MBPT and the GW approximation

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1 MBPT and the GW approximation Matthieu Verstraete Université de Liège, Belgium European Theoretical Spectroscopy Facility (ETSF) Benasque - TDDFT /60

2 Outline 1 Green s functions I: G0 W0 2 Green s functions II: beyond G0 W0 2/60

3 Outline 1 Green s functions I: G0 W0 2 Green s functions II: beyond G0 W0 3/60

4 Why are we here? Electronic structure and TDDFT Predict ground state geometric and electronic structure But also materials properties Predict or explain some experimental results... How do experiments work? 4/60

5 What experimentalists really do Perturb a system (smash it, burn it, bombard it...) Look at the reponse Change a parameter and start over There is no way to access only ground state data Pray that the perturbation is small... contacted CNT device luminescent lanthanides 5/60

6 Photoemission Direct Photoemission Inverse Photoemission 6/60

7 Plethora of methods Schrödinger equation Hartree Fock CI, CAS, MP2, etc... (TD)DFT MBPT 7/60

8 Why do we need more than DFT? adapted from M. van Schilfgaarde et al., PRL 96 (2006). 8/60

9 Why do we need more than DFT? Band gap problem (XC discontinuity) Weak forces (VdW, H bonds...) Dissociation limit of molecules DFT is a ground state theory excited states are complex unknown functionals of ρ 9/60

10 Alternatives Q-chemistry: lots of experience, precision (bad scaling, basis set problems...) TDDFT: good scaling, intuitive (unknown functionals, memory, nonlocality...) MBPT well established formalism, systematic (can be heavy, or even intractable) 10/60

11 Alternatives Q-chemistry: lots of experience, precision (bad scaling, basis set problems...) TDDFT: good scaling, intuitive (unknown functionals, memory, nonlocality...) MBPT well established formalism, systematic (can be heavy, or even intractable) Someday, TDDFT will overcome... maybe 10/60

12 One-particle Green s function The one-particle Green s function G Definition and meaning of G: ig(x1, t1 ; x2, t2 ) = hn T ψ(x1, t1 )ψ (x2, t2 ) Ni 11/60

13 One-particle Green s function The one-particle Green s function G Definition and meaning of G: ig(x1, t1 ; x2, t2 ) = hn T ψ(x1, t1 )ψ (x2, t2 ) Ni for t1 > t2 ig(x1, t1 ; x2, t2 ) = hn ψ(x1, t1 )ψ (x2, t2 ) Ni for t1 < t2 ig(x1, t1 ; x2, t2 ) = hn ψ (x2, t2 )ψ(x1, t1 ) Ni (1) (2) 12/60

14 One-particle Green s function t1 > t2 hn ψ(x1, t1 )ψ (x2, t2 ) Ni t1 < t2 hn ψ (x2, t2 )ψ(x1, t1 ) Ni 13/60

15 One-particle Green s function What is G? Definition and meaning of G: i h G(x1, t1 ; x2, t2 ) = i < N T ψ(x1, t1 )ψ (x2, t2 ) N > Insert a complete set of N + 1 or N 1-particle states. This yields X G(x1, t1 ; x2, t2 ) = i fj (x1 )fj (x2 )e iεj (t1 t2 ) j [θ(t1 t2 )θ(εj µ) θ(t2 t1 )θ(µ εj )]; where: εj = fj (x1 ) = E(N + 1, j) E(N), εj > µ E(N) E(N 1, j), εj < µ hn ψ (x1 ) N + 1, ji, εj > µ hn 1, j ψ (x1 ) Ni, εj < µ 14/60

16 One-particle Green s function What is G? - Fourier transform Fourier Transform: G(x, x0, ω) = X j fj (x)fj (x0 ) ω εj + iηsgn(εj µ). Spectral function: A(x, x0 ; ω) = X 1 ImG(x, x0 ; ω) = fj (x)fj (x0 )δ(ω εj ). π j 15/60

17 One-particle Green s function Spectral function 16/60

18 One-particle Green s function Spectral function 17/60

19 One-particle Green s function Quasiparticles G(x, x0, ω) = X Φj (x)φ j (x0 ) j ω Ej. 18/60

20 Photoemission Direct Photoemission Inverse Photoemission One-particle excitations poles of one-particle Green s function G 19/60

21 One-particle Green s function One-particle Green s function From one-particle G we can obtain: one-particle excitation spectra ground-state expectation value of any one-particle operator: e.g. density ρ or density matrix γ: ρ(r, t) = ig(r, r, t, t + ) γ(r, r0, t) = ig(r, r0, t, t + ) ground-state total energy 20/60

22 One-particle Green s function Straightforward? G(x, t; x0, t 0 ) = i < N T ψ(x, t)ψ (x0, t 0 ) N > N > =??? Interacting ground state! Perturbation Theory? Time-independent perturbation theories: messy. Textbooks: adiabatically switched on interaction, Wick s theorem, expansion (diagrams). Lots of diagrams... 21/60

23 One-particle Green s function Straightforward? G(x, t; x0, t 0 ) = i < N T ψ(x, t)ψ (x0, t 0 ) N > N > =??? Interacting ground state! Perturbation Theory? Time-independent perturbation theories: messy. Textbooks: adiabatically switched on interaction, Wick s theorem, expansion (diagrams). Lots of diagrams... 21/60

24 One-particle Green s function Straightforward? G(x, t; x0, t 0 ) = i < N T ψ(x, t)ψ (x0, t 0 ) N > N > =??? Interacting ground state! Perturbation Theory? Time-independent perturbation theories: messy. Textbooks: adiabatically switched on interaction, Wick s theorem, expansion (diagrams). Lots of diagrams... 21/60

25 One-particle Green s function Perturbation theory starts from what is known to evaluate what is not known, hoping that the difference is small... Let s say we know G0 (ω) that corresponds to the Hamiltonian h0 Everything that is unknown is put in Σ(ω) = G0 1 (ω) G 1 (ω) This is the definition of the self-energy 22/60

26 One-particle Green s function Equation of motion To determine the 1-particle Green s function: Z i h h0 (1) G(1, 2) = δ(1, 2) i d3v (1, 3)G2 (1, 3, 2, 3+ ), i t1 Do the Fourier transform in frequency space: Z [ω h0 ]G(ω) + i vg2 (ω) = 1 where h0 = vext is the independent particle Hamiltonian. The 2-particle Green s function describes the motion of 2 particles. 23/60

27 One-particle Green s function Equation of motion To determine the 1-particle Green s function: Z i h h0 (1) G(1, 2) = δ(1, 2) i d3v (1, 3)G2 (1, 3, 2, 3+ ), i t1 Do the Fourier transform in frequency space: Z [ω h0 ]G(ω) + i vg2 (ω) = 1 where h0 = vext is the independent particle Hamiltonian. The 2-particle Green s function describes the motion of 2 particles. 23/60

28 One-particle Green s function Perturbation theory starts from what is known to evaluate what is not known, hoping that the difference is small... Let s say we know G0 (ω) that corresponds to the Hamiltonian h0 Everything that is unknown is put in Σ(ω) = G0 1 (ω) G 1 (ω) This is the definition of the self-energy Thus, Z [ω h0 ]G(ω) Σ(ω)G(ω) = 1 to be compared with Z [ω h0 ]G(ω) + i vg2 (ω) = 1 24/60

29 One-particle Green s function Trick due to Schwinger (1951): introduce a small external potential U(3), that will be made equal to zero at the end, and calculate the variations of G with respect to U δg(1, 2) = G2 (1, 3; 2, 3) + G(1, 2)G(3, 3). δu(3) 25/60

30 Hedin s equation Hedin s equations Σ =igw Γ G =G0 + G0 ΣG δσ Γ =1 + GGΓ δg P = iggγ W =v + vpw L. Hedin, Phys. Rev. 139 (1965) 26/60

31 GW bandstructure: photoemission additional charge 27/60

32 GW bandstructure: photoemission additional charge reaction: polarization, screening GW approximation 1 polarization made of noninteracting electron-hole pairs (RPA) 2 classical (Hartree) interaction between additional charge and polarization charge 27/60

33 Hedin s equation and GW GW approximation Σ =igw Γ G =G0 + G0 ΣG Γ =1 P = iggγ W =v + vpw L. Hedin, Phys. Rev. 139 (1965) 28/60

34 Hedin s equation and GW GW approximation Σ =igw G =G0 + G0 ΣG Γ =1 P = igg W =v + vpw L. Hedin, Phys. Rev. 139 (1965) 29/60

35 GW in practice Standard perturbative G0 W0 H0 (r)ϕi (r) + Vxc (r)ϕi (r ) = i ϕi (r) Z H0 (r)φi (r) + dr0 Σ(r, r0, ω = Ei ) φi (r0 ) = Ei φi (r) First-order perturbative corrections with Σ = igw : Ei i = hϕi Σ Vxc ϕi i Hybertsen and Louie, PRB 34 (1986); Godby, Schlüter and Sham, PRB 37 (1988) 30/60

36 GW results M. van Schilfgaarde et al., PRL 96 (2006). 31/60

37 GW results dots experiment - dashed lines LDA - solid lines GW A. Marini et al., PRL 88 (2002) - Cu bulk. 32/60

38 GW results I.D. White et al., PRL 80 (1998) - Al(111): potential 33/60

39 GW and Hartree-Fock Hartree-Fock Σ(12) = ig(12)v (1+ 2) GW Σ(12) = ig(12)w (1+ 2) v infinite range in space W is short ranged v is static W is dynamical Σ is nonlocal, hermitian, static Σ is nonlocal, complex, dynamical 34/60

40 Hartree-Fock is pretty good... 35/60

41 ... but not in solids 36/60

42 PES in Hartree-Fock 37/60

43 PES in GW 38/60

44 GW and Hartree-Fock 39/60

45 GW and Hartree-Fock 40/60

46 Implementations Recip space G(g, g 0, ω) = Space Time X φ (g)φnk (g 0 ) nk ω εnk + iη nk X P(q, ω) = G(x, x 0 ; iτ ) = Mnn 0 k ;q Mnn0 k ;q nn0 k X φ nk (x)φnk (x 0 )eτ ( n iη) nk f0 f ε0 ε P(x, x 0 ; iτ ) = ig(x, x 0 ; iτ )G(x 0, x; iτ ) Mnn0 k ;q = hφn0 k +q ei(q+g)r φnk i (g, g 0 ; ω) = (1 v (g)p(g, g 0 ; ω)) W (g, g ; ω) = Z Σ(ω) = i 0 0 (g, g ; ω)v (g ) W (g, g 0 ; ω) = 1 (g, g 0 ; iω)v (g 0 ) dω 0 G(ω ω 0 )W (ω 0 ) Σ(x, x 0 ; iτ ) = ig(x, x 0 ; iτ )W (x 0, x; iτ ) Hybertsen and Louie, PRB 34 (1986) (g, g 0 ; iω) = (1 v (g)p(g, g 0 ; iω)) 1 H.N. Rojas et al. PRL (1995) 41/60

47 Outline 1 Green s functions I: G0 W0 2 Green s functions II: beyond G0 W0 42/60

48 Hedin s equation and GW GW approximation Σ =igw G =G0 + G0 ΣG Γ =1 P = igg W =v + vpw L. Hedin, Phys. Rev. 139 (1965) 43/60

49 Good things about full GW Ingredients are intuitive : P, ε, G0 No need for functional derivation Conservation laws for E, N, and (angular) momentum Baym and Kadanoff Phys Rev (1961); ibid (1962) 44/60

50 The origin of the pieces Conservation comes from existence of a total energy functional Σ follows from this choice by / G Total energy diagrams for HF, GW: E= G = (G0 1 + Σ) 1 = 45/60

51 Galitskii-Migdal We do not have Ψ or a simple operator form for E How can we calculate the total energy from G? From eq. of motion for Ψ in Heisenberg form: E = <T >+<V > Z i 1 2 = dx lim lim i G(x, x 0 ; t t 0 ) x 0 x t 0 t + 2 t 2 x Z 1 =i dp 0p Σ(p, ) G(p, ) 2 C Galitskii and Migdal JETP 34(7) 96 (1958) 46/60

52 Example applications 2 metallic slabs (jellium) calculate Etot (d) Long range interaction of charge fluctuations gives VdW LDA decays exponentially García-González PRL (2002) 47/60

53 Flavors of SC: 1 - the full Monty Full Self Consistency Solve Dyson, get new G, iterate Very heavy, mainly done for the HEG Good energy, bad spectrum 48/60

54 Flavors of SC: 1 - the full Monty Full Self Consistency Solve Dyson, get new G, iterate Very heavy, mainly done for the HEG Good energy, bad spectrum 48/60

55 Flavors of SC: 1 - the full Monty Full Self Consistency Solve Dyson, get new G, iterate Very heavy, mainly done for the HEG Good energy, bad spectrum 48/60

56 2 - The half Monty Partial SC On W: reasonable, WLDA is usually the problem Simplest is eigen-energy-only SC Or on G only (not very useful - DFT G is good and not any easier) 49/60

57 3 - The fake Monty QuasiParticle SC Hermitianize self-energy Get new 1 particle states from simpler equation Only calculate QP corrections etc... at last iteration best G which comes from pure QP van Schilfgaarde PRL (2006) 50/60

58 Example applications II Don t need spectrum for Etot use iτ formalism Energy aafo volume for Na... comparable to GGA... and spectrum is worse... Kutepov et al. PRB 80, (2009) 51/60

59 Better starting points You don t really want to do SC... Basic problem is localization of LDA states So: Hartree-Fock (too large gap) COHSEX (static Σ, similar to HF) metagga, hybrids, EXX (pretty close!) 52/60

60 Example applications III VO2 forms chains of dimers (Anti)bonding gap Metallic in LDA Bad, W: G0 W0 is helpless Gap in COHSEX, G0 W0 ok Gatti et al. PRL 99, (2007) 53/60

61 Example applications III VO2 forms chains of dimers (Anti)bonding gap Metallic in LDA Bad, W: G0 W0 is helpless Gap in COHSEX, G0 W0 ok Gatti et al. PRL 99, (2007) 53/60

62 Example applications III VO2 forms chains of dimers (Anti)bonding gap Metallic in LDA Bad, W: G0 W0 is helpless Gap in COHSEX, G0 W0 ok Gatti et al. PRL 99, (2007) 53/60

63 Vertex corrections Γ=1+ Σ GGΓ G Challenges: How do you do Σ/ G? Find simple form for Γ LDA vertex ADA vertex 54/60

64 LDA vertex LDA vertex With ΣLDA = Vxc ΓLDA is explicit and local G0 WLDA improves spectra G0 WLDA ΓLDA worsens them! Hybertsen PRB (1986) del Sole PRB (1994) Morris et al. PRB 76, (2007) 55/60

65 ADA vertex ADA vertex Local ΓLDA is pathological So, make Γ a bit non-local Average density locally, then evaluate Vxc Hubbard local field factor in GG0 Restores good ionization potentials Morris et al. PRB 76, (2007) 56/60

66 Why is G0 W0 any good? Virtues of G0 W0 Not too expensive Good starting points Small perurbation Variational principle Improves both X and C over DFT (cancelation of errors?) Compensation of SC and vertex effects in spectrum? 57/60

67 Beyond all that Tougher cases Neutral excitations (see BSE lectures) Need semi-core states for transition metals (AE vs psp) Strong correlation, ladder diagrams, DMFT Complex magnetism (again ladder diagrams) 58/60

68 Acknowledgements Rex Godby Martin Stankovski Pablo García González Peter Bokes Matteo Gatti 59/60

69 Acknowledgements Rex Godby Martin Stankovski Pablo García González Peter Bokes Matteo Gatti 59/60

70 Bibliography Some references L. Hedin Phys Rev 139, A796 (1965). L. Hedin and S. Lunqdvist Sol State Phys 23 (1969). F. Aryasetiawan and O. Gunnarsson Rep Prog Phys 61, 237 (1998). L. Hedin J. Phys Cond Matt 11, R489 (1999) W.G. Aulbur et al. Sol State Phys 54, 1 (2000) G. Mahan G. Vignale and G. Giuliani Properties of the electron liquid () 60/60

The GW approximation

The GW approximation Matteo Gatti European Theoretical Spectroscopy Facility (ETSF) LSI - Ecole Polytechnique & Synchrotron SOLEIL - France matteo.gatti@polytechnique.fr - http://etsf.polytechnique.fr