1 Recent developments in our Quasi-particle self-consistent GW （ QSGW) method Takao Kotani, tottori-u OUTLINE 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)

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 H 0 ”, which reproduces “Quasiparticle”. We inevitably need some self-consistency  How? 1.Theory

3 Generating functional and the Legendre transformation Foundation of Density Functional 1 Then solve 1.Theory The HK theorem (and so on) made things too complicated… See 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 2 1.Theory 0 1 Long-range part Short-range part Dynamical case  Effective action formalism  [n,A,B,…] It is very general; you can derive TDLDA, Fluid dynamics, Rate eq., Dynamical Eliashberg eq… 0 1 NOTE: Keep n for the coupling constant α An another connection path

5 Problem in DF In the Kohn-Sham construction, it only uses local potential. Foundation of Density Functional 3 1.Theory Onsite non-locality. No orbital moment. Important for localized electrons. Offsite non-locality. A simplest example is H 2.  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 G 0 Start from some non-interacting one-body Hamiltonian H e,g.  H LDA Polarization function W in the RPA Self-energy G 0  n  V H also 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 LDA Many other problems, become severe when LDA is poor … see PRB B74, (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]] ]). W and Γare given as a functional of G. 1.Theory (renomalization factor) X  Thus, you can not set  if we use G Difficulty 1. Z-factor cancellation 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 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  ”. 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 H 0 (or ) to describe “Best quasi-particle picture”. or “Best division H = H 0 + (H –H 0 ) “.  Self-consistency Quasiparticle Self-consistent GW (QSGW) See PRB (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/ q 2 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 blue QSGW: green O : Experiment m * (LDA) = m * (QSGW) = m * (expt) = Ga d level well described Gap too large by ~0.3 eV Band 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 1 st- order transtion; so called, Excitonic Insulator). Scaled  LDA+U 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 : *This is to evaluate errors in QSGW

23 SW calculation on QSGW: J.Phys.C20 (2008) 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, PRB (2002) for Z=57  They concluded that LDA (or GGA) is good enough. We now re-examine it. Apply QSGW to La 1-x Ba x MnO 3. 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 E fermi Black:QSGW Red:LDA ARPES experiment *Liu et al: t2g is 1eV deeper than LDA Chikamatu et al: observed flat dispersion at E fermi -2eV t2g eg

27 PRB55,4206 Im 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 - t2gSmall AF They don’t change our conclusion!

30 Thus we have a puzzle. We think we need to include magnon-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 LaBaMnO 3, 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 part onsite part = true part –counter part (by Solar and Willams) augmented wave 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. (local orbital  exact at ) 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.

38 PMT=MTO+APW Use MTO and APW as basis set simultaneously.

39 MTO (smooth Hankel) 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 orbital 3.Frozen core 4.Coarse real space mesh for smooth density (charge density)

41 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. 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

end