Post on 25-Dec-2015
Merging First-Principles and Model Approaches
Ferdi AryasetiawanResearch Institute for Computational Sciences, AIST,
Tsukuba, Ibaraki 305-8568 – Japan
ISSP 2007.07.25
Collaborators:
Antoine Georges and Silke Biermann (Ecole Polytechnique, France)
Takashi Miyake and Rei Sakuma (RICS-AIST)
Outline
•Motivations: The need to go beyond one-particle picture in correlated materials•Previous works: LDA+U, LDA+DMFT•The GW approximation: success and difficulties•Combining GW and DMFT: A first-principles scheme for correlated materials•A simplified implementation: Application to ferromagnetic nickel
First Part
Second Part (if time permits)
Constrained RPA:Calculating the Hubbard U from first-principles
Fujimori PRL 69, 1796 (1992)
Difficultto treatwithin
one-particletheory
Spectral evolution as a function of U/bandwidth
Georges et al Rev. Mod. Phys.1996
(Dynamical Mean-Field Theory)
Can use LDA
Can use LDA+U
metal
insulator
La/YTiO3
cubicorthorhombic
[1] Solovyev, Hamada, and Terakura, PRB 53, 7158 (1996) (LDA+U: correct magnetic structure of La/YTiO3)[2] Pavarini, Biermann, Poteryaev, Lichtenstein, Georges, and Andersen, PRL 92, 176403 (2004) (LDA+DMFT: consistent description of metal-insulator transition)
These materials sharesimilar electronic structure in LDA
and they are all predicted to be metals
ExperimentallySrVO3 and CaVO3 are metals
LaTiO3 and YTiO3 are insulators
Typical electronic structure of correlated materials:Partially filled narrow 3d or 4f band across the Fermi level
By slight distortion or pressure the ratio of U/bandwidth changes and the materials can undergo, e.g., phase transitions (metal-insulator).
competition between kinetic energy (bandwidth) and U.
The main actiontakes place here
D
U
U=0
U/D>1
Competition between kinetic energy and U(itineracy and localisation)
satellite
QP
LowerHubbard band
UpperHubbard band
U/D>>1
When U is small it is preferable for the electrons to delocalise metal
When U is large it is costly for the electronsto hop localised (Mott insulator)
Electrons prefer to “spread” themselves to lower their kinetic energy.
For intermediate U it is a mixture oflocalised and delocalised electrons.
D
U
i
ij iiij nnUcc
DH
4
U=0
U/D>1
Mapping to a Hubbard model
satellite
QP
LowerHubbard band
UpperHubbard band
U/D>>1
More realistic models take intoaccount the underlying one-electron
band structureLDA+U and LDA+DMFT
To treat systems with strong on-site correlations:Previous works: LDA+U and LDA+DMFT
''
.
'',
'
.
'''
'
.
''
'''',
)(2
1
2
1
miim
correl
im
tingdoublecounmiim
imim
correl
mim
imm
imm
imim
correl
imm
imm
miim
all
im
LDAmiim
ccH
nnJU
nnU
ccHH
Ad-hocdouble-counting
term
Adjustable U
Anisimov et al, J. Phys. Condens. Matter 9, 7359 (1997) (LDA+U, LDA+DMFT)Lichtenstein and Katsnelson, PRB 57, 6884 (1998) (LDA+DMFT)
The Goal
To construct a consistent theoretical scheme that can describethe electronic structure of correlated materials from first-principles.
“Consistent” means that the scheme should be capable ofdescribing the continuous transition from metal to insulator, i.e.,
it can describe both the itinerant and localised characters of the electrons.
One of the main features of correlated electrons is that they haveboth itinerant and localised characters.
First-Principles Methods:•Local Density Approximation (LDA) (ground states)•GW method (excited states)
Model Approaches:•Exact diagonalisation (Lanczos)•Quantum Monte Carlo (QMC)•Dynamical Mean-Field Theory (DMFT)•LDA+U and LDA+DMFT
Can treat strong correlationsbut need adjustable parameters
Insufficient to treatcorrelated materials
Combine First-Principles and Model Approaches: GW+DMFT
Electronic structure ofcorrelated materialsfrom first-principles
spintronicselectronictransport
nanotech.opticaldevices
Possible applications
The GW approximation
Hartree-Fock approximation:
)'()'()(
)'();',()',(
rrvrr
rrvtrriGrrocc
kkk
x
GW approximation:
);',();',();',( trrWtrriGtrrGWxc
- - - - - -v
G
= = = = =
G
W is a screened interaction
Lars Hedin, Phys. Rev. 139, A796 (1965)
vW 1
);',(Re)'()();',(Re *kn
occ
knknknSEX rrWrrrr
'
)';',(Im')'()();',(Re
0
*
knknknknCOH
rrWdrrrr
Screened exchange: from the poles of G
Correlation hole: interaction between an electron and its screening hole
Since W<v, it reduces the Hartree-Fock band gap.
In addition we have another term arising from the poles of W
COHSEXGWxc
0),0;',()'(2
1 knifrrWrr
From Hybertsen and Louie, PRB34, 5390 (1986)
Assessing the GWA using the Polaron Hamiltonian:
)( bbgccbbccH p
iiD
pp
11)(
i
g
DgGdi
p
2
2 )'()'('2
)(
,0
)(!
12)/()(
2
nn
g
n
geA p
n
p
p
pg /2
iG
1)(
Exact solution:
pPlasmons satellites
2
42
pp
QP
ggE
0)(Re
p
p
p
Sat
ggE
22 2
Quasi-particle and satellite energy
Error in QP energy is smallerthan error in satellite energy
i
g
p
2
)(
)(Im)(Re
1)(
iwG
There is no spin dependence in W, only in G: Exchange is taken into account properly, but not correlation between the same spin.
Collective excitations only arise from RPA: (1-Pv)= 0 (plasmons). Satellites arising from local correlations (atomic multiplets) are probably not fully captured.
The Hubbard Hamiltonian cannot be transformed into a polaron-type Hamiltonian for which GW is good.
Usually start from a single Slater determinant: Difficult to treat systems that are inherently dominated by a few Slater determinants.
Self-consistent GW appears to give poor spectra (?)
Some difficulties with the GWA
GW
Band insulator
Mott-Hubbard insulator
Iminfinite
Removes QP and transfers
weight to Hubbard bands
(QP life-time ~ 1/Im)
Imfinite
Shifts and broadens
Quasi-Particle
Band insulator vs Mott-Hubbard insulator
UDMFT
Strongly correlated systems are problematic for LDA: LDA often predicts (anti-ferromagnetic) insulators to be metals.
DMFT: Map the lattice to an impurity embedded in a “bath”.
“bath”
A. Georges et al, Rev. Mod. Phys. 68, 13 (1996)G. Kotliar and D. Vollhardt, Physics Today (March 2004)
Dynamical
Even the GW approximation may not be sufficient.
U
A la Fukuyama
)()()'()'('
nndUccddSeff
-10G
Effective dynamics for an impurity problem
with the dynamical mean-field -10G
bath
)'( 0G
The Coulomb interaction is fully taken into account in one site (impurity), the rest of the sites (medium) is treated as an effective field
Dynamical Mean-Field Theory (DMFT)A. Georges et al, Rev. Mod. Phys. 68, 13 (1996)
)()()( 1 iGii impimp -1
0G
k k impklatticeloc ii
ikGiG)(
1),()(
)()0()()( locSimp GcTcGeff Self-consistency:
effSimp cTcG )0()()( Green’s function
Self-energy
Restore the lattice periodicity
Comparison between GW and DMFT
DMFT(Dynamical Mean-Field Theory)
GW
iGWOnsite, no k-dependence Full k-dependence
Onsite interactions are summed to all orders
Approximate (RPA), good for long-range correlations
Model Hamiltonian
Adjustable parameters
Real systems
No parameters
Can treat systems with strong onsite correlations (3d and 4f)
Good for sp systems but has problems with 3d and 4f systems
Basic physical idea of GW+DMFT
0
R )()( 0000 DMFT
)()( 00 GWRR
The onsite self-energy is taken to be DMFT.
The off-site self-energy is approximated by GW .
P. Sun and G. Kotliar, PRB 66, 85120 (2002)S. Biermann, F.A. and A. Georges, PRL 90, 86402 (2003)
General Framework
Generalized Luttinger-Ward functional*
*Almbladh, von Barth and van Leeuwen, Int. J. Mod. Phys. B 13, 535 (1999)
WPPVW
W
GGG
G
2,0
,0
11
10
1
],[]1/[2
1ln
2
1]1/[ln],[ 0 WGVWtrWtrGGtrGtrWG
][]1/[ln][ 00 GGGtrGtrG
Luttinger-Ward functional
][][]ln[][ 00 GGtrGtrG
Approximation for
)(),(),(),(
)(),(),(),(
iPikPikPikP
iikikik
impk
GWGW
impk
GWGW
PWG
2,
],[],[ '' RRRRsiteonimp
RRRRsiteoffGW WGWG
2
1
2
1],[ trGWGWGGW wwww
Conservingapproximation
Self-Consistency Conditions
impk
loc
kimploc
WkPkVW
GkkGG
11
110
)]()([
)]()([
UUUW
cTcG
impimp
Simp eff
)0()()( 00
The Hubbard U is screened within the impurity model such that the screened U is equal to the local W
k
impimpGW WGk )()(),(
Additionalcondition:
Determines U
11
11
impimp
impimp
impimp
WUP
UUUW
G
G
Self-consistency loop
impq
GWGW
impq
GWGW
PqPkPkP
qkk
)()()(
)()()(
kloc
kloc
kPkVW
kkGG
11
110
)]()([
)]()([
imploc
imploc
PWU
G
11
11G
Impurity: given Weiss field G and U
Combine GW and imp
Check self-consistency: Gloc=Gimp? Wloc=Wimp?
New Weiss field G and U
Simplified GW+DMFT scheme
1
1
)])0(2
1(
)(),(),([)(
onsitexcimp
nimpk
noffsiteGWnHnloc
VTr
iikikGiG
Offsite self-energy: corrected by GW
Onsite self-energy: corrected by DMFT
Non-self-consistent GW + DMFT (static impurity model)
Test: Application to ferromagnetic nickel(Simplified GW+DMFT scheme)
Correlation problems in nickel within LDA:
LDA Exp.
3d band width 4 eV 3 eV
Exchange splitting 0.6 eV 0.3 eV
Satellite none 6 eV
GW
3.2 eV
0.6 eV
Very weak
(if any)
GW+DMFT
(Experimental data fromBuenemann et al, cond-mat/0204142)
majority
minority
majority
minorityexp
----- LDA
Nickel band structure
Too large LDA bandwidth
S. Biermann et al, PRL 90, 86402 (2003)
GW+DMFT density of states of nickel
no 6 eV satellite in LDA
Exchange splitting too large by 0.3 eV in LDA
Next target:
From V. Eyert
metal
insulator
From Miyake
Koethe et al
From Biermann et al, PRL 94, 026404 (2005)
Non-local self-energy is important a good test for GW+DMFT
Single impurity DMFT does notopen up a gap in M1
Preliminary GW result from Sakuma and Miyake
O2p
a_1g
One-shot GW gives too small gap. need DMFT? self-consistency?
Weak satellites
The self-energy around theLDA energy is not linear.
Summary
•GW+DMFT scheme:
-Potentially allows for ab-initio electronic structure calculations of correlated systems with partially filled localized orbitals and describes phase transitions.
-Unlike LDA+DMFT,*The Hubbard U is determined self-consistently*Avoids double counting.
Challenges
•Global self-consistency•Solving the impurity problem with an energy-dependent U•Treat all orbitals on equal footing