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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/

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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

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• 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.

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• 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

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• 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)

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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

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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

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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).

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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

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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

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•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.

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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

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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

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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.

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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

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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

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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

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NiO

Black:QSGW Red:LDA Blue: e-only

2.Application

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MnO

Black:QSGW Red:LDA Blue: e-only

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NiO MnO Dos

Red(bottom): expt

Black:t2g Red:eg

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NiO MnO dielectric

Black:Im eps Red:expt

2.Application

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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 : LDA

xc xc xc, ' 0.8 , ' 0.2V V V r r r r r

*This is to evaluate errors in QSGW

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SW calculation on QSGW:J.Phys.C20 (2008) 295214

Effective interaction is determined so at to satisfy, q 0 limit.

2.Application

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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

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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.

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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

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PRB55,4206Im part of dielectric function

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Spin wave

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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!

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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.

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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

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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

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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

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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’.

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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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PRB49,17424

Augmentation is very effective

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Good for localized basis Cu(3d), O(2p).But not for extended states.

Not so systematic.

( ) 0 where e<0e h

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PMT=MTO+APW

Use MTO and APW as basis set simultaneously.

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MTO (smooth Hankel)

Hankelre

r

Smooth Hankel

‘Smooth Hankel’ reproduces deep atomic states very well.

3.PMT

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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)

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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

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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

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