Prospects of LDA+DMFT Silke Biermann Centre de Physique Th ´ eorique 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
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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