Transcript 1.Theory

Recent developments in our
Quasi-particle self-consistent GW（QSGW) method
Takao Kotani, tottori-u
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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 QSGW
2. 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/
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1.Theory
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?
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1.Theory
Foundation of Density Functional 1
The HK theorem (and so on) made things too complicated…
• Generating functional and the Legendre transformation
 W [ J ]
  ( Hˆ  Jnˆ )
e
 Tr[e
W
E[ n ]  W  J
J
]
Then solve
E
0
n
One to one correspondence,
 2W
n(r )  J  r  can be shown, because
 0 (convexity)
 J (1) J (2)
•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.
See http://pmt.sakura.ne.jp/wiki/
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1.Theory
Foundation of Density Functional 2
• Adiabatic connection
E
E[n]  E0 [n]   d 

0
E0 [n] is non-interacting part.
1
1
0
NOTE: Keep n for the
coupling constant α
0
1
(Instead of E0 [n], you can use E0.3HF [n] or so.
It may give a foundation for hybrid functionals...)
An another connection path
• Dynamical case  Effective action formalism G[n,A,B,…]
It is very general; you can derive
TDLDA, Fluid dynamics, Rate eq.,
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Dynamical Eliashberg eq…
1.Theory
Foundation of Density Functional 3
• Problem in DF
In the Kohn-Sham construction, it only uses local potential.
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
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reduction by fxc for the Coulomb interaction)
1.Theory
GW approximation starting from G0
Start from some non-interacting one-body Hamiltonian H0.

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1. H 0    Veff  r, r '  G0 
2
  H0
2
2.   i G0  G0
3.
G0
Polarization function
W   v  1   v  v
1
e,g.  H LDA
1
2
e
v(r, r ') 
r r'
W in the RPA
4.   i G0W Self-energy
G0  n VH also
G0
W
G0
2
H  r, r ',     V H  r   V ext  r  +  r, r ', 
2
 G 
1
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H
1.Theory
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, …
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Results from G LDA W LDA Approximation
Bandgaps too small
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!
Sol. State Comm.
121, 461 (2002).
Bands, magnetic
moments in MnAs
worse than LDA
Many other problems, become severe
when LDA is poor … see
PRB B74, 245125 (2006)
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1.Theory
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]] ]).
 E[G]
1
0  G
G
  [G]
  i GW G
W and Γare given as a functional of G.
W
G
G
Difficulty 1. Z-factor cancellation
Zi
1
G
 (incoherent part)
G  at q  0,  0
  i
Z
(renomalization factor) X G  1
Thus, you can not set G1 if we use G
Difficulty 2. If you use RPA like formula,   i G  G,
This only contains QP weights by ZxZ.
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This is wrong from the view of independent-particle picture
1.Theory
Comment: Replace a part of  with some
accurate ?
self-energy
  GW  (OnsiteDMFT  OnsiteGW )
Polarization without onsite polarization
   RPA   Onsite-RPA
 a b   a 0  0 b 
Symbolically, this is    *   
 * 

b d   0 0   b d 
This can be NOT positive definite at 0 !
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”.
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For DMFT or so,we need to set up “physically well-defined model”.
1.Theory
How to construct accurate method beyond DF?
•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
We must respect physics; the Landau-Silin’s QP idea.
(but the QP is not necessarily mathematically well-defined.
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1.Theory
Quasiparticle Self-consistent GW (QSGW)
  1H 0 ) to describe
“Best quasi-particle picture”.
or “Best division H = H0 + (H –H0) “.
 We determine H0 (or G0
 Self-consistency
GW
 r ,r ', Vxc  r ,r '
( A)
(B )
G0 
G
G G0
See PRB76 165106(2007)
In (B), we determine Vxc so as to reproduce “QP” in G.
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1.Theory
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 Y Y.
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.
A difficulty was in the interpolation of ...
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2.Application
Application of the QSGW
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.
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2.Application
Results of QSGW : sp bonded systems
LDA: broken blue
QSGW: green
O: Experiment
Na
GaAs
m* (LDA) = 0.022
m* (QSGW) = 0.073
m* (expt) = 0.067
Gap too large by ~0.3 eV
Band dispersions ~0.1 eV
Ga d level well described
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2.Application
Optical Dielectric constant 
QSGW
Diagonal line
20%-off line
 is universally ~20% smaller than
experiments.
“Empirical correction:” scale W by 0.8
LDA gave good agreement because; “too narrow gap”+”no excitonic effect”
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2.Application
LDA+U
GdN
QSGW
Scaled 
QSGW
Scaled 
Up is red;down is blue
Scaled   0.2LDA＋0.8GW (to correct systematic error in QSGW)
Conclusion: GdN is almost at Metal-insulator transition
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(our calculations suggest 1st-order transtion; so called, Excitonic Insulator).
2.Application
NiO
Black:QSGW Red:LDA Blue: e-only
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MnO
Black:QSGW Red:LDA Blue: e-only
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Black:t2g Red:eg
NiO MnO Dos
Red(bottom): expt
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2.Application
NiO MnO dielectric
Black:Im eps
Red:expt
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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 :
Vxc  r, r '  0.8 Vxc r, r '  0.2 V LDA xc r 
*This is to evaluate errors in QSGW
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2.Application
SW calculation on QSGW:
J.Phys.C20 (2008) 295214
Effective interaction is determined so at to satisfy, q 0 limit.23
2.Application
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.
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Z=57-x
eg-O(Pz)
One-dimentional bands
t2g-O(Px,Py)
Two-dimentional bands
Results in the QSGW look
reasonable.
t2g are mainly different
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Black:QSGW
Red:LDA
1eV
1eV
LDA
QSGW
t2g
eg
Efermi
ARPES experiment
*Liu et al: t2g is 1eV deeper
than LDA
•Chikamatu et al: observed
flat dispersion at Efermi-2eV
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Im part of dielectric function
PRB55,4206
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Spin wave
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Why is the SW so large in the QSGW?
Exchange coupling =
eg(Ferro) - t2g(AntiFerro)
very huge cancellation
Large t2g - t2g




Small AF
Lattice constant
Empirical correction on QSGW
Rhombohedral case
Dielectric function
They don’t change our conclusion!
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Thus we have a puzzle.
We think we need to include
magnon-phonon interaction (MPI).
Magnon
Jahn-Teller phonon
This is suggested by Dai et al PRB61,9553(2000).
But we need much larger MPI than it suggested.
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2.Application
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.
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3.PMT
APW+MTO (PMT) method
Linear method with
Muffin-tin orbital +
Augmented Plane wave
*Very efficient
*Not need to set parameters
*Systematic check for convergence.
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Linear method
One particle potential V(r)
smooth part + onsite part
onsite part = true part –counter part
(by Solar and Willams)
Basis set {Fi (r)}
augmented wave
Hamiltonian H ij  Fi    V Fj ,
Overlap matrix Oij  Fi Fj
Diagonalization (H   O)  0
iteration
Electron density n(r)
smooth part + onsite part
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Key points in linear method
* Envelop function is augmented within MT.
Augmentation by 1 at 1 , and 2 at  2 (or  and  )
 Exact solution at these energies if we use infinite number of APWs.
eigenfunction error  (  1 )(   2 )
eigenvalue error  (  1 ) 2 (   2 )2
(local orbital  exact at 1 ,  2 and  3 )
In practice, ‘too many APW’ causes ‘linear dependency problem’.
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Good for Na(3s), high energy bands.
But not so good for Cu(3d), O(2p)
Systematic.
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Augmentation is very effective
PRB49,17424
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(  e)h  0 where e<0
Good for localized basis Cu(3d), O(2p).
But not for extended states.
Not so systematic.
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PMT=MTO+APW
Use MTO and APW as basis set simultaneously.
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3.PMT
MTO (smooth Hankel)
e  r
Hankel 
r
Smooth Hankel
‘Smooth Hankel’ reproduces deep atomic states very well.39
1. Hellman Feynman force is already implemented(in
principle, straightforward) . Second-order correction.
2. Local orbital
3. Frozen core
4. Coarse real space mesh for smooth density (charge
density)
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3.PMT
Result
Use minimum basis; parameters for smooth Hankel are
determined by atomic calculations in advance.
For example,
Cu 4s4p3d + 4d (lo)
O 2s2p
are for MTO basis.
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3.PMT
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.
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