1: Sheldon L. Glashow (Ouroboros) [1] 1 v(r) u(r, r ) ( e 2 / r r ) H 2 [2] H = ( dr ψ σ + (r) 1 2 ) σ 2m r 2 + v(r) µ ψ σ (r) + 1 dr dr ψ σ + (r)ψ +

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1 cm cm cm 1cm

2 1: Sheldon L. Glashow (Ouroboros) [1] 1 v(r) u(r, r ) ( e 2 / r r ) H 2 [2] H = ( dr ψ σ + (r) 1 2 ) σ 2m r 2 + v(r) µ ψ σ (r) + 1 dr dr ψ σ + (r)ψ + σ (r )u(r, r )ψ 2 σ (r )ψ σ (r) (1) σσ ψ σ (r) m σ µ Φ 0 E 0 ( Φ 0 H Φ 0 / Φ 0 Φ 0 + µn) n(r) ( Φ 0 σ ψ+ σ (r)ψ σ (r) Φ 0 / Φ 0 Φ 0 ) N ( n(r) dr) N 10 CI: Configuration Interaction) [3] N 100 (DMC: Diffusion Monte Carlo) [4] CI DMC N 2 H Φ 0 1 v(r) ṽ(r) 1 H 0 H 0 ( dr ψ σ + (r) 1 2 ) σ 2m r 2 + ṽ(r) µ ψ σ (r) (2) H 0 0 H H = H 0 + H 1 H 0 n = ϵ n n H 0 { n } H 1 2

3 H 1 (t) = e ih0t H 1 e ih0t Φ 0 Φ 0 = 0 + n n H [ ] + =T t exp i dte 0+t H 1 (t) 0 (3) ϵ n 0 0 ϵ n [5] T t T T t [H 1 (t 1 ) H 1 (t i ) H 1 (t n )] t i H 1 (t i ) n (3) Φ 0 H 1 0 Φ 0 = F 0 F 2 [6] [7] (CBF: Correlated-Basis Function) [8] FHNC(Fermi Hypernetted Chain) [9] [10] CC (Coupled- Cluster) [11] EPX(Effective-Potential Expansion) [12] Φ 0 (3) EPX CC E 0 = 0 H Φ 0 / 0 Φ v(r) E 0 n(r) CC EPX CBF FHNC n(r) 1 BBGKY (Bogoliubov-Born-Green-Kirkwood-Yvon) [13] n(r) 2 n 2 (r 1, r 2 ) n 2 (r 1, r 2 ) 3 n 3 (r 1, r 2, r 3 ) n(r) n(r) DFT: Density Functional Theory)[14] DFT n 2 (r 1, r 2 ) n 3 (r 1, r 2, r 3 ) E 0 n(r) (2) ṽ(r) n(r) 0 σ ψ+ σ (r)ψ σ (r) 0 1 n(r) BBGKY n(r) DFT ṽ(r) ṽ(r) ṽ(r) 3

4 BBGKY DFT ṽ(r) E xc [n(r)] ṽ(r) E xc [n(r)] DFT E xc [n(r)] (LDA: Local Density Approximation) (GGA: Generalized Gradient Approximation) E xc [n(r)] DFT DFT E 0 n(r) n 2 (r 1, r 2 ) DFT DFT n(r) n 2 (r 1, r 2 ) 1 2 DFT n 2 (r 1, r 2 ) DFT [15] 1 [16] Φ 0 = F 0 4

5 Φ n F F H 0 n Φ n = F n HF(Hartree- Fock) [17] G R σσ (r, r ; t) t = 0 r σ r σ t (> 0) r σ r σ G R σσ (r, r ; t) iθ(t) {ψ σ (r, t), ψ + σ (r )} (4) θ(t) ψ σ (r, t) e iht ψ σ (r)e iht e βω tr(e βh ) T β 1/T Ω Ω T ln[tr(e βh )] ARPES: Angle-Resolved Photo- Emission Spectroscopy [18] G R σσ (r, r ; t) G R σσ (r, r ; ω) 1 A σσ (r, r ; ω) G R σσ (r, r ; ω) = = dt e iωt 0+t G R σσ (r, r ; t) de A σσ (r, r ; E) ω + i0 + E A σσ (r, r ; ω) = 1 π Im GR σσ (r, r ; ω) (6) (5) t t + e 0+t ω ω + i0 + ω de A σσ (r, r ; E) = δ σ,σ δ(r r ) (7) (Sum rule) (5) N G R σσ (r, r ; t) N ) 5

6 G R σσ (r, r ; t) ψ σ (r, τ) e Hτ ψ σ (r)e Hτ G σσ (r, r ; τ) T τ ψ σ (r, τ)ψ + σ (r ) (8) G σσ (r, r ; τ) T τ τ T (8) G σσ (r, r ; τ + β) = G σσ (r, r ; τ) τ G σσ (r, r ; τ) β G σσ (r, r ; τ) = T ω p e iω pτ G σσ (r, r ; iω p ) (9) ω p p p = 0, ±1, ±2, ω p = πt (2p+1) β G σσ (r, r ; iω p )= dτ e iωpτ G σσ (r, r ; τ)= de A σσ (r, r ; E) 0 iω p E (10) G σσ (r, r ; iω p ) (5) ω iω p ω + i0 + G R σσ (r, r ; ω) G σσ (r, r ; iω p ) G(iω p ) = 1/(iω p H) G σσ (r, r ; iω p ) = rσ 1 iω p H r σ (11) G(iω p ) G { rσ } 1.7 G(iω p ) H u(r, r ) H 0 (2) ṽ(r) v(r) u(r, r ) H 1 H 0 G 0 (iω p ) G 0 (iω p ) = 1/(iω p H 0 ) (iω p H 0 )G 0 (iω p ) = 1 ( iω p ) 2m r 2 v(r) + µ G σσ,0(r, r ; iω p ) = δ(r r )δ σσ (12) 1 v(r) G 0 (iω p ) G(iω p ) G 0 (iω p ) 1 iω p H = H 1 iω p H 0 iω p H 0 iω p H G(iω p ) = G 0 (iω p ) + G 0 (iω p ) Σ(iω p ) G(iω p ) Σ(iω p ) (13) Σ σσ (r, r ; iω p ) = rσ H 1 r σ (14) (14) 2 H 1 1 6

7 ψ σ + (r)ψ + σ (r )ψ σ (r )ψ σ (r) ψ σ + (r)ψ σ (r) ψ + σ (r )ψ σ (r ) ψ σ + (r)ψ σ (r ) ψ + σ (r )ψ σ (r) ψ + ψ G(τ = 0 + ) Σ σσ (r, r ; iω p ) δ(r r )δ σσ Σ H (r) + δ σσ Σ F (r, r ) (15) 1 Σ H (r) Σ H (r) dr u(r, r ) T G σσ (r, r ; iω p )e iωp0+ σ ω p = dr u(r, r )n(r ) (16) 1 Σ F (r, r ) Σ F (r, r ) u(r, r )T G σσ (r, r ; iω p )e iω p0 + (17) ω p Σ(iω p ) Σ H Σ(iω p ) [ Σ(iω p ) Σ H ] Σ F 1 ω p G G = G 0 + G 0 (Σ H + Σ) G (18) Σ (18) G 1.8 (15) HF (14) G(τ) (8) τ H H 0 U(τ) e H 0τ e Hτ ψ σ (r, τ) = U(τ) 1 e H 0τ ψ σ (r)e H 0τ U(τ) ψ σ (r, τ) e H 0τ ψ σ (r)e H 0τ H 1 (τ) = e H 0τ H 1 e H 0τ U(τ) S S(τ, τ ) U(τ)U(τ ) 1 S(τ, τ ) = 1 S(τ, τ ) τ = e H 0τ (H 0 H)e Hτ U(τ ) 1 = H 1 (τ)s(τ, τ ) (19) S [ S(τ, τ ) = T τ exp τ dτ 1 H 1 (τ 1 ) ] τ (20) Ω e βω = tr(e βh ) = e βω 0 S(β, 0) 0 (21) Ω 0 e βω 0 = tr(e βh 0 ) 0 e βω 0 tr(e βh0 ) 7

8 S(β, 0) (8) G σσ (r, r ; τ) = T τ [S(β, 0)ψ σ (r, τ)ψ + σ (r )] 0 S(β, 0) 0 (22) (21) (22) (22) G σσ (r, r ; τ) ψ σ (r, τ) ψ + σ (r ) c G σσ (r, r ; τ) = T τ [S(β, 0)ψ σ (r, τ)ψ + σ (r )] 0c (23) G σσ (r, r ; iω p ) 1.9 (23) (20) S(β, 0) H 1 G 0 G (0) G 0 1 G (1) G (1) = G 0 Σ H [G 0 ] G 0 + G 0 Σ F [G 0 ] G 0 (24) Σ H [G 0 ] Σ F [G 0 ] (16) (17) Σ H Σ F G G 0 2 (1) 2 G (2) 2 (2) Σ 2a [G 0 ] Σ 2b [G 0 ] G (2) = G 0 Σ 2a [G 0 ] G 0 + G 0 Σ 2b [G 0 ] G 0 +G 0 Σ H [G 0 Σ H [G 0 ] G 0 ] G 0 + G 0 Σ H [G 0 Σ F [G 0 ] G 0 ] G 0 +G 0 Σ F [G 0 Σ H [G 0 ] G 0 ] G 0 + G 0 Σ F [G 0 Σ F [G 0 ] G 0 ] G 0 +G 0 Σ H [G 0 ] G 0 Σ H [G 0 ] G 0 + G 0 Σ H [G 0 ] G 0 Σ F [G 0 ] G 0 +G 0 Σ F [G 0 ] G 0 Σ H [G 0 ] G 0 + G 0 Σ F [G 0 ] G 0 Σ F [G 0 ] G 0 (25) (24) Σ 1 [G 0 ] Σ H [G 0 ] + Σ F [G 0 ] (25) G 0 Σ 1 [G (1) ] G 0 + G 0 Σ 1 [G 0 ] G (1) (26) Σ 2 [G 0 ] 1 Σ H [G 0 ] Σ F [G 0 ] n n Σ n [G 0 ] G Σ n [G 0 ] G 0 G Σ n [G] G = G (n) = G 0 + G 0 n=0 n=1 Σ n [G] G (27) 8

9 Σ Σ Σ Σ H F 2a 2b u G u G 0 0 G 0 G 0 G 0 G u u G 0 u 0 G 0 u 2: (1) (2) Σ H [G] Σ F [G] Σ H Σ F (27) (18) Σ Σ = Σ F [G] + Σ n [G] (28) n=2 Σ 1.10 LW: Luttinger-Ward Φ[G 0 ] [19] Φ[G 0 ] n Φ n [G 0 ] δφ n [G 0 ] δg 0 Σ n [G 0 ] (29) 1 ( ) Φ[G 0 ] = Φ n [G 0 ] = 2n tr G 0 Σ n [G 0 ] n=1 n=1 (30) tr( ) T ω p Φ[G 0 ] 3 3 Φ[G 0] = : Φ[G 0 ] Φ[G 0 ] G 0 Φ[G 0 ] G 0 1 Σ H [G 0 ] 2 Σ F [G 0 ] Σ 2a [G 0 ] Σ 2b [G 0 ] n 9

10 2n G 0 (30) 1/2n Φ n [G 0 ] Σ n [G 0 ] Φ[G 0 ] G 0 G Φ[G] Σ Σ H + Σ = Σ n [G] = n=1 n=1 δφ n [G] δg = δφ[g] δg (31) Φ[G] Φ[G] Ω LW Ω [ ] Ω = tr e iωp0+ ln( G(iω p ) 1 ) + G(iω p )(Σ H + Σ(iω p )) + Φ[G] (32) (18) Σ H + Σ = G 1 0 G 1 G (32) G Ω[G] G δω[g] δg = G δ ( ) G 1 + Σ H + Σ Σ H Σ + δφ δg δg (33) (18) (31) δω[g]/δg = 0 G Ω[G] LW u(r, r ) [20] Φ[G] (31) Σ[G] Σ[G] (18) [21] 1.11 LW Φ[G] Σ [22] 5 G Σ Π W Γ 5 Γ e e ( eγ) LW Σ 4 1 (18) Σ Π W 5 Γ Ĩ Σ/ G 5 G Σ Π W Γ 10

11 G0 (1) G : = + (2) Σ : Σ (4) W : = + Π Γ = (3) Π : Π = Γ Γ (5) Γ : δσ = + δg u + Γ Σ 4: 5 Ĩ 1.12 GW G 0 u LW u G 0 G G u W G W u W W 4 5 Γ = 1 Σ = GW Σ GW HF u W HF Φ[G] GW Σ GW [23] G G 0 Π Σ GW G 0 W 0 GW G 0 W 0 11

12 1.13 GWΓ Σ Γ WI: Ward Identity Γ = 1 Σ 0 GW WI ~ Σ = GWΓ ~ W ~ u W= ~ 1+(u+ I ) Π WI Π WI WI Σ G= 1 G -1 Σ Γ Π = GGΓ WI WI 0 G Γ WI = -1-1 G -G -1-1 G 0 -G 0 5: WI Σ Γ [24] GWΓ 5 GWΓ G 0 W 0 Σ Γ G 0 W 0 GW GWΓ G 0 W 0 1 GWΓ GW TL: Tomonaga-Luttinger [25] TL GWΓ

13 [1] UTPhysics [2] h = k B = 1 [3] P.-O. Löwdin, Adv. Chem. Phys. 2, 207 (1959) 2007 [4] B. L. Hammond, W. A Lester, Jr. and P.J. Reynolds, Monte Carlo Methods in Ab Initio Quantum Chemistry (World Scientific, 1994); W. M. C. Foulkes, L. Mitas, R. J. Needs, and G. Rajagopal, Rev. Mod. Phys. 73, 33 (2001). [5] [6] R. Jastrow, Phys. Rev. 98, 1479 (1955). [7] M. C. Gutzwiller, Phys. Rev. Lett. 10, 159 (1963). [8] E. Feenberg: Theory of Quantum Fluids (Academic, NY, 1969). [9] S. Fantoni and S. Rosati, Nuovo Cimento 25, 593 (1975). [10] S. F. Boys and N. C. Handy, Proc. Roy. Soc. A 309, 195; 209; 310, 43; 46; 311, 309 (1969). [11] F. Coester and H. Kümmel, Nucl. Phys. 17, 477 (1960). [12] Y. Takada, Phys. Rev. A 28, 2417 (1983); Phys. Rev. B 35, 6923 (1987). [13] G. Sandri, Ann. Phys. (NY) 24, 332 (1963); E. Gozzi and M. Reuter, Phys. Rev. E 47, 726 (1993). [14] P. Hohenberg and W. Kohn, Phys. Rev. 136, 864 (1964); W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965); W. Kohn, Rev. Mod. Phys. 71, 1253 (1999). [15] L. D. Landau, Sov. Phys. JETP 3, 920 (1956); 5, 101 (1957); 8, 70 (1959). [16] D. Pines: Elementary excitations in solids (Benjamin, 1963). [17] Y. Takada, Phys. Rev. B 43, 5979 (1991). [18] II ); 1998); D. W. Lynch and C. G. Olsen, Photoemission Studies of High-Temperatue Superconductors (Cambridge University Press, 1999); S. Hufner: Photoelectron Spectroscopy (Springer, Berlin, 2003); W. Schülke: Electron Dynamics by Inelastic X-Ray Scattering (Oxford University Press, 2007). [19] J. M. Luttinger and J. C. Ward, Phys. Rev. 118, 1417 (1960). [20] P. Nozières and J. M. Luttinger, Phys. Rev. 127, 1423 (1962); J. M. Luttinger and P. Nozières, Phys. Rev. 127, 1431 (1962). [21] G. Baym and L. P. Kadanoff, Phys. Rev. 124, 287 (1961); G. Baym, Phys. Rev. 127, 1391 (1961). [22] L. Hedin, Phys. Rev. 139, A796 (1965). [23] GW F. Aryasetiawan and O. Gunnarsson, Rep. Prog. Phys. 61, 237 (1998); W. G. Aulbur, L. Jönsson, and J. W. Wilkins, in Solid State Physics, edited by H. Ehrenreich and F. Spaepen (Academic, New York, 2000), Vol. 54, p.1. [24] Y. Takada, Phys. Rev. Lett. 87, (2001); Y. Takada and H. Yasuhara, Phys. Rev. Lett. 89, (2002); S. Ishii, H. Maebashi, and Y. Takada, unpublished. [25] I. E. Dzyaloshinskii and A. I. Larkin, Soviet Phys. JETP 38, 202 (1974); W. Metzner, C. Castellani, C. D. Castro, Adv. Phys. 47, 317 (1998). 13

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