . – p.44/69
What can we calculate ?
phonons
Example:
-Pu
Dai et al., Science 2003
. – p.45/69
What can we calculate ?
local susceptibilities
0.0 0.5 1.0 1.5 2.00.0
0.5
1.0
0.0
0.5
1.0
Ni
Fe
χ-1M
eff2 /3
Tc
M(T
)/M
(0)
T/Tc
A. Lichtenstein, M. Katsnelson, G. Kotliar,
PRL 2001
. – p.46/69
What can we calculate ?
total energy
local spectral functions
k-resolved spectral functions
phonons
local susceptibilities
optical properties
. – p.47/69
Optical conductivityIn the Hubbard model within DMFT
from Rozenberg et al., 1995
d
f f
tr
k
k
k
k
see also: Pruschke et al., Blümer PhD thesis, Oudovenko et al., Haule et al.
. – p.48/69
Optical conductivityThe example of insulating VO :
from J.M. Tomczak, PhD thesis, 2007(for full formalism Poster by J. Tomczak)
0
1
2
3
4
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5
Re
σ(ω
) [1
03 (Ω
cm)-1
]
ω [eV]
Experiment LDA+CDMFT ’upfolded’ E || [100]E || [010]E || [x0z]
. – p.49/69
LDA+DMFT – current statusMany successful applications:
Correlation effects in transition metals
Mott transitions in oxides
Volume collapse transitions in rare earth/actinidesystems
etc ...
Still (at least partly) open questions/challenges:
influence of full charge self-consistency
basis sets lecture by F. Lechermann
what’s U ? (Is there a U? On what energy scale?)
... and nuisances: double counting. – p.50/69
What’s U in a solid?... an answer from RPA:
Divide where = polarization of thecorrelated orbitals (e.g. 3d orbitals)Then:
where that does not include 3d-3d screening:
Identify = !F. Aryasetiawan, M. Imada, A. Georges, G. Kotliar, S.B., A. I. Lichtenstein PRB 70 195104 (2004)
. – p.51/69
U and W in a solidScreened Coulomb interaction from RPA
and
for Nickel:
0 10 20 30 40 50 60
ω (eV)
0
10
20
30
40
Re
<dd|
W|d
d> (e
V)
W Eg
Wr E
g
Paramagnetic Ni
(Aryasetiawanet al., 2004)
. – p.52/69
What’s U in a solid?
... what about an answer beyond RPA ??
. – p.53/69
Can we calculate ...... from a (dynamical) impurity model?
Question of representability !
DMFT: calculated from impurity model
What about ?
Self-consistency requirement:
= of the solid
= of the solid
“GW+DMFT”
S.B., Aryasetiawan, Georges, PRL 2003
condmat
. – p.54/69
The GW approximation(Hedin, 1965)
dynamically screened Coulomb interaction
GW successful for sp-metals, semiconductors ...
(Reviews:Onida et al., Rev. Mod. Phys. 2002;
Aryasetiawan et al., Rep. Prog. Phys. 1998)
. – p.55/69
A functional point of view[Almbladh et al. 1999]
Tr
Tr
Tr
Tr
Free energy
is a functional of
one-electron Green’s function
the screened Coulomb interaction
= bare (Hartree) Green’s function
. – p.56/69
Approximations to
?GW:
Extended DMFT (“E-DMFT”):
E-DMFT for local part + GW for nonlocal part:
NB: “local” = “onsite” is a basis-set dependent notion!
. – p.57/69
Extended DMFT ...... maps a lattice problem
onto a single-site (Anderson impurity) problem
with a dynamical interaction
. – p.58/69
Self-consistency loop
. – p.59/69
Challenges and questions
Global self-consistency?
Choice of orbitals? Hamiltonian?
Treat all orbitals – localized and delocalized – onequal footing? Downfolding?
How to solve the dynamical impurity model?here: static approximation
. – p.60/69
A simplified implementationNon-selfconsistent GW + local from static impuritymodel
Tr
Nonlocal part: correct Hartree by GW
Local part: correct LDA by DMFT
. – p.61/69
Simplified GW+DMFTNi band structure
−8
−6
−4
−2
0
Γ
Ene
rgy
[eV
]
XX Minority Majority
Circles: GW+DMFTDashed: LDATriangles: photoemission data
(Bünemann et al. 2002, Mårtensson et al., 1984)
. – p.62/69
Simplified GW+DMFT:Spectral function of Ni
0
2
4
6
−10 −5 0 50
2
4
6
−10 −5 0 5E−EF
[eV]
ρ(E)
6eV
Majority and minority spins
Satellite at 6eV correct!
. – p.63/69
Conclusion and perspectivesCombination of LDA and DMFT ...
a useful tool for electronic structure calculationsof strongly correlated materials, such as
transition metals, their oxides, f-electron materials
VO (Correlation-induced Peierls transition)
Ce O , a Mott insulator
Prospects ?
technical and physical questions concerningimplementations
calculate more quantities, more materials ... !
Beyond LDA+DMFT ? GW+DMFT
Plenty of work left! . – p.64/69
References of recent work
VO : Correlation-induced Peierls transitionS.B., A. Poteryaev, A. Georges, A. Lichtenstein, Phys. Rev. Lett. 94, 026404 (2005)
J. M. Tomczak, SB, Journal of Phys. Cond. Mat., in press
J.M. Tomczak, F. Aryasetiawan, SB, condmat07040902
Ce O : A Mott insulator – questions ofcharge-selfconsistencyL. Pourovskii, B. Amadon, S. Bierman, A. Georges arXiv:0705.2161
U and GW+DMFT:F. Aryasetiawan, M. Imada, A. Georges, G. Kotliar, S.B., A. I. Lichtenstein PRB 70195104 (2004),
S. Biermann, F. Aryasetiawan, A. Georges, PRL 2003, condmat 2004
LDA+DMFT - recent Reviews:D. Vollhardt, G. Kotliar, Physics Today 2004
S. B., in Encyclop. of Mat. Science. and Technol., Elsevier 2005
. – p.65/69
GW+DMFT: local part
calculated from local impuritymodel:
. – p.66/69
GW+DMFT (contd)Combine local self-energy and polarization
with non-local self-energy and polarization:
. – p.67/69
Self-consistency condition
Update Weiss field and impurity interaction:
Iterate until self-consistency ...
. – p.68/69
NickelEvidence for many-body effects:
Exp. LDA GW
LDA+DMFT
bandwidth 3.3 eV 4.5 eV 3.5 eV OK!x-splitting 0.3 eV 0.6 eV 0.6 eV OK!satellite at 6eV NO! NO! YES!
large quasi-particle widthsOpen question: Fermi surface pocket ?
(1) Aryasetiawan, 1992(2) Lichtenstein et al., 2001
. – p.69/69
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