Today: Thursday, April 30. Examples of applications of the Baym-Kadanoff theory for the polarization propagator: Approximations for the self-energy:

Transcription

1 Thursday, April 30 Today: Examples of applications of the Baym-Kadanoff theory for the polarization propagator: Free propagator Random Phase Approximation Extended (2p2h) RPA Approximations for the self-energy: GW method Faddeev RPA equations

2 Response and self-consistent equations One finds the Bethe-Salpeter equation: α β Diagramatically: L = α β + α μ κ K ph L β ν λ γ δ γ δ γ δ

3 Response and self-consistent equations Finally, one finds: single-particle motion in medium p-h interaction excitation of the medium N.B. These equations are all exact! interaction of a particle with medium Note: L. Hedin [Phys. Rev. 139, A796 (1965)] has derived an analogous set of (five) equations, which put in evidence the polarizability and dielectric matrix. This formulation leads to the GW approximation. It is better known in solid state.

4 Response and self-consistent equations in medium p-h interaction Bethe-Salpeter equation Conservation laws: if is a conserving approximation of the self-energy in the sense of Baym-Kadanoff (lecture n.3, Apr.13) then also derived from the above equations, satisfies the symmetries of the Hamiltonian (for the excited states). This approximation scheme requires self-consistency [G. Baym and L. P. Kadanoff, Phys. Rev. 124, 287 (1961); G. Baym, Phys. Rev. 127, 1391 (1962)]

5 Approximations for the polarization propagator in medium p-h interaction Bethe-Salpeter equation Possible self-energies: free p-h propagator random phase approximation (RPA) extended (2p2h) RPA ( ERPA )

6 Approximations for the polarization propagator =0 means no residual interaction at all. Particles do not interact with each other (not even at mean-field level!) and the system is in its unperturbed state (a Slater determinant). The excited states are described by H 0 and the polarization propagator is constructed from the unperturbed : α β γ δ

7 Lindhard function Free Fermi gas The polarization propagator of the free Fermi gas is: This contains no interactions among particles but is it already good enough to get an idea of correlations due to the Pauli principle. The response to excitations is: 1 k F k i See what happens when transferring a momentum q

8 Lindhard function Free Fermi gas The operator for transferring a momentum hq is with matrix elements: Thus, with Lindhard function

9 Lindhard function Free Fermi gas The range of allowed excitation energies stays on a ridge around (hq) 2 /2m and spread proportionally to k F : for For a free Fermi gas: ω q

10 Lindhard function Free Fermi gas q<2k F : q>2k F : q q for k F k F After calculating the integral:

11 Lindhard function Free Fermi gas The range of allowed excitation energies stays on a ridge around (hq) 2 /2m and is spread proportionally to k F : ω [Pic. source: Dickhoff & Van Neck, Many-vody Theory Exposed! ] q

12 Response function in Nuclear Physics This is a schematic plot of the response function for nuclear systems: Note: Q 2 q 2 - ω 2 real photons are on-shell: Q 2 =0 electrons: Q 2 0 [Source: S. Boffi et. al. Electromagnetic Response of Atomic Nuclei. Oxford Univ. Press, 1996.]

13 Approximations for the polarization propagator Random phase approximation (RPA) Particles interact with each other but only at the mean-field level (1 st order): this leads to Hartree-Fock single particle orbits. The corresponding p-h interaction is also a 1 st order: α β These are the RPA equations γ δ

14 Approximations for the polarization propagator Ring expansion of RPA The RPA equations correspond to the following series of diagrams: TDA RPA Pauli correlations are partially neglected, but one assumes (=hopes) that the missing corrections cancel each other randomly ( RPA)

15 Approximations for the polarization propagator Beware: there are two definitions of RPA! 1) V contains only the direct term: α β γ δ 2) V has both direct and exchange terms: Hartree potential Hartree-Fock potential γ δ α β α γ β δ N.B.:This is sometimes called Generalized RPA (GRPA), e.g. in atomic physics, and other times simply RPA (in nuclear physics).

16 Approximations for the polarization propagator Use RPA to evaluate the electron-electron interaction in the medium: W = v = + Lindhard function П (0) (ω) Then: the in-medium interaction depends on energy!

17 Approximations for the polarization propagator Use RPA to evaluate the electron-electron interaction in the medium: The bare interaction is: with: In this case, this is NOT antisymmetrized it depends only on the momentum transferred in the t channel. The Lindhard function is: RPA eq.: And in the limit of small momenta: The coulomb interaction in the electron gas is screened at long distances and behave like a Yukawa force!!!

18 One-hole spectral function -- example independent particle picture σ red S (h) correlations p 1/2 0p 3/2 0s 1/2 E m [MeV] Saclay data for 16 O(e,e p) [Mougey et al., Nucl. Phys. A335, 35 (1980)] S ( h) A 1 A 2 ( pm, Em) = Ψn c Ψ p 0 m n distribution of momentum (p m ) and energies (E m ) δ ( E m ( E A 0 E A 1 n ))

19 Dressing RPA: RPA ph states described in terms of MF orbits Dressed RPA account for the spectral distribution

20 Dressing RPA: The DRPA equations are: account for the spectral distribution However: implies: but NOT: The missing term is a Pauli exchange contribution and a phonon mediated interaction

21 Approximations for the polarization propagator v becomes energy dependent and non-trivial Deriving with respect to means cutting lines in its (self-consistent) diagram representation: Intermediate 2p2h states

22 Approximations for the polarization propagator The Extended RPA (ERPA) equation has an energy dependent Kernel, that describes intermediate 2p2h propagation:

23 ERPA Calculations for 48 Ca Excitation spectrum of 48 Ca Transition densities: [Brand et al. Nucl. Phys. A509, 1 (1990)]

24 RPA/ERPA Calculations for 48 Ca RPA(1p1h) ERPA(2p2h) GQR GDR Gamow -Teller [Brand et al. Nucl. Phys. A509, 1 (1990)]

25 Quenching of GT and 2p2h configurations measured GT strength in 90 Zr compared to models with explicit 2p2h R ± GT(ω) [MeV -1 ] quenching problem: strength seen above the GT giant resonance ERPA calcs. other nuclei? ω [MeV] M. Ichimura et al., Prog. Part. Nucl. Phys 56, 446 (2006)

26 Baym-Kadanoff: hierarchy of conserving approx. Σ K (ph ) self-consistent RPA Extended RPA [Brand et al., Nucl. Phys. A509, 1 (90)] Baym-Kadanoff theory [PRC124, ( 61) 287; PRC137, ( 62) 1379] is used as GUIDANCE!! Two-phonon BSE [Phys. Rev. C86, (2003)]

27 One- and two-phonons in 16 O (anharmonicity) p-h state GMR splitting of (0 +,2 +,4 +,6 + ) multiplet about right (but yet wrong ordering)

28 Anharmonicity effects? Pauli exch. Vpp Π (ph) Π (ph) Π (ph) Π (ph)

29 One- and two-phonons in 16 O (anharmonicity) p-h state GMR splitting of (0 +,2 +,4 +,6 + ) multiplet about right (but yet wrong ordering)

30 Solving (D)RPA and ERPA in discrete bases Consider the most general case of a dressed (== fragmented) propagator, which lead to the Dressed RPA equation. This will be the case in a fully self-consistent calculation. The Lehmann representations for the single particle and the free particle-hole propagators are, α β γ δ With the notation:

31 Solving (D)RPA and ERPA in discrete bases The full polarization propagator is: α П(ω) β γ δ with: The (D)RPA equation and the normalization for the residues are obtained in the same way as for Dyson Eq. case. The eigenvalue Eq. is obtained by extracting the residues of each single solution:

32 Solving (D)RPA and ERPA in discrete bases There exist two forms of RPA equation (with the same solution) The diagrammatic forms of the above equations are, α β П(ω) γ δ α β = + γ δ П(ω) γ δ α β α β α β П(ω) = + γ γ δ δ γ δ α П(ω) β

33 Solving (D)RPA and ERPA in discrete bases There exist two forms of RPA equation (with the same solution) Expand in series around the pole : Normalization condition:

34 Solving (D)RPA and ERPA in discrete bases Cast DRPA into matrix form: The eigenvalue equation is: Define the vectors X and Y as:

35 Solving (D)RPA and ERPA in discrete bases Cast DRPA into matrix form: After some algebra Eigenvalue eq.: Normalization: Closure:

36 Solving (D)RPA and ERPA in discrete bases Two main ways to solve the Extended RPA (ERPA) equation: 1. By matrix inversion: Need to do a matrix inversion for each value of ω Chose a finite width to spread the strength: Г is finite If many poles are needed, then it requires inversions wit a small width Г and at many ω. Better matrix diagonalization. 1. One single diagonalization (as seen before ): Get everything at once, but a HUGE dimension for the eigenvalue problem

37 Solving (D)RPA and ERPA in discrete bases The Extended RPA (ERPA) equation has an energy dependent Kernel, that describes intermediate 2p2h propagation:

38 Solving (D)RPA and ERPA in discrete bases The Extended RPA (ERPA) equation has an energy dependent Kernel, that describes intermediate 2p2h propagation: Take the forward contribution, as an example: Then:

39 Part II: approximations for the self-energy GW approximation Faddeev Random Phase Approximation (FRPA)

40 Approximations for the Self-energy Diagrams of some common approximations for the self- energy: -2 nd -ph rings -pp ladders GW approach; particle-vibration coupling Pairing-like effects; Used for nuclear matter Keep in mind that in certain cases these two interfere and should not be used separately from each other -R 2p1h This is the most complete, including both pp and ph correlations and their interference

41 The GW method Consider the self-energy of the uniform electron gas and use only direct matrix elements of V (i.e. not antisymmetrized). The Hertree term is k k The Fock contribution k-q k k q The Hartree correlations simply give the electrostatic repulsion which is a constant term, so we only consider the Fock part

42 The GW method Want to correct the interaction for the effects of the medium. Take the second order correction to the Coulomb force: v 2 v + This imply: k-q q k k k k-q q k + k-q k Problem: k q This term diverges, and the divergence become worse going to higher orders

43 The GW method The divergence is due to the long-range part of the Coulomb interaction: q -2 q -2 [Picture from Mattuck]

44 The GW method The divergence is due to the long-range part of the Coulomb interaction: q -2 q -2 Need to resum the full RPA series: The screening from RPA avoids the infrared divergence!

45 The GW method The GW self-energy is: This is named in different ways, according to weather the propagator and the one used calculating the in-medium interaction ( or ) are unperturbed or self-consistent: G 0 W 0 = GW 0 = GW =

46 The GW method for the electron gas Self-consistent GW calculations electron gas were achieved only in the last years, see: B. Holm and U. von Barth, Phys. Rev. B57, 2108 (1998). B. Holm, Phys. Rev. Lett. 83, 788 (1999). P. García-González and R. W. Godby, Phys. Rev B63, (2001). Y. Dewulf, D. Van Neck, and M. Waroquier, Phyr. Rev. B (05).

47 The GW method for the electron gas Numerical implementation (Holm and von Barth). Write the single-particle propagator in terms of its spectral function, and expand in a sum of Gaussians: [B. Holm and U. von Barth, Phys. Rev. B57, 2108 (1998)]

48 The GW method for the electron gas Results for G 0 W 0 to GW Note: r s is the radius of the mean volume occupied by each electron (in Borh s radii): In practice, it is used to label the density. [B. Holm and U. von Barth, Phys. Rev. B57, 2108 (1998)]

49 The GW method for the electron gas Correlation energies (==tot. energy HF) of the electron gas from quantum Monte Carlo and GW approaches [P. García-González and R. W. Godby, Phys. Rev B63, (2001)]

50 The GW method for the electron gas Ionization energies for atoms: [S. Verdonck, et al., Phys. Rev A74, (2006)] 2 nd order unperturbed 2 nd order self-consistent GGW== generalized GW contains the Pauli exchange term of the interaction in the calculation of W (GRPA) it is the most complete but it gives poor results...

51 Two words on GW vs DFT Density functional theory (DFT) summarized in two words For confined systems (e.g. electrons in an atom or crystal) there exist a universal (==the same for any system!) energy functional of the density E[ρ]: if one knows the exact density, it is immediate to extract the energy. in the Kohn-Sham formulation of DFT the density is expanded in a Slater determinant; this leads to a one-body Schroedinger-like equation, no matter the number of particles N. In can therefore solved very easily. The E[ρ] functional is proven to exist but it is not known. If it was known, it would be possible to calculate for any system the exact energy, density and first ionization potential (and only the first!!!). In practice, one uses a phenomenological potential. Results for the energies are usually very good, first ionizations and cases with substantial long-range correlations (e.g. Van der Walls) can be poor. [W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965). P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964)]

52 Two words on GW vs DFT For a comparison between Green s function and DFT see the review of Onida et al., [Rev. Mod. Phys. 74, 601 (2002)] For the single-particle spectrum: DFT GW It is fast to solve Calculations are more cumbersome Kohn-Sham orbitals and energies do not have a physical meaning associated (except the first ionization) Kohn-Sham orbitals and energies usually give a good input for G 0 W 0 calculations. Band gaps in insulators and semiconductors are usually underestimated. Single-particle properties are directly related to experimental quantities G 0 W 0 can give quite accurate s-p spectra. The more elaborated GW does better on energies but ruins the s-p spectra Note that GW is NOT a conserving approx. (Baym-Kadanoff)

53 Two words on GW vs DFT Example of band calculation for copper [Onida et al., Rev. Mod. Phys. 74, 601 (2002)]

54 Graphite graphene and graphane Some allotropes of carbon: Graphene is a one-atom thick sheet of Graphite [K.S.Novoselof, et al., Science 306, 666 (2004).] a) diamond; b) graphite; c) lonsdaleite; d-f ) fullerenes (C60, C540, C70); g) amorphous carbon; h) carbon nanotube. [Picure from Wikipedia]

55 Graphite graphene and graphane Graphene is a good candidate material for constructing future electronics components one wish to turn it into a semiconductor, while keeping it two-dimensional [Source: Wikipedia] Graphene is a one-atom-thick planar sheet of sp 2 - bonded carbon atoms that are densely packed in a honeycomb crystal lattice. The carbon-carbon bond length in graphene is approximately nm. Graphene is the basic structural element of some carbon allotropes including graphite, carbon nanotubes and fullerenes. Measurements have shown that graphene has a breaking strength 200 times greater than steel, making it the strongest material ever tested. It is a good conductor of heat and electricity..

56 Graphite graphene and graphane Graphane is Hydrogenated graphene. It was: 1) predicted theoretically based on DFT-GGA calculations [J. O. Sofo, et al. Rev. B 75, (2007). D. W. Boukhvalov, et al., Phys. Rev. B 77, (2008).] graphene 2) Recently systetized [D.C.Elias,et al. Science 323, 320 (2009)] boat 3) The band-gap is not known but it is predicted by DFT-GGA to be a semiconductor 4) GW calculations instead suggest that it is an insulator! [arxiv: v1] graphane chair [Picture: arxiv: v1]

57 Graphite graphene and graphane Graphane is predicted to be: - a semiconductor by DFT-GGA calculations. -an insulator in GW [arxiv: v1]

58 Faddeev RPA method The following two diagram can be equally important. However summing them would not work well: -They both contain, which would be over counted -They would not interfere NO!!!!! So, the following is NOT good:

59 Faddeev RPA method Thus, to include both ladder and ring correlations one must calculate the full 2p1h/2h1p propagator In general this is exact if one can calculate the full 6-points Green s function (see lecture of Apr. 13 th ): μ ν λ μ ν λ μ ν λ R (2p1h/2h1p) = g 2p1h - g 2p1h-1p α β γ α β γ g 1p-2p1h α β γ

60 Faddeev RPA method The full 2p1h/2h1p polarization propagator also satisfies a Bethe-Salpeter-like equation: However, this depends on 4-tmes (3 frequancies) and it is much more complicatde than the p-h Bethe-Salpeter.

61 Faddeev RPA method The full 2p1h/2h1p polarization propagator also satisfies a Bethe-Salpeter-like equation: However, this depends on 4-tmes (3 frequancies) and it is much more complicatde than the p-h Bethe-Salpeter.

62 Faddeev RPA method The full 2p1h/2h1p polarization propagator also satisfies a Bethe-Salpeter-like equation: Strategy: solve each pp and ph channel separately, by solving the (simpler) DRPA equations. Then couple to a third line and mix the corresponding amplitudes Faddeev eqs.!!

63 Faddeev equations for the 2h1p motion Strategy: solve each pp and ph channel separately, by solving the (simpler) DRPA equations. Then couple to a third line and mix the corresponding amplitudes Faddeev eqs.!! References: CB, et al., Phys. Rev. C63, (2001); Phys. Rev. A76, (2007)

64 Coupling single particle to collective modes Non perturbative expansion of the self-energy: Extended Hartree Fock 2p1h/2h1p configurations Explicit correlations enter the three-particle irreducible propagators: particle hole Both pp/hh (ladder) and ph (ring) response included Pauli exchange at 2p1h/2h1p level References: CB, et al., Phys. Rev. C63, (2001); Phys. Rev. C65, (2002); Phys. Rev. A76, (2007)

65 FRPA: Faddeev summation of RPA propagators Both pp/hh (ladder) and ph (ring) response included Pauli exchange at 2p1h/2h1p level All order summation through a set of Faddeev equations where: TDA RPA References: CB, et al., Phys. Rev. C63, (2001); Phys. Rev. A76, (2007)

66 Correlations & model space (RPA and SM). Shell Model s-d-g 0f-1p (0g9/2) 1s-0d 0p 0s Open-shell nuclei require explicit configuration mixing: shell model Faddeev-RPA describes well the coupling to collective modes including those outside the reach of the shell model apply at shell closures!! RPA / QRPA

67 Self-consistent Green s function approach pp-rpa ph-rpa optical potential FULL self consistency in mid size bases in now POSSIBLE: 16 O, 8 shells ~ CB, Phys. Lett. B643, 268 (2006)

68 Faddeev RPA method Example of sole ladder or ring and full mixing Atom of Ne (10 electrons problem) Phys. Rev. A76, (2007)

69 Binding energy simple cases [C. B., to be published] V low- k, Λ=1.9 fm -1 binding energy E 2 ( k ) ε F A 1 αβ ( h) 0 = ω ωδαβ Sβα ( ω) 2 d + αβ 2m [Migdal-Galitski-Koltun (2NF)] Atoms: HF FRPA Exp. He: He: CB and van Neck, arxiv: v1 [physics.chem-ph] Be: Ne: Mg:

70 Quasiparticle spectrum of 16 O (i.e. 17 F) experiment without 3-, 1- and (7.1MeV) 3 - (6.1MeV) 0 + (6.0MeV) spectrum of 16 O 0 + (g.s.)

71 Quasiparticle spectrum of 16 O (i.e. 17 F) particle on the first 0+ excited state experiment 1 - (7.1MeV) 3 - (6.1MeV) 0 + (6.0MeV) without 3- and 1- spectrum of 16 O without 3-, 1- and (g.s.)

72 Quasiparticle spectrum of 16 O (i.e. 17 F) particle on the first 0+ excited state coupling a proton to 3- and 1 phonons in 16 O experiment SCGF/Fadd without 3- and 1- without 3-, 1- and (7.1MeV) 3 - (6.1MeV) 0 + (6.0MeV) spectrum of 16 O 0 + (g.s.)

73 Results for the hole spectral function of 16O p shell experiment s shell d shell experiment C.B. and WD, PRC65, (02) Experiment from NIKHEF, Leuschner et. al., PRC59, 655 (94) Results from Faddeev expansion and SCGF -40 ε - k (MeV) ε- k (MeV) May-09

74 Results for the hole spectral function of 16O p shell experiment s shell d shell experiment C.B. and WD, PRC65, (02) Experiment from NIKHEF, Leuschner et. al., PRC59, 655 (94) Results from Faddeev expansion and SCGF -40 ε - k (MeV) ε- k (MeV) (d 5/2 p 1/2-1 ) p 1/2-1 d 5/2 3-2-May-09

75 Results for the hole spectral function of 16O p shell experiment s shell d shell experiment C.B. and WD, PRC65, (02) Experiment from NIKHEF, Leuschner et. al., PRC59, 655 (94) Results from Faddeev expansion and SCGF -40 ε - k (MeV) ε- k (MeV) p4h? 3 - (d 5/2 p 1/2-1 ) p 1/2-1 d 5/2 3-2-May-09

76 Treating short-range corr. with a G-matrix The short-range core can be treated by summing ladders outside the model space: G Qˆ ( ω ) = V + V G ( ω ) 2 2 ω ( k + k ) / 2m iη a b + Q P Σ (ω) = + + G(ω) F-RPA + = + F-RPA + G(ω) (long-range effects)

77 Treating short-range corr. with a G-matrix The short-range core can be treated by summing ladders outside the model space: = G(ω) Near E F : long-range / SM-like physics stronger eff. interaction Deeply bound orbits : binding! the HF mean-field is weaker

78 Single neutron levels around 16 O with FRPA Theory(MeV) Exp.[MeV] (AV 18 ) spin-orbit: E d3/2 -E d5/ E p1/2 -E p3/ [CB, Phys. Lett. B643, 268 (2006)] p-h gap: E d3/2 -E p1/ E s1/2 -E p1/ particle-hole gap accurate with a G-matrix with ω-dependence p 3/2 -p 1/2 spin-orbit splitting close to the VMC estimates 3.4MeV [S. Pieper et al. PRL70 ( 93) 2541, using AV 14 ]

79 Convergence of valence orbits in 56 Ni νp 3/2 single-particle energies almost converged (within ~1MeV) for 10 oscillator shells Little dep. on the oscillator parameter νf 7/2 N3LO interaction + monopole corr. [CB, M.Hjorth-Jensen, arxiv: ]

80 Particle and hole spectral distribution of 56 Ni 55 Ni 57 Ni Spect. factors: N3LO interaction + monopole corr. [CB, M.Hjorth-Jensen, arxiv: ]