Applications: LDA+DMFT scheme - I

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Applications: LDA+DMFT scheme - I A. Lichtenstein niversity of Hamburg In collaborations with: A. Poteryaev (Ekaterinburg), M. Rozenberg, S. Biermann, A. Georges (Paris) L. Chioncel (Graz), I.

1 Applications: LDA+DMFT scheme - I A. Lichtenstein niversity of Hamburg In collaborations with: A. Poteryaev (Ekaterinburg), M. Rozenberg, S. Biermann, A. Georges (Paris) L. Chioncel (Graz), I. di Marco, M. Katsnelson (Nijmegen) E. Pavarini (Jülich), O.K. Andersen (Stuttgart) G. Kotliar (Rutgers), S. Savrasov (Devis), A. Rubtsov (Moscow) F. Lechermann, H. Hafermann, T. Wehling, C. Jung, M. Karolak (Hamburg)

2 Outline-I From Atom to Solids: Multiplets in solids Functionals: MFT, DFT, SDF Multiorbital DMFT for General Lattice Matrix version LDA+DMFT Scheme Impurity solvers: CT-QMC Examples of LDA+DMFT

3 3d - 4f open shells materials Strongly Correlated Electron Systems Control parameters Bandwidth (/W) Band filling Dimensionality La 1-x Ca x MnO 3 CMR FM Dopant Concentration x <<W Charge fluct. >>W Spin fluct. Temperature (K) Kondo Mott-Hubbard Heavy Fermions High-Tc SC Spin-charge order Colossal MR I II IIIb IVb Vb VIb VIIb VIIIb Ib IIb III IV V VI VII H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe Cs Ba La* Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn Fr Ra Ac** Rf Db Sg Bh Hs Mt Lanthanides * Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Actinides ** Th Pa Np Pu Am Cm Bk Cf Es Fm Md No Lr Degrees of freedom Charge / Spin Orbital Lattice Nd 2-x Ce x CuO 4 La 2-x Sr x CuO 4 AF AF Pseudogap 'Normal' Metal SC SC Dopant Concentration x

From Atom to Solids - How to incorporate atomic physics in the band structure? - How good is a local approximation? - What is a best solution for atomic problem in effective medium?

4 From Atom to Solids - How to incorporate atomic physics in the band structure? - How good is a local approximation? - What is a best solution for atomic problem in effective medium? - What is different from one band Hubbard model? - How to solve a complicated Quantum multiorbital problem? - What is the best Tight-Binding scheme for realistic Many-Body calculation for solids?

5 Theory of everything: t vs. Multiband Hubbard model (<im jm >=δ ij δ mm ) Coulomb inraatomic interaction Matrix elements of electron-electron interactions: Exact diagonalization of atom: t ij = gives multiplets! Solution with hoppings t ij in solids is unknown!

6 Strong correlations in real systems Real? Multiplets in solids Hub-I =< m m V m m > mm 1 2mm ee 3 4 Local moments above Tc E A.L. and M. Katsnelson, PRB (1998)

7 Hubbard-I approximation

8 Disolving Multiplets in 3d-alkali system PES: Fe on alkali metals C. Carbone (Trieste), O. Rader (BESSY), et al DMFT imp.: 5-band in 1-bath

9 Functionals: MFT- DFT- DMFT G. Kotliar et. al. (22), A. Georges (24) Weiss Mean-Field Theory (MFT) of classical magnets Kohn Density Functional Theory (DFT) of inhomogeneous electron gas in solids Dynamical Mean-Field Theory (DMFT) of strongly correlated electgron systems

10 The Euclidian Action: x=(r,τ,σ) Many-body System

11 Functionals: general consideration The one-electron Green's function Introduction of the source (constraining field) Functional derivative:

12 Functionals: Legendre Transformation The Functionals of Green's function The partition function Z Self-Energy: Constraining field J=Σ the inverse of the exact Green's function

13 Baym-Kadanoff Functional Exact representation of Φ Different Functionals: DFT: G=ρ SDF: G=G(iω) BKF: G=G(k,iω) J=V=V h +V xc J=Σ loc (iω) J=Σ(k,iω)

14 DFT-Density Functional Theory Inhomogeneous electron gas in solids ( =e 2 / r-r ): Energy functional with constrained density <n (r) > =ρ (r) Stationarity in λ insures that: Construct a functional of ρ (r) only:

15 DFT: reference system Non-interacting electrons in effective potential (t=- 2 /2): Minimization with respect to λ (r): Coupling constant trick:

16 DFT - Functional where density-density correlations defined as: Exact relations for density functional: Local density approximation (LDA):

17 Exchange parameters and Functionals

18 Exchange interactions from DFT Heisenberg exchagne: Magnetic torque: Exchange interactions: Spin wave spectrum: Non-collinear magnetism : M. Katsnelson and A. L., Phys. Rev. 61, 896 (2)

19 Hubbard model for correlated electrons H = ij t ij c + iσ c jσ + i n i n i /t Chemical potential t

20 Dynamical Mean Field Theory G ( τ τ ) Σ Σ Σ Σ Σ Σ Σ Σ Σ A.Georges, G.Kotliar, W.Krauth and M.Rozenberg, Rev. Mod. Phys. 68, 13 (1996) G. Kotliar and D. Vollhardt, Physics Today 57, 53 (24)

21 DMFT: Self-Consistent Set of Equations Σ Σ Σ Gˆ i BZ 1 r Ω ( ω ) = Gˆ ( k, i ) n k r ω n Σ Σ ˆ ˆ 1 1 G n ( iω ) = G ( iω ) + Σ( iω ) n ˆ n Σ Σ Σ QMC ED τ τ Quantum Impurity Solver DMRG FLEX IPT G( ττ ) Σˆ new ( ) 1( ) 1 iω = G iω G ( iω ) n ˆ n ˆ n

22 DMFT solution for the Hubbard model H = ij + ( tij + μδij ) ci σcjσ + i n i n i t Hans Bethe METAL Quasi-particle peak Lower HB ~ pper HB INSLATOR Transition from paramagnetic metal to paramagnetic insulator on the Bethe lattice A.Georges, G.Kotliar, W.Krauth and M.Rozenberg, Rev. Mod. Phys. 96

23 Atomic physics From Atom to Solid Bands effects (LDA) N(E) N(E) n d SL> E F d n+ 1 E E F E N(E) LHB QP HB E F E LDA+DMFT

24 LDA+DMFT LDA Models approaches Materials-specific (structure, Z, etc.) Fast code packages Fails for strong correlations Input parameters unknown Computationally expensive Systematic many-body scheme BZ 1 r ( ) [ ˆ( ) ˆ ( ) ˆ ( )] 1 ω = I μ + iω H k Σ iω Gˆ i n Ω H r k ˆ r r ( k ) = Hˆ LDA( k ) ˆ dc n n Multi band Quantum Monte Carlo

25 Flow diagram for the LDA+DMFT approach: σ = G 1 σ, bath G 1 σ, loc Band problem (LDA) Quantum impurity problem G = [ ] 1 G Σ σ, σ σ, loc LDA κ 1 [ G ] QMC, : σ loc G DMFT self-consistency = G 1 1 σ, bath σ, loc + σ

26 Specific features of realistic-dmft Matrix form of multiorbital bath Green functions G mm (ω) General form of the electron-electron interactions mm m m Screened (GW) Coulomb interaction - frequency dependent (ω) Accurate Wannier description of one-electron band structure Complicated solution of LDA+DMFT Quantum-Impurity Model

27 Matrix form of Bath Green Function eg eg g t g t g t mm G m G m G m G m G m m m m m m G ' Simple case of d-orbital GF in the cubic lattice: Genaral case of non-cubic lattice- Matrix form of G mm (ω) Small cluster in DMFT e.g. double Bethe lattice: 1 2 ^ ( ) ( ) ( ) ( ) = G G G G G ω ω ω ω

28 Coulomb vertex The Coulomb interaction matrix for t 2g : J J J J m J J J J m J J J J m J J J J m J J J J m J J J J m m m m m m m ij ' 2 1) ( 2}/ ) / 2 ( ) 3 1)[( ( { J N N J J N N N av = = + + =

29 Multi-band QMC-scheme (Hirsch-Fye) Discrete HS-transformation (Hirsch, 1983) 1 1 exp{ Δτ [ n n ( n + n )]} = exp{ ( )} mm' m m' m m' λ S n n mm' mm' m m' 2 2 S mm ' = ± 1 1 β 2 1 λ mm ' = arccos h exp Δτ mm ' _ Note _ or 2 Δτ = < L W Number of Ising fields: N = M(2M 1) Green Functions: 1 Gmm' ( τ, τ ') = Gmm' ( τ, τ ', S) detg Z mm ' ( τ ) G ( τ, τ ', S) = G ( τ, τ ') V ( τ ) δ δ V σ 1 1 mm' mm ' m mm ' ττ ' ( τ ) = λ S ( τ ) σ m mm ' mm ' mm ' m ' mm' H = t c c + n n + 1, m < m ' = 1, m > m ' S ij σ + i ij iσ jσ mm' mm' m m' 1 τ m L β mm Δτ m τ

30 Sign problem and DMFT Determinant ratio in one-band model: ˆ 1 det[ Gnew] Rσ = = 1 + [1 G (, )](exp( 2 ) 1) ˆ 1 ii ττ V i det[ G ] Green function in arbitrary Ising fields: ˆ ˆ ˆ V' V ˆ V' G' = G+ (1 G)( e 1) G' [1 + (1 G)( e 1)] G = [ 1] G 1 -G β τ β < τ < β < ( τ ) < 1 G ii -1

31 DOS DMFT-SCF Bethe lattice bath-green function: G -1 ( i ω n ) = i ω n + μ t 2 G ( i ω n ) E F Energ y

32 DMFT-SCF Hilbert transform (perovskite lattice): ) ( ) ( ) ( ) ( ) ( ) ( 1 n n n n n i i G i i i z z d N z G ω ω ω ω μ ω ε ε ε Σ + = Σ + = = 1 - G Energ y DOS E F t 2g

33 DMFT-QMC calculation: Sr 2 RuO 4 QMC for 3-orbitals: 15 auxiliary fields 3 sweeps 128 imaginary times T=15 mev Max-Ent for DOS LDA+DMFT, A. Liebsch and A.L., PRL(2)

34 Spectral function ARPES and DMFT ARPES (A. Damascelli, et al PRL2) LDA+DMFT, A. Liebsch,et al PRL(2) DMFT LDA Sr 2 RuO 4 Van Hove=1 mev m * /m=

35 DMFT-SCF LDA+DMFT (orthogonal LMTO-TB): ) ( ˆ ), ˆ ( ) ˆ ( ) ˆ ( ), ( ) ( ) ( ) ( ), ( 1 ' ' ' ' 1 ' α ω α ω ω ω ω μ ω ω α + = = Σ + = i k G i G i k G i G i k H i i k G IBZ k n O n BZ k n LL n LL n DMFT LL LDA LL n n LL h Ene rg y DOS E F sp d dd dd pd d dp pp p d ps s H k +Σ = +Σ H H H H H H H H H s s s s ) ˆ( ) ( ˆ ω r Correlated d-states: FBZ-integration IBZ-integration+symmetrization

36 DMFT for a general lattice Bath Green function for multiband case: from the cavity construction: G ij () is the Green function with eliminated -site: sing the Fourier transform:

37 DMFT for a general lattice: GF GF in DMFT: Here: Taking into account that t(k)= we have: and sing this formulas we obtained: Finally for a bath GF we have: A.L. and M. Katsnelson, PRB (1998)

38 LDA+DMFT: Local Dynamics V. Anisimov, et al. J. Phys. CM 9, 7359 (1997) A. L. and M. Katsnelson PRB, 57, 6884 (1998) LDA+ Static mean-field approximation Energy-independent potential LDA+DMFT Dynamic mean-field approximation Energy-dependent self-energy operator Vˆ = inlm σ mm σ > V σ mm < inl m σ Applications: Insulators with long-range spin-,orbital- and charge order Σˆ( ε) = mm σ short range spin and orbital order inlmσ > Σ( ε) σ mm < inlm σ Applications: Paramagnetic, paraorbital strongly correlated metals Cluster LDA+DMFT approximation A. Poteryaev, A. L., and G. Kotliar, PRL 93, 8641 (24) S. Biermann, A. Poteryaev, A. L., and A. Georges PRL 94, 2644 (25)

39 General Projection formalism for DA+DMFT DELOCALIZED S,P-STATES L> CORRELATED D,F-STATES G> G. Trimarchi et al. arxiv: , JPCM (28) B. Amadon et al. arxiv: , PRB (28)

40 SCF-LDA+DMFT F. Lechermann, et al, PRB (27)

41 LDA+: static mean-filed approximation LDA+ functional: E One-electron energies: ε = = ε + ( n ) i n LDA 2 i i Occupied states: Empty states: n n i E = E + n n - n (n -1) = 1 = LDA i j d d 2 ij 2 ε i =ε LDA ε =ε + i i LDA Mott- Hubbard gap ε LDA ε n LDA d V. Anisimov et al, PRB, 44, 943 (1991), JPCM 9, 767 (1997)

42 Full-potential LDA+: a problem E LDA+ = LDA + - DC = + - No = + - OK! Spherical RI-LDA+ Interchange possible! A.L. et al. Springer Series in Materials Science. Volume. 54 (23) S. Dudarev et. al. PRB 57, 155 (1998)

43 Static limit: LDA+ Rotationally invariant LDA+ functional Local screend Coulomb correlations LDA-double counting term (n σ =Tr(n mm σ ) and n=n +n ): Occupation matrix for correlated electrons:

44 Slater parametrization of Multipole expansion: Coulomb matrix elements in Y lm basis: Angular part 3j symbols Slater integrals:

45 Average interaction: and J Average Coulomb parameter: Average Exchange parameter: For d-electrons: Coulomb and exchange interactions:

46 Orbital degrees of freedom e g orbitals Mn (3+) = 3d 4 Cubic Crystal field splitting 5x d 2x 3x e g t 2g Spins Atomic Hunds rule t 2g orbitals

47 Charge transfer TMO insulators N(E) d n-1 W L p Zaanen-Sawatzky-Allen (ZSA) phase diagram Δ E F n+1 d E W M p-metal V 2 O 5 CuO Charge-Transfer E g ~ Δ Insulator LaMnO 3 V 2 O 3 NiO FeO d-metal Mott-Hubbard E g ~ TiO (W M +W L )/2 Δ

In KCuF 3 Cu +2 ion has d 9 configuration Experimental crystal structure antiferro-orbital order Orbital order: KCuF 3 with a single hole in e g doubly

48 In KCuF 3 Cu +2 ion has d 9 configuration Experimental crystal structure antiferro-orbital order Orbital order: KCuF 3 with a single hole in e g doubly degenerate subshell. LDA+ calculations for undistorted perovskite structure hole density of the same symmetry A.L. et al, Phys. Rev.B 52, R5467 (1995);

49 Electronic structure of TMO: LDA+ Density of States (states/ev formula unit) LSDA = 5e V = 9e V = 13eV MnO Energy (e V) 3d MnO O 2p DOS 3d LSDA = 5e V = 9e V = 13eV NiO NiO Ene rg y (e V) w(q), mev Spin-wave Spectrum NiO I. Solovyev G Z F G L = 13 LDA exp

50 Constrained LDA calculation of and J Gunnarsson-1989 supercell with cutting hybridisation Norman-1995 estimation of screening parameter

52 Constrain GW calculations of F. Aryasetiawanan et al PRB(24)

53 Wannier - GW and effective (ω) T. Miyake and F. Aryasetiawan Phys. Rev. B 77, (28) C-GW GW

54 Continuous Time QMC formalism: (ω) Partition function and action for fermionic system with pair interactions Z = Tr( Te S ) S = t c c drdr + w c c c c drdr dr dr r' 1r' 2 r1 r2 ' ' ' r' + r + + r r' rr r' r' r = {,,} τ i s dr Splitting of the action into Gaussian part and interaction ' ' ' ' ( ) β = dτ i s S = S + W ( r' r r r r r α ' ) r S = tr + r' w 2 rr + w 1 2 r2r dr 1 2dr 2 cr' c drdr + + ( α )( α ) r' 1r' 2 r1 r1 r2 r2 W = wrr c ' 12 r c 1 r c 1 r c 2 r dr dr dr dr 2 ' ' ' ' ' ' α -- additional parameters, which are necessary to minimize the sign problem r r ' A. Rubtsov et al Phys Rev B 72, (25)

55 Continuous Time QMC formalism Formal perturbation-series: Z = dr dr'... dr dr' Ω ( r, r',..., r, r' ) 1 1 2k 2k k 1 1 2k 2k k= k ( 1) r' r' r' r' r... r k( r1, r' 1,..., r2k, r' 2k) Z wrr... wr r Dr'... r' Ω = k! 1 2 2k1 2k 1 2k 12 2k12k 1 2k + r1 r1 + ( α )...( α ) D = T c c c c r... r r r r r r r r r 1 2k 2k 2k '... 1 ' 2k ' 1 ' 1 ' 2k ' 2k Since S is Gaussian one can apply the Wick theorem The Green function can be calculated as follows In practice efficient calculation of a ratio is possible due to fast-update formulas g D can be presented as a determinant g r r1 r1 r2k r2k ( α )...( α ) 1 1 2k 2k + r1 r1 + 2k 2k ( r' αr' )...( r' αr' ) Tc c c c c c r r' r' r' r' r' r' ( k) = r r T c c c c 1 1 2k 2k ratio of determinants A. Rubtsov and A.L., JETP Lett. 8, 61 (24)

56 Random walks in the k space Z= Z k-1 + Z k + Z k+1 +. decrease k-1 k+1 Acceptance ratio increase Step k-1 k D w D k1 k Distribution Step k+1 w k + 1 D D k+ 1 k Maximum at βn 2 k

57 CT-QMC: fast update k -> k+1 N 2 operations

58 Metal-Insulator transition: Bethe lattice =3 = Density of states for β=64: =2; =2.2; =2.4; =3 G(iω) DOS =2 = Energy 4 3 coexistence of the metallic and insulating solutions: =2.4, β=64, W=2 Σ(iω) G(iω) iω iω iω

59 Dynamical screening in Hubbard model ( ω) = V 2 Ω ω + Ω 2 2 () τ = δτ () + V() τ Compare to Exact solution G ( iω) ( iω μ) = =3, V=2, Ω=1, β=8.4 =2, V=.7, Ω=4, β=4.3.3 G(τ).2 G(τ) τ τ

60 Co on Cu: 5d-orbitals CT-QMC calculation CT-QMC G(τ) LDA τ DOS for Co atom in Cu =4, b = 1 (T ~ 1/4 W) E. Gorelov et al, to be published

Advantages of the CT-QMC method non-local in time

interactions: multi-band systems, E-DMFT Auxiliary field

introduce large number of auxiliary fields, while CT-QMC

auxiliary spins in the Hirsch scheme Short-range interactions

61 Advantages of the CT-QMC method non-local in time interactions: dynamical Coulomb screening non-local in space interactions: multi-band systems, E-DMFT Auxiliary field (Hirsch) algorithm is time-consuming since it s necessary to introduce large number of auxiliary fields, while CT-QMC scheme needs almost the same time as in local case Number of auxiliary spins in the Hirsch scheme Short-range interactions Local in time interactions Long-range interactions Non-local in time interactions

62 Miracle of continuous time QMC Path Integral: Weak coupling expansion (A. Rubtsov et al) Strong coupling expansion (P. Werner et al)

63 Comparison of different CT-QMC Σ Σ Σ Σ Σ Σ Σ Σ τ G( ττ ) τ

64 Double counting in LDA+DMFT Analytic models Around mean field Fully localized limit Constraint on particle number Tr G Tr G! = Tr! = Tr G G LDA Constraint on self-energy! Tr Re Σ ( ) = Tr Re Σ! ( ) =

65 Choice of double counting in LDA+DMFT Shift of chemical potential for correlated state G 1 ( k, ) ( i ) H ( k) [ E ] [ ( ) ] ω = ω + μ LDA + dc δμc Σ ω δμc Natural choice Edc = δμc : G 1 ( k, ω) = ( iω+ μ) H ( k) Σc( ω) G Transformations: 1 1 c G = δμ LDA 1 c( ) c 1 c( ) c Σ ω = G G =Σ ω δμ Condition for δμ c (Friedel SR) Tr G [ ] = Tr[ G] c G(τ). -.5 Ni Ferro E g -up G τ G

66 DC-test for LDA+DMFT: NiO LDA part includes entire Hilbert space H LDA = ε c + c k k k k Coulomb interaction acting on correlated subspace only 1 σσ' 1 1 Hint = ij niσ nj σ' 2 ijσ 2 2 Double counting H DC = μ D iσ n i σ

67 NiO a charge transfer system LDA band structure (paramagnetic) Ni-3d orbitals as correlated subspace O-2d orbitals as uncorrelated subspace

68 NiO double counting (ev) (ev) Total particle number (color encoded) as function of chemical potential μ and double counting μ DC

69 Peak positions and spectral weights I Fitting of Green functions to DOS with 3 δ-peaks at ε i with spectral weight Z i ( ) τ = Z e ε τ [ Θ( τ ) n ( )] G i ε i i F i

70 Peak positions and spectral weights II Tr AMF Re Σ! ( ) = FLL Tr Tr! G = Tr! G = Tr G G LDA Energy of peaks in spectral function of Ni-3d and O-2p orbitals as function of double counting. Line thicknesses correspond spectral weight of each peak.

71 Spectral functions and double counting Mott insulator μ DC = 21eV Charge transfer insulator Almost metallic μ DC 26eV

72 Itinerant ferromagnetism Stoner Heisenberg Spin-fluctuation T= T<T c T>T c

73 Magnetism of metals: LDA+DMFT Exchange interactions in metals Finite temperature 3d-metal magnetism 1,2 1, M(T) and χ(t): LDA+DMFT 1,2 1, M(T)/M() Global spin flip,8,6,4 M(T) Ni Fe χ(t),8,6,4 χ -1 M eff 2 /3Tc,2,2,,,2,4,6,8 1, 1,2 1,4 1,6 1,8 2, 2,2, T/T c A. L., M. Katsnelson and G. Kotliar, PRL 87, 6725 (21)

74 Ferromagnetism of transiton metals Ferromagnetic Ni DMFT vs. LSDA: M(T)/M() LDA+DMFT with ME J. Braun, et al PRL (26) 1,2 1,,8,6,4,2 M(T) M(T) and χ(t): LDA+DMFT Ni Fe,,,,2,4,6,8 1, 1,2 1,4 1,6 1,8 2, 2,2 T/T c χ(t) 3% band narrowing 5% spin-splitting reduction -6 ev sattelite 1,2 1,,8 χ -1 M eff 2 /3Tc,6,4,2 Density of states, ev <S(τ)S()> PES Ni: LDA+DMFT (T=.9 Tc) τ, ev -1 DMFT LDA Energy, ev E F A. L, M. Katsnelson and G. Kotliar, PRL (21)

75 Ferromagnetic Iron: Spectral Function

76 Realistic DMFT for Co(111) surface First model surface DMFT M.Potthoff (1999) FPLMTO+DMFT: Igor di Marco PRB (27)

77 Orthorhombic 3d 1 Perovskites SrVO 3 Metal m*/m=2.7 CaVO 3 Metal m*/m=3.6 LaTi TiO 3 YTiO 3 Insulator Gap=.2eV Insulator Gap=1.eV

78 LDA-NMTO results: DOS all metallic in LDA Crystal-field splittings w/in t 2g multiplet: (14,2) mev for LaTiO3 ; (2,33) mev for YTiO3

79 Orbital ordering in MTiO 3 La Violet: Oxygen Orange: M YTiO3 Occup. LDA DMFT LaTiO Y YTiO E. Pavarini et al. PRL (24)

80 LDA+DMFT: DMFT: comparison with experiments SrVO 3 CaVO 3 m*/m=2.2 Exp=2.7 LDA + DMFT m*/m=3.5 Exp=3.6 LaTiO 3 YTiO 3 Exp.2 Exp 1.

81 Conclusions LDA+DMFT is a perfect scheme for realistic description of electronic structure of correlated electron materials Matrix DMFT formalism can be used for general lattice or cluster compounds

Introduction to SDFunctional and C-DMFT

Introduction to SDFunctional and C-DMFT A. Lichtenstein University of Hamburg In collaborations with: M. Katsnelson, V. Savkin, L. Chioncel, L. Pourovskii (Nijmegen) A. Poteryaev, S. Biermann, M. Rozenberg,