Post on 05-Jan-2016
1
Recent developments in our Quasi-particle self-consistent GW ( QSGW) method Takao Kotani, tottori-u----- OUTLINE ------1.Theory
Criticize other formalisms. Then I explain QSGW. *Foundation of DF. *Problems in methods, DF(LDA,OEP), one-shot GW. *Some comments *Basic idea for QSGW2. Application Doped LaMnO3 (with H.Kino). * It gives serious doubts for results in LDA(GGA).
3. A new linearized method to calculate one-body eigenfuctions. * PMT= L(APW+MTO) http://pmt.sakura.ne.jp/wiki/
2
1.Theory I will criticize theories below.• Density Functional (DF)
Formalism. It is limited in cases.
Even in OEP (like EXX+RPA, it is limited.
Some comments.• One-shot GW from LDA. Not so good in cases.• Full self-consistent GW (I think), hopeless.
• Quasiparticle Self-consistent GW(QSGW).
Look for the “best one-body part H0”, which reproduces “Quasiparticle”.
We inevitably need some self-consistency How?
1.Theory
3
• Generating functional and the Legendre transformation
Foundation of Density Functional 1
2
One to one correspondence,
( ) can be shown, because 0 (convexity) (1) (2)
Wn J
J J
r r
Then solve
ˆ ˆ[ ] ( )e Tr[e ]W J H Jn
1.Theory
[ ]W
E n W JJ
0E
n
The HK theorem (and so on) made things too complicated…
See http://pmt.sakura.ne.jp/wiki/
•Convex anywhere, even if you add other order parameters.•But E[n] in LDA is really convex?* “finite system infinite system” and “Legendre transformation” are not commute.
4
• Adiabatic connection
Foundation of Density Functional 21.Theory
1
0
0
[ ] [ ]E
E n E n d
0[ ] is non-interacting part.E n
0 1
Long-range
part
Short
-range p
art
• Dynamical case Effective action formalism [n,A,B,…] It is very general; you can derive TDLDA, Fluid dynamics, Rate eq., Dynamical Eliashberg eq…
0 0.3 HF(Instead of [ ], you can use [ ] or so.
It may give a foundation for hybrid functionals...)
E n E n
01
NOTE: Keep n for thecoupling constant α
An another connection path
5
• Problem in DF In the Kohn-Sham construction, it only uses local potential.
Foundation of Density Functional 31.Theory
Onsite non-locality.
No orbital moment. Important for localized electrons.
Offsite non-locality.
A simplest example is H2. Local potential
can hardly distinguish “bonding” and “anti-bonding”.
Required for semiconductor.
My conclusion Even in EXX+RPA or so, it is very limited. For the total energy, “adiabatic connection” is problematic (in cases it needs to connect metal and insulator).A comment: TDLDA is really good? Or it is happened to be good? (too narrow gap +no excitonic effect+ additional reduction by fxc for the Coulomb interaction)
6
GW approximation starting from G0
Start from some non-interacting one-body Hamiltonian H0.
1.
2.
3.
4.
2
0 eff 00
1, '
2H V G
H
r r e,g. H LDA
0 0iG G Polarization function 11
2
1
( , ')'
W v v v
ev
r rr r
W in the RPA
0i G W Self-energy
0G
W
2
, ', + , ',2
H extH V V r r r r r r
0G
0G
G0 n VH also
1 G
H
1.Theory
7
Limitations of “one-shot GW from LDA”
* Before Full-potential GW, people believes “one-shot GW is very accurate to ~0.1 eV”. But, Full-potential GW showed this is not
correct.
* “one-shot G W” is essentially not so good for many correlated systems, e.g. NiO, MnO, …
1.Theory
8
Results from G LDA W LDA Approximation
Bands, magnetic moments in MnAs worse than LDAMany other problems, become
severe when LDA is poor … seePRB B74, 245125 (2006)
If LDA has wrong ordering, e.g. negative gap as in Ge, InN, InSb,
G LDA W LDA cannot undo wrong topology. Result: negative mass conduction band!
Bandgaps too small
Sol. State Comm. 121, 461 (2002).
9
Full self-consistent GW too problematic Start from E[G], which is constructed
in the same manner as E[n]. (There are kinds of functionals, e.g.,
E[ G[Σ[G]] ]).
Difficulty 2. If you use RPA like formula, ,iG G
W and Γare given as a functional of G.
iGW
G
W
1 at q 0, 0
Z
1.Theory
[ ] 10
[ ]
E GG
G G
(renomalization factor) X Thus, you can not set if we use G
Difficulty 1. Z-factor cancellation
(incoherent part)i
i
ZG
This only contains QP weights by ZxZ.This is wrong from the view of independent-particle picture
10
Comment: Replace a part of with some accurate
1.Theory
Onsite Onsite( )GW DMFT GW
RPA Onsite-RPA
Generally speaking, this kinds of procedures (add something and subtract something) can easily destroy analytical properties “Im part>0”, and/or “Positive definite property at ”.
* *
0 0Symbolically, this is
0 0
a b a b
b d b d
Polarization without onsite polarization
For DMFT or so,we need to set up “physically well-defined model”.
self-energy
This can be NOT positive definite at
11
•non-locality is important.
•One-shot GW is not so good
•Full self-consistent GW is hopeless.
•Within GW level.
• Treat all electrons on a same footing.
Quasiparticle self-consistent GW(QSGW) method
How to construct accurate method beyond DF?
We must respect physics; the Landau-Silin’s QP idea.
1.Theory
(but the QP is not necessarily mathematically well-defined.
12
We determine H0 (or ) to describe
“Best quasi-particle picture”. or “Best division H = H0 + (H –H0) “.
Self-consistency
Quasiparticle Self-consistent GW (QSGW)
001HG
xc, ', , '
( ) 0VG G r r r r
B0 ( )
GWG G A
See PRB76 165106(2007)
In (B), we determine Vxc so as to reproduce “QP” in G.
1.Theory
13
Our numerical technique
1. All-electron FP-LMTO (including local orbitals).
(now developing PMT-GW…)
2. Mixed basis expansion for W. it is virtually complete to expand
3. No plasmon pole approximation
4. Calculate from all electrons
5. Careful treatment of 1/q2 divergence in W.FP-GW is developed from an ASA-GW code by F.Aryasetiawan.
1.Theory
A difficulty was in the interpolation of
14
Application of the QSGW
2.Application
Doped LaMnO3. * QSGW gives serious doubt for results in
LDA.
*Spin Wave experiments no agreement. Our conjecture: Magnon-Phonon interaction should be very important.
At first, I show results for others, and then LaMnO3.
15
GaAs
LDA: broken blueQSGW: greenO: Experiment
m* (LDA) = 0.022m* (QSGW) = 0.073m* (expt) = 0.067
Ga d level well described
Gap too large by ~0.3 eVBand dispersions ~0.1 eV
Na
Results of QSGW : sp bonded systems
2.Application
16
Optical Dielectric constant
is universally ~20% smaller than
experiments.“Empirical correction:” scale W by 0.8
LDA gave good agreement because; “too narrow gap”+”no excitonic effect”
QSGW
2.Application
Diagonal line
20%-off line
17
GdN
Scaled LDA + GW (to correct systematic error in QSGW)
Conclusion: GdN is almost at Metal-insulator transition (our calculations suggest 1st-order transtion; so called, Excitonic Insulator).
Scaled
LDA+U
QSGW
QSGW
Scaled
2.Application
Up is red;down is blue
18
NiO
Black:QSGW Red:LDA Blue: e-only
2.Application
19
MnO
Black:QSGW Red:LDA Blue: e-only
20
NiO MnO Dos
Red(bottom): expt
Black:t2g Red:eg
21
NiO MnO dielectric
Black:Im eps Red:expt
2.Application
22
QSGW gives reasonable description for wide range of materials. Even for NiO, MnO
• ~20% too large dielectric function • Corresponding to this fact,
A little too large band gap
A possible empirical correction : LDA
xc xc xc, ' 0.8 , ' 0.2V V V r r r r r
*This is to evaluate errors in QSGW
23
SW calculation on QSGW:J.Phys.C20 (2008) 295214
Effective interaction is determined so at to satisfy, q 0 limit.
2.Application
24
Doped LaMnO3 (J.Phys.C, TK and H.Kino)
* Solovyev and Terakura PRL82,2959(1999)
* Fang, Solovyev and Terakura PRL84,3169(2000)
* Ravindran et al, PRB65 064445 (2002) for Z=57
They concluded that LDA (or GGA) is good enough.
We now re-examine it.
Apply QSGW to La1-xBaxMnO3.
Z=57-X, virtual crystal approx. Simple cubic. No Spin-orbit.
2.Application
25
Z=57-x
t2g are mainly different
eg-O(Pz)One-dimentional
bands
t2g-O(Px,Py)Two-dimentional
bands
Results in the QSGW look reasonable.
26
LDA
QSGW
1eV 1eV
Efermi
Black:QSGWRed:LDA
ARPES experiment*Liu et al: t2g is 1eV deeper than LDA•Chikamatu et al: observed flat dispersion at Efermi-2eVt2g
eg
27
PRB55,4206Im part of dielectric function
28
Spin wave
29
Why is the SW so large in the QSGW?
Lattice constant Empirical correction on QSGW Rhombohedral case Dielectric function
Exchange coupling = eg(Ferro) - t2g(AntiFerro) very huge cancellation
Large t2g - t2g Small AF
They don’t change our conclusion!
30
Thus we have a puzzle. We think we need to includemagnon-phonon interaction (MPI).
Jahn-Teller phonon
Magnon
This is suggested by Dai et al PRB61,9553(2000). But we need much larger MPI than it suggested.
31
Conclusion 2•QSGW works well for wide-range of materials
•Even for NiO and MnO, QSGW’s band picture describes optical and magnetic properties.
•As for LaBaMnO3, QSGW gives serious difference from LDA. The MPI should be very laege.
2.Application
32
APW+MTO (PMT) method
Linear method with Muffin-tin orbital + Augmented Plane wave*Very efficient*Not need to set parameters*Systematic check for convergence.
3.PMT
33
One particle potential V(r)
Electron density n(r)
smooth part + onsite partonsite part = true part –counter part(by Solar and Willams)
Basis set { ( )}iF r
augmented waveHamiltonian ,
Overlap matrix
ij i j
ij i j
H F V F
O F F
Diagonalization ( ) 0H O
smooth part + onsite part
Linear method
iteration
34
Key points in linear method
* Envelop function is augmented within MT.Augmentation by Exact solution at these energies if we use infinite number of APWs.
1 1 2 2 at , and at (or and )
(local orbital exact at ) 1 2 3 , and
1 2
2 21 2
eigenfunction error ( )( )
eigenvalue error ( ) ( )
In practice, ‘too many APW’ causes ‘linear dependency problem’.
35
Good for Na(3s), high energy bands.But not so good for Cu(3d), O(2p)
Systematic.
36
PRB49,17424
Augmentation is very effective
37
Good for localized basis Cu(3d), O(2p).But not for extended states.
Not so systematic.
( ) 0 where e<0e h
38
PMT=MTO+APW
Use MTO and APW as basis set simultaneously.
39
MTO (smooth Hankel)
Hankelre
r
Smooth Hankel
‘Smooth Hankel’ reproduces deep atomic states very well.
3.PMT
40
1. Hellman Feynman force is already implemented(in principle, straightforward) . Second-order correction.
2. Local orbital3. Frozen core 4. Coarse real space mesh for smooth density (charge
density)
41
Result
Use minimum basis; parameters for smooth Hankel aredetermined by atomic calculations in advance.
For example, Cu 4s4p3d + 4d (lo) O 2s2p are for MTO basis.
3.PMT
42
43
44
45
46
47
Conclusion 3*We have developed linearized APW+MTO method (PMT).
*Shortcomings in both methods disappears.
*Very effective to apply to e.g, ‘Cu impurity in bulk Si or SiO2’.
*Flexibility to connect APW and MTO.
* Give reasonable calculations just from crystal structure.
* In feature, our method may be used to set up Wannier functions.
3.PMT
48
------ end -------------