Theories for Dynamical Response of Many-Electron Systems Yasutami Takada 19 7 9 I make a review of theoretical frameworks of calculating the dynamical properties of many-electron systems in solids. Emphasis is put on the accurate first-principles evaluation of spectral functions characterizing elementary excitations in both the Green s-function approach and the time-dependent density functional theory. 1 dynamical response 1) 1
DMFT: Dynamical Mean-Field Theory 2) (LDA: Local Density Approximation) (LDA+DMFT) 3) 4) 192 194 5 196 DFT: Density Functional 2
Theory 5) n(r) E xc [n(r)] DFT E xc [n(r)] LDA DFT 6) E xc [n(r)] QMC: Quantum Monte Carlo fixed-node approximation 7) DFT QMC 8) 3
2 2.1 2.2 9) 2 4eV 1 1keV X 4
+e 1/2 l 2 4eV ARPES: Angle-Resolved Photo-Emission Spectroscopy 1meV E f Φ hω E i 1: t = r σ t(> ) r σ ψ σ (r) ψ + σ (r ) ψ σ (r, t)ψ + σ (r ) H ψ σ (r, t) e iht ψ σ (r)e iht e βω tr(e βh ) 1) T β 1/T Ω Ω T ln[tr(e βh )] 5
ρ(x) (a) e - t = -ε ρ(x) (b) t = -ε e - ρ(x) r' t = x ρ(x) t = r x ρ(x) r' t > e - x ρ(x) e- t > r x r' r x r' r x 2: (a) (b) N N + 1 t = r σ t r σ ψ σ (r, t)ψ + σ (r ) N 1 N ψ + σ (r, t)ψ σ (r) ψ + σ (r, t) ψ + σ (r, t) ψ + σ (r )ψ σ (r, t) ψ σ (r, t)ψ + σ (r ) ψ + σ (r )ψ σ (r, t) G σσ (r, r ; t) iθ(t) {ψ σ (r, t), ψ + σ (r )}. (1) θ(t) i {A, B} A B AB + BA 6
2.3 X ω (ev) 1 2 1 1 1-1 1-2 ( k, ω ) q = k - k1 ω = ω - ω 1 1 3 1-3 1 6 1 7 1 8 1 9 q (cm-1) 1 1 ( k, ω ) 3: ω q ω 1keV k 1keV ω q ω 1 (= ω ω) k 1 (= k q) R(q, ω) e e iq r iωt H (t) H (t) = e iωt V (q) dre iq r ρ(r) (2) V (q) = 4πe 2 /q 2 ρ(r) σ ψ + σ (r)ψ σ (r) ω ω 7
R(q, ω) H R(q, ω) = 2πV (q) 2 S(q, ω) (3) S(q, ω) S(q, ω) = dt 2π eiωt dre iq r dr e iq r ρ(r, t)ρ(r ) (4) N N ± 1 N ω 5 1keV X ω q d 2 σ/dωdω X A A A A A (e 2 /mc 2 )ρ(r) m Thomson d 2 σ dωdω = ( e 2 mc 2 ) 2 ( ω 1 ω ) ( ϵ ϵ 1 ) 2 S(q, ω). (5) ϵ ϵ 1 X S(q, ω) V (q) q q( q ) X q 1 Å 1 1eV S(q, ω) (4) 3 (1) (4) 8
(Rayleigh-Ritz-Schrödinger) DFT i Ψ(t) = [H + H (t)]ψ(t) H total (t)ψ(t) (6) t H total (t) Ψ(t) (6) (TDDFT: Time- Dependent Density Functional Theory) 11) A t1 t dt Ψ(t) i t H total(t) Ψ(t) (7) (6) δa = TDDFT (RPA: Random-Phase Approximation) H V (r) H total (t) V (r, t) t DFT TDDFT DFT LDA TDLDA H Φ n HΦ n = E n Φ n (8) H (t) Ψ(t) e iht Φ(t) (9) 9
Φ(t) (6) i Φ(t) t = e iht H (t)e iht Φ(t) (1) t H (t) Φ(t) Φ n H (t) H (t) Φ(t) Ψ(t) t Φ(t) Φ n i dt e iht H (t )e iht Φ n (11) t Ψ(t) e ient Φ n i dt e ih(t t) H (t )e ih(t t) e ient Φ n [ ] = 1 i dt e iht H (t t )e iht e ient Φ n (12) A H (t) H (t) = e iωt A (13) H (t) B B Ψ(t) B Ψ(t) Φ n B Φ n = i = i dt e iω(t t ) Φ n [B, e iht Ae iht ] Φ n dt e iω(t t ) Φ n [e iht Be iht, A] Φ n (14) [A, B] A B AB BA t Φ n B B = e iωt Q BA (t) dt e iωt Q BA (t ) (15) Q BA (t) iθ(t) [B(t), A] (16) B B(t) = e iht Be iht TDDFT TDLDA 1
(16) A B ρ(r) Q ρρ (r, r ; t) iθ(t) [ρ(r, t), ρ(r )]. (17) Q ρρ (r, r ; t) Q ρρ (q, ω) = dte iωt dre iq r dr e iq r Q ρρ (r, r ; t) (18) Q ρρ (q, ω) S(q, ω) S(q, ω) = 1 1 1 e βω π Im Q ρρ(q, ω) (19) ω q QMC 4 N 1 U U V N N N ± 1 N 11
N TDDFT 12) (Luttinger-Ward) (Baym-Kadanoff) (Hedin) GW (SEROT: Self-Energy Revision Operator Theory) GWΓ 13) TDDFT TDDFT SEROT [1] 1987 [2] A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg: Dynamical meanfield theory of strongly correlated fermion systems and the limit of infinite dimensions, Rev. Mod. Phys. 68 (1996) 13. [3] K. held, I. A. Nekrasov, N. Blümer, V. I. Anisimov, and D. Vollhardt: Realistic modeling of strongly correlated electron systems: An introduction to the LDA+DMFT approach, Int. J. Mod. Phys. B 15 (21) 2611. [4] 23 (1988) 1. [5] P. Hohenberg and W. Kohn: Inhomogeneous electron gas, Phys. Rev. 136 (1964) B864; W. Kohn and L. J. Sham: Self-consistent equations including exchange and correlation effects, Phys. Rev. 14 (1965) A1133; W. Kohn: 12
Nobel lecture: Electronic structure of matter wave functions and density functionals, Rev. Mod. Phys. 71 (1999) 1253. [6] Kohn 34 (1999) 148 [7] D. M. Ceperley: Microscopic simulations in physics, Rev. Mod. Phys. 71 (1999) S438. [8] 1999 [9] 29 (1994) 25, 183, 743; 3 (1995) 12 929. [1] h = k B = 1 [11] E. Runge and E. K. U. Gross: Density-functional theory for time-dependent systems, Phys. Rev. Lett. 52 (1984) 997; E. K. U. Gross, J. F. Dobson, and M. Petersilka: Density functional theory of time-dependent phenomena, R. F. Nalewajski Topics in Current Chemistry: Density Functional Theory II 181 (Springer, Berlin, 1996 ) 81; M.E. Casida Time-dependent density-functional response theory for molecules, D.P. Chong Recent Advances in Density Functional Methods Part I (Singapore, World Scientific, 1995 ) 155 Time-Dependent Density Functional Response Theory of Molecular Systems: Theory, Computational Methods and Functionals, J. M. Seminario Recent Developments and Applications of Modern Density Functional Theory (Elsevier, Amsterdam, 1996 ) 391. [12] D. Pines: Elementary excitations in solids, (Benjamin, 1963 RPA 4 [13] Y. Nambu: Quasiparticles and gauge invariance in the theory of superconductivity, Phys. Rev. 117 (196) 648. J. R. Schrieffer: Theory of superconductivity, Benjamin, 1964 13
Theories for Dynamical Response of Many-Electron Systems Yasutami Takada 19 7 9 Based on the formal framework developed from very elementary quantum mechanics and statistical mechanics, the concepts of the self-energy and the polarization function are explained. 1 G σσ (r, r ; t) Q ρρ (r, r ; t) 1
2 G σσ (r, r ; τ) G σσ (r, r ; t) 2.1 H { n } H n = E n n (1) G σσ (r, r ; t) G σσ (r, r ; ω) = nm dte iωt G σσ (r, r ; t) i dte iωt {ψ σ (r, t), ψ + σ (r )} e β(ω E n) (e β(e n E m ) + 1) n ψ+ σ (r ) m m ψ σ (r) n ω + i + + E m E n (2) T β = 1/T Ω T ln(tr e βh ) t t + e +t ω ω + i + ω A σσ (r, r ; ω) A σσ (r, r ; ω) 1 π Im G σσ (r, r ; ω) = e β(ω En) (e βω +1) n ψ + σ (r ) m m ψ σ (r) n δ(ω+e m E n ) (3) nm G σσ (r, r ; ω) G σσ (r, r ; t) G σσ (r, r ; ω) = G σσ (r, r ; t) = iθ(t) de A σσ (r, r ; E) ω + i + E (4) de e iet A σσ (r, r ; E) (5) { n } dea σσ (r, r ; E) = δ σ,σ δ(r r ) (6) (Sum rule) ω lim G σσ (r, ω r ; ω) = δ σ,σ δ(r r ). (7) ω 2
ω 2.2 G σσ (r, r ; τ) G σσ (r, r ; τ) T τ ψ σ (r, τ)ψ + σ (r ) θ(τ) ψ σ (r, τ)ψ + σ (r ) + θ( τ) ψ + σ (r )ψ σ (r, τ) (8) ψ σ (r, τ) e Hτ ψ σ (r)e Hτ G σσ (r, r ; τ) = dea σσ (r, r ; E)e Eτ [ θ(τ)f( E) + θ( τ)f(e)] (9) f(e) = (1 + e βe ) 1 G σσ (r, r ; τ + β) = G σσ (r, r ; τ) (1) τ G σσ (r, r ; τ) β G σσ (r, r ; τ) = T ω p e iωpτ G σσ (r, r ; iω p ) (11) ω p p p =, ±1, ±2, ω p = πt (2p + 1) G σσ (r, r ; iω p ) = β dτ e iω pτ G σσ (r, r ; τ) = de A σσ (r, r ; E) iω p E (12) G σσ (r, r ; ω) G σσ (r, r ; iω p ) ω iω p ω + i + 2.3 B ρρ (r, r ; ω) B ρρ (r, r ; ω)= e β(ω En) (e βω 1) n ρ(r ) m m ρ(r) n δ(ω+e m E n ) (13) nm 3
Q ρρ (r, r ; t) Q ρρ (r, r ; ω) = dt e iωt Q ρρ (r, r ; t) = de B ρρ(r, r ; E) ω + i + E (14) ω (ω E) 1 ω 1 + Eω 2 ω 1 [ρ(r), ρ(r )] = ω 2 lim Q ρρ(r, r ; ω) = [[ρ(r), H], ρ(r )]. (15) ω ω 2 (15) f Q ρρ (r, r ; τ) Q ρρ (r, r ; τ) T τ ρ(r, τ)ρ(r ) θ(τ) ρ(r, τ)ρ(r ) θ( τ) ρ(r )ρ(r, τ) (16) B ρρ (r, r ; ω) Q ρρ (r, r ; τ) = deb ρρ (r, r ; E)e Eτ [θ(τ)n( E) θ( τ)n(e)] (17) n(e) = (e βe 1) 1 Q ρρ (r, r ; τ) β ω q = 2πT q q =, ±1, ±2, Q ρρ (r, r ; τ) = T ω q e iωqτ Q ρρ (r, r ; iω q ) (18) Q ρρ (r, r ; iω q ) = β dτ e iω qτ Q ρρ (r, r ; τ) = de B ρρ(r, r ; E) iω p E (19) Q ρρ (r, r ; ω) Q ρρ (r, r ; ω) = Q ρρ (r, r ; iω q ) iωq ω+i + (2) Q ρρ (r, r ; iω q ) [1] Q ρρ (r, r ; ω) 3 G σσ (r, r ; τ) H 4
H H = σ + 1 2 ( dr ψ σ + (r) 1 σσ ) 2m 2 r + v(r) ψ σ (r) dr dr ψ σ + (r)ψ + σ (r )u(r, r )ψ σ (r )ψ σ (r) (21) v(r) u(r, r ) G σσ (r, r ; τ) σ = σ σ 3.1 (8) G(r, r ; τ) τ G(r, r ; τ) τ = δ(τ) {ψ σ (r), ψ + σ (r )} T τ e Hτ [H, ψ σ (r)]e Hτ ψ + σ (r ) (22) δ(r r ) T τ (8) u(r, x) G(r, r ; τ) τ ( + δ(τ)δ(r r ) + 1 ) 2m 2 r + v(r) G(r, r ; τ) = dx u(r, x) T τ ψ + σ (x, τ)ψ σ (x, τ)ψ σ(r, τ)ψ σ + (r ) σ = = β dx u(r, x) dτ δ(τ τ ) T τ ψ σ (r, τ) ψ + σ (x, τ )ψ σ (x, τ )ψ σ + (r ) σ β dx u(r, x) dτ δ(τ τ ) T τ ψ σ (r, τ)ρ(x, τ )ψ σ + (r ) (23) δ(τ τ ) ρ(x)[ σ ψ+ σ (x)ψ σ (x)] τ τ τ (23) (11) ( iω p + 1 ) 2m 2 r v(r) G(r, r ; iω p ) dx Σ(r, x; iω p )G(x, r ; iω p ) = δ(r r ) (24) (23) Σ(r, x; iω p ) (23) 5
(24) δ(τ τ ) Σ(r, x; iω p )G(x, r ; iω p ) β β = u(r, x) dτe iω pτ dτ T e iω q(τ τ ) T τ ψ σ (r, τ)ρ(x, τ )ψ σ + (r ) (25) ω q (25) ω q ω p (= ω p + ω q ) x z Σ(r, z; iω p )G(z, r ; iω p ) = T ω p u(r, z) β β dτe iω p τ dτ e i(ω p ω p )τ T τ ψ σ (r, τ)ρ(z, τ )ψ σ + (r ) (26) G(z, r ; iω p ) G 1 (r, x; iω p ) dr G(z, r ; iω p ) G 1 (r, x; iω p ) = δ(z x) (27) (26) G 1 (r, x; iω p ) r z Σ(r, x; iω p ) = T β β dz dr u(r, z) dτe iω p τ dτ e i(ω p ω p )τ ωp T τ ψ σ (r, τ)ρ(z, τ )ψ + σ (r ) G 1 (r, x; iω p ) (28) dy δ(r y ) = dy dy G(r, y; iω p )G 1 (y, y ; iω p ) (29) Σ(r, x; iω p ) = T ωp dz dy u(r, z)g(r, y; iω p ) Λ (y, z, x; iω p, iω p ) (3) Λ (y, z, x; iω p, iω p ) Λ (y, z, x; iω p, iω p ) = β β dy dx dτe iω p τ dτ e i(ω p ω p )τ G 1 (y, y ; iω p ) T τ ψ σ (y, τ)ρ(z, τ )ψ σ + (x ) G 1 (x, x; iω p ) (31) ψ σ (r, τ) r δ(r y ) y 6
3.2 (3) (31) T τ ψ σ (y, τ)ρ(z, τ )ψ + σ (x ) T τ ψ σ (y, τ)ρ(z, τ )ψ + σ (x ) = σ T τ ψ σ (y, τ)ψ + σ (z, τ )ψ σ (z, τ )ψ + σ (x ) T τ ψ σ (y, τ)ψ + σ (x ) ρ(z, τ ) + T τ ψ σ (y, τ)ψ + σ (z, τ ) T τ ψ σ (z, τ )ψ + σ (x ) = G(y, x ; τ) ρ(z) + G(y, z; τ τ )G(z, x ; τ ) (32) ρ(z, τ ) = e Hτ ρ(z)e Hτ = ρ(z) τ (8) τ = + ρ(z) = σ G(z, z; + ) = σ T ω p G(z, z; iω p )e iωp+ (33) G(z, z; iω p ) (32) (31) Λ (y, z, x; iω p, iω p ) (32) Λ H (y, z, x; iω p, iω p ) Λ F (y, z, x; iω p, iω p ) Λ H (y, z, x; iω p, iω p ) = βδ ωp,ω p ρ(z) dy dx G 1 (y, y ; iω p )G(y, x ; iω p )G 1 (x, x; iω p ) = βδ ωp,ω p ρ(z) G 1 (y, x; iω p ) (34) = Λ F (y, z, x; iω p, iω p ) dx dy G 1 (y, y ; iω p )G(y, z; iω p )G(z, x ; iω p )G 1 (x, x; iω p ) = δ(y z)δ(z x) (35) Λ (y, z, x; iω p, iω p ) = Λ H (y, z, x; iω p, iω p )+Λ F (y, z, x; iω p, iω p ) (3) Σ(r, x; iω p ) Σ H (r, x) Σ F (r, x) Σ H (r, x) = dz dy u(r, z)g(r, y; iω p ) ρ(z) G 1 (y, x; iω p ) = δ(r x) dz u(r, z) ρ(z) (36) Σ F (r, x) = T dz dy u(r, z)g(r, y; iω p )δ(y z)δ(z x) ω p = u(r, x)t ω p G(r, x; iω p ) (37) 7
iω p Σ H (r, x) Σ F (r, x) iω p (37) ω p (33) e iω p + Σ(r, x; iω p ) (24) Σ H (r, x) Σ F (r, x) v(r) V (r) v(r) + dz u(r, z) ρ(z) (38) V (r) u(r, z) = e 2 / r z Σ F (r, x) 3.3 (24) Σ(r, x; iω p ) G(r, r ; iω p ) u(r, x) (3) G(r, r ; iω p ) Σ(r, x; iω p ) δ(r x) iω p Σ(r, x; iω p ) Σ H (r, x) Σ(r, x; iω p ) (31) Λ (y, z, x; iω p, iω p ) Λ H (y, z, x; iω p, iω p ) Λ (y, z, x; iω p, iω p ) 8
Λ F (y, z, x; iω p, iω p ) Λ (y, z, x; iω p, iω p ) 3.4 Λ (y, z, x; iω p, iω p ) (31) ρ(z, τ ) j µ (z, τ ) Λ µ (y, z, x; iω p, iω p ) j µ (z) j µ (z) ( 1 ψ σ + (z) [ ] ) ψ σ (z) ψ σ + (z) ψ σ (z), (µ = x, y, z) (39) σ 2mi z µ z µ j µ (z) = Λ µ Λ µ (iω p iω p ) Λ (y, z, x; iω p, iω p ) (31) T τ ψ σ (y, τ)ρ(z, τ )ψ + σ (x ) T τ ψ σ (y, τ)ρ(z, τ )ψ + σ (x ) / τ T τ ψ σ (y, τ)ρ(z, τ )ψ σ + (x ) τ = δ(τ τ )δ(z y )G(z, x ; τ) δ(τ )δ(z x )G(y, z; τ) + T τ ψ σ (y, τ)e Hτ [H, ρ(z)]e Hτ ψ σ + (x ) (4) H ρ(z) [H, ρ(z)] = i µ=x,y,z j µ (z) z µ (41) ( e iht ρ(z)e iht) + ( e iht j µ (z)e iht) = (42) t z µ µ=x,y,z (4) (41) (31) (iω p iω p )Λ (y, z, x; iω p, iω p ) i Λ µ (y, z, x; iω p, iω p ) z µ µ=x,y,z = δ(z x)g 1 (y, z; iω p ) δ(z y)g 1 (z, x; iω p ) (43) Λ (y, z, x; iω p, iω p ) ω p = ω p Λ Λ 9
4 4.1 Q ρρ (r, r ; iω q ) β Q ρρ (r, r ; iω q ) = dτ e iωqτ T τ ρ(r, τ)ρ(r ) = σ β dτ e iω qτ T τ ψ σ (r, + )ρ(r, τ )ψ + σ (r ) (44) (31) β dτ e iω qτ T τ ψ σ (r, τ)ρ(z, τ )ψ + σ (r ) = T ω p e iω p τ dx dy G(r, y; iω p ) Λ (y, z, x; iω p, iω p + iω q )G(x, r ; iω p + iω q ) (45) Q ρρ (r, r ; iω q ) = T e iωp+ dx dy G(r, y; iω p ) σ ω p Λ (y, r, x; iω p, iω p + iω q )G(x, r ; iω p + iω q ) (46) Q ρρ (r, r ; ω) ω q > Λ Λ (46) Λ Λ 4.2 r r W (r, r ; iω q ) u(r, r ) u(r, x) x y u(y, r ) u(r, r ) x y Q ρρ (x, y; iω q ) W (r, r ; iω q ) W (r, r ; iω q ) = u(r, r ) + dx dy u(r, x)q ρρ (x, y; iω q )u(y, r ) (47) 1
(3) u(r, z) W (r, z; iω p iω p ) Γ (y, z, x; iω p, iω p ) Λ (y, z, x; iω p, iω p ) = Γ (y, z, x; iω p, iω p ) + dz dz Q ρρ (z, z ; iω p iω p ) u(z, z )Γ (y, z, x; iω p, iω p ) (48) (47) (48) dz u(r, z)λ (y, z, x; iω p, iω p ) = dz W (r, z; iω p iω p )Γ (y, z, x; iω p, iω p ) (49) Γ (y, z, x; iω p, iω p ) (46) Λ (y, z, x; iω p, iω p ) Π(r, r ; iω q ) Π(r, r ; iω q ) = T e iω p + dx dy G(r, y; iω p ) σ ω p Γ (y, r, x; iω p, iω p + iω q )G(x, r ; iω p + iω q ) (5) (48) Q ρρ (r, r ; iω q ) = Π(r, r ; iω q ) dz dz Q ρρ (r, z; iω q )u(z, z )Π(z, r ; iω q ) (51) W (r, r ; iω q ) W (r, r ; iω q ) = u(r, r ) dx dy W (r, x; iω q )Π(x, y; iω q )u(y, r ) (52) 4.3 Q ρρ (r, r ; iω q ) Λ (y, z, x; iω p, iω p ) Π(r, r ; iω q ) Γ (y, z, x; iω p, iω p ) Π Γ 11
(1) W(r,r';iω ) W = q + u u Λ G G u (2) Γ (y,z,x;iω,ω ) = + Γ p' p Λ = + Λ Γ Γ (3) Π (r,r';iω ) q Λ = Γ + Λ Γ 1: ϕ ext (r, t ) n ind (r, t) n ind (r, t) = dr dt Q ρρ (r, r ; t t )ϕ ext (r, t ) (53) n ind = Q ρρ ϕ ext n ind = Πϕ eff n ind (r, t) = dr dt Π(r, r ; t t )ϕ eff (r, t ) (54) ϕ eff ϕ eff = ϕ ext /(1 + uπ) (51) Q ρρ = Π Q ρρ uπ Q ρρ = Π/(1 + uπ) (54) ϕ ext D E 12
E ϕ eff D E Q ρρ (r, r ; iω q ) Π(r, r ; iω q ) (5) Γ (y, z, x; iω p, iω p ) Λ (y, z, x; iω p, iω p ) (2) Γ (y, z, x; iω p, iω p ) Λ (y, z, x; iω p, iω p ) u Γ (y, z, x; iω p, iω p ) Λ (y, z, x; iω p, iω p ) u (43) Λ Λ µ Γ Γ µ 5 G(r, r ; iω p ) Σ(r, x; iω p ) ( iω p + 1 ) 2m 2 r v(r) dz u(r, z) ρ(z) G(r, r ; iω p ) dx Σ(r, x; iω p )G(x, r ; iω p ) = δ(r r ), (55) Σ(r, x; iω p ) = T dy dz G(r, y; iω p )W (r, z; iω p iω p ) ω p Γ (y, z, x; iω p, iω p ) (56) Π(r, r ; iω q ) (5) W (r, r ; iω q ) (52) [2] 13
(31) (31) [1] ω Q ρρ (r, r ; ω) ω q > ω Q ρρ (r, r ; iω q ) iω q ω + i + 1999 122 [2] L. Hedin: New Method for Calculating the One-Particle Green s Function with Application to the Electron-Gas Problem, Phys. Rev. 139 (1965) A796 Appendix A G. Baym and L. Kadanoff; Conservation Laws and Correlation Functions, Phys. Rev. 124 (1961) 287 S. Engelsberg and J. R. Schrieffer: Coupled Electron-Phonon Systems, Phys. Rev. 131 (1963) 993 Appendic B 14
Theories for Dynamical Response of Many-Electron Systems Yasutami Takada 16 5 7 We shall explain the Luttinger-Ward formalism, the Baym-Kadanoff conserving approximation, and the Hedin s GW approximation with emphasis of physical backgrounds and implications rather than mathematical details. 1 G(r, r ; iω p ) Σ(r, r ; iω p ) Q ρρ (r, r ; iω q ) Π(r, r ; iω q ) Γ (y, z, x; iω p,iω p ) Γ e e ( eγ ) 1
(1) Φ[G] [1] (2) [2] (3) [3] [4] (7.1) H G(r, r ; iω p ) G(iω p ) Σ(iω p )G(iω p ) (r, r ) ( Σ(iωp )G(iω p ) ) r,r = dx Σ(r, x; iω p )G(x, r ; iω p ) iω p G(iω p ) G Tr G(iω p ) Tr G T ω p 2 2.1 H u(r, r ) H U G (iω p ) (9.1) U = ( ) iω p + 1 2m 2 r v(r) G (r, r ; iω p )=δ(r r ) (1) v(r) G (iω p ) U G(iω p ) U 2
G(r, r ; τ) τ H H τ U e τh e τh U G(r, r ; τ) G(r, r ; iω p ) β [ ( β G(r, r ; iω p )= dτe iωpτ T τ exp dτ U(τ ) ) ψ σ (r,τ)ψ σ + (r ) ] c (2) H τ c G(r, r ; τ) ψ σ (r,τ) ψ + σ (r ) 2.2 (2) G U G () G G (1) G (1) = G Σ H [G ] G + G Σ F [G ] G (3) Σ H [G ] Σ F [G ] (7.16) (7.17) Σ H Σ F G G 1 (1) First-Order Skeleton Diagrams for the Self-Energy (1a) Σ : Hartree Term (1b) H ΣF : Fock Term (2) Second-Order Skeleton Diagrams for the Self-Energy (2a) Σ2a : Direct Term (2b) Σ2b: Exchange Term 1: 3
G (2) 1 Σ 2a [G ] Σ 2b [G ] G (2) = G Σ 2a [G ] G + G Σ 2b [G ] G +G Σ H [G Σ H [G ] G ] G + G Σ H [G Σ F [G ] G ] G +G Σ F [G Σ H [G ] G ] G + G Σ F [G Σ F [G ] G ] G +G Σ H [G ] G Σ H [G ] G + G Σ H [G ] G Σ F [G ] G +G Σ F [G ] G Σ H [G ] G + G Σ F [G ] G Σ F [G ] G (4) (3) Σ 1 [G ] Σ H [G ]+Σ F [G ] (4) G Σ 1 [G (1) ] G + G Σ 1 [G ] G (1) (5) Σ 2 [G ] Σ H [G ] Σ F [G ] n n Σ n [G ] G Σ n [G ] G G Σ n [G] G = G (n) = G + G n= n=1 Σ n [G] G (6) Σ H [G] Σ F [G] Σ H Σ F 2.3 (1) G (9.1) G = G + G (Σ H +Σ)G (7) iω p + 2 r /2m v(r) (6) (7) Σ Σ=Σ F [G]+ Σ n [G] (8) 4 n=2
2.4 Φ[G ] Φ[G ] n Φ n [G ] δφ n [G ] δg Σ n [G ] (9) 1 Φ[G ]= Φ n [G ]= n=1 n=1 2n Tr ( G Σ n [G ] ) (1) Φ[G ] 2 Φ[G ] G Φ[G ] Φ[G ] = + + + + + + + + + +... 2: Φ[G ] G Σ H [G ] Σ F [G ] Σ 2a [G ] Σ 2b [G ] n 2n G (1) 1/2n Φ n [G ] Σ n [G ] Φ[G ] G G Φ[G] Σ Σ H +Σ= Σ n [G] = n=1 n=1 δφ n [G] δg = δφ[g] δg (11) 5
2.5 Φ[G] Φ[G] Ω( T ln[tr e βh ]) Ω { Ω= Tr e iωp+ ln ( G(iω p ) 1) + G(iω p ) ( Σ H +Σ(iω p ) )} +Φ[G] (12) (7) Σ H +Σ=G 1 G 1 G (12) G Ω[G] G δω[g] δg = G δ ( G 1 +Σ H +Σ ) Σ H Σ+ δφ δg δg (13) (7) (11) δω[g]/δg = G Ω[G] U [5] 3 3.1 (11) Φ[G] ` ` ` Φ[G] 6
Φ[G] Φ[G] Φ[G] Φ[G] =Σ H [G]+Σ F [G] Φ[G] Φ[G] Φ[G] Φ[G] 3.2 Φ[G] G Q ρρ Φ[G] (8) Σ (7.11) Λ Γ 3 Γ Q ρρ Σ Γ (7.23) Σ Γ 7
Γ = + + + + + + + + +... 3: Γ Φ[G] Φ[G] Φ[G] G (11) Σ[G] Σ[G] (7) G Σ[G] G Ĩ[G] Ĩ[G] Ĩ[G] = δσ[g] δg ( = δ2 Φ[G] ΦH [G] ) δgδg Φ[G] Φ H [G] Φ[G] Φ H [G] 4 Ĩ[G] G Γ Γ = + I Γ (14) 4: Γ Φ[G] Φ H [G] Φ[G] Λ Γ Q ρρ 5 Φ[G] Φ H [G] 2 Ĩ[G] Ĩ[G] 4 Γ 3 8
= + + + I + + + 5: Ĩ[G] Φ[G] Φ[G] G Q ρρ δσ(r 1 τ 1 ; r 2 τ 2 ;[G]) δg(r 1τ 1; r 2τ 2) = δ 2( Φ[G] Φ H [G] ) δg(r 1 τ 1 ; r 2 τ 2 )δg(r 1τ 1; r 2τ 2) = δσ(r 1τ 1; r 2τ 2;[G]) δg(r 1 τ 1 ; r 2 τ 2 ) ` ` ` ` Φ[G] ` ` ` ` ` ` (15) 3.3 Φ[G] Ω (12) (9) G G Ω[G] Ω[G] G Φ[G] (13) (11) δω[g]/δg = G Ω[G] G G G + δg G Σ H +Σ=G 1 G 1 δ 2 Ω[G] δgδg = δ(g 1 ) δg + δ2 Φ δgδg 9 (16)
GG 1 =1 G (δ(g 1 )/δg) G = 1 Λ Λ =1+G (δ 2 Φ[G]/δGδG) G Λ Λ 1 =1 G (δ 2 Φ[G]/δGδG) G G δ2 Ω[G] δgδg G = Λ 1 (17) G 1 (8.3) Q ρρ Tr { δg δ2 Ω[G] δgδg δg} = Tr { δg Q 1 ρρ δg } (18) δg Q ρρ Q ρρ 3.4 [6] Φ[G] = + + +... + + +... + + +... 6: Φ[G] 6 Φ[G] 1
Fluctuation Exchange: FLEX 2 FLEX Φ[G] 7 7: FLEX FLEX FLEX 4 4.1 U U U Φ[G] U Ũ Ũ U U U (8.4) (8.9) W Ũ 11
r s U r s /(1 + r s ) W 1.9 <r s < 5.6.66 <r s /(1 + r s ) <.85 4.2 U W Φ[G] G G Σ Π W Γ 8 (9.1) (7) (1) G : = + + Σ (2) Σ : Σ = Γ (3) Π : Π = Γ (4) W : = + Π (5) Γ : δσ Γ = + Γ δg 8: (9.2) (8.7) W (r, r ; iω q ) r r iω q iω q (8.9) 12
4 Ĩ (14) G Σ Π W Γ 4.3 Σ G W Σ Π Γ G U G G G G G G U 8 (4) U W G Σ[G, W ] Π[G, W ] Γ [G, W ] 8 (2) (3) (5) W W Γ [G, W ] 8 (5) Γ () [G] =1 8 (3) Π[G, W ] Π () [G] = GG 8 (2) Σ[G, W ] Σ (1) [G, W ]= GW 9 Γ [G, W ] 8 (5) Γ () δσ (1) [G, W ]/δg 9(1) Γ (1) [G, W ] Π (1) [G, W ] Σ (2) [G, W ] 9(2) 13
() O(W ) : () Γ = () (1) Π = ; Σ = 1 (1) O(W ) : (1) Γ = (1) (2) Π = ; Σ = 2 (2) (2) O(W ) : Γ = + + + + + 9: Γ [G, W ] Π[G, W ] Σ[G, W ] W Γ (2) [G, W ] Π (2) [G, W ] Σ (3) [G, W ] Γ (3) [G, W ] 4.4 Σ[G, W ]= n=1 Σ (n) [G, W ] W W U Σ[G, W ]=Σ (1) [G, W ]= GW G W GW W Π () [G] W = u/(1 + u Π () [G]) G Σ 9() Σ (1) 1(1b) Σ F GW 14
U W (7.23) Π () Q ρρ GW [7] G G Π () Σ (1) G G G v(r) 5 FLEX GW 199 [1] J. M. Luttinger and J. C. Ward, Phys. Rev. 118 (196) 1417. [2] G. Baym and L. P. Kadanoff, Phys. Rev. 124 (1961) 287; G. Baym, Phys. Rev. 127 (1961) 1391. 15
[3] L. Hedin, Phys. Rev. 139 (1965) A796. [4] (1) (2) 1999 3.3 Σ H Σ [5] P. Nozières and J. M. Luttinger, Phys. Rev. 127 (1962) 1423; J. M. Luttinger and P. Nozières, Phys. Rev. 127 (1962) 1431. [6] N. E. Bickers, D. J. Scalapino, and S. R. White, Phys. Rev. Lett. 62 (1989) 961; N. E. Bickers and S. R. White, Phys. Rev. B 43 844 (1991). [7] GW F. Aryasetiawan and O. Gunnarsson, Rep. Prog. Phys. 61 (1998) 237; W. G. Aulbur L. Jönsson, and J. W. Wilkins, in Solid State Physics, edited by H. Ehrenreich and F. Spaepen (Academic, New York, 2), Vol. 54, p.1 16
Theories for Dynamical Response of Many-Electron Systems Yasutami Takada 17 8 11 Basic concepts in constructing the self-energy revision operator theory are explained with emphasis on its close connection with both the Baym-Kadanoff conserving approximation and the Hedin s GW approximation. Based on this exact theoretical framework a practical approximation scheme named the GWΓ method is introduced and its usefulness is illustrated by its application to the homogeneous electron gas. 1 G Q ρρ Σ Π G Q ρρ 1
Γ ( Σ GW Φ[G] G W u/(1 + uπ) u Σ Π 2 2.1 1 Φ[G] G Φ input [G] Φ input [G] Φ input [G] Φ[G] Σ Σ 2
Φ input[g] δφ input[g] Σ[G] = δg -1-1 G = G Σ[G] 2 δ Φ input[g] I = δg δg Γ = 1 + GIGΓ Π = GGΓ 1: [1] 1 Σ[G] G Σ G Γ 2 (a) Σ G Γ 2 (b) (a) Σ (b) Σ G Γ G Γ 2: (a) Σ 1 G Γ (b)σ G Γ Φ[G] 3
Φ[G] Σ δφ[g]/δg Φ[G] 2.2 S {Σ[G]} F S Σ input [G] F [ Σ input [G] ] Σ output [G] S (1) (7.1) H F S Σ input [G] Ĩinput = δσ input [G]/δG Ĩinput Γ input = 1+GĨinputGΓ input Γ input Π input = GGΓ input W input = u/(1 + uπ input ) Σ output [G] = GW input Γ input Σ input [G] Σ output [G] Σ input [G] Σ input [G] ( S Σ input [G] Σ output [G] F 2.3 F [1] (i) Σ input [G] = Σ F [G] m m F Σ output [G] Σ (m) [G] Σ (m) [G] Σ (m+1) [G](= F[Σ (m) [G]]) 4
(ii) Σ (m) [G] u (m + 1) (iii) Σ input [G] F F[Σ input [G]] Σ F [G] F m [Σ input [G]] u m Σ input [G] F Σ[G] = lim m F m [Σ input [G]] (2) Σ[G] (i) (iii) Σ[G] Σ input [G] (2) F[Σ[G]] = Σ[G] (3) F S Σ[G] G Σ Γ F Φ[G] F S 2 (a) (b) F Γ Σ Γ F δφ[g]/δg Σ Γ 5
3 GW F Σ input [G] Σ[G] (2) (5) (1) F Σ[G] Σ input [G] Σ[G] Σ[G] G W GW W Σ Σ[G] F Σ[G] F F Γ GW Γ GW 4 4.1 Σ F δσ input [G]/δG F Σ 6
[2] n(r) E xc [n(r)] n(r) E xc [n(r)] δσ input [G]/δG F [3] p iω p σ G σ (p; iω p ) G(p) p T ω p p σ 4.2 F Ĩinput ( input Ĩ(p + q, p; p + q, p ) Γ (p + q, p) Ĩ(p + q, p; p + q, p ) Γ ν (p + q, p) ν = x y z i = ν γ i (p + q, p) [ 1 i = (2p i + q i )/2m i = ν ] m Γ i (p + q, p) = γ i (p + q, p) + p Ĩ(p + q, p; p + q, p )G(p )G(p + q)γ i (p + q, p ) (4) iω q Γ (p + q, p) ν=x,y,z q ν Γ ν (p + q, p) 7
= G (p + q) 1 Σ(p + q) G (p) 1 + Σ(p) (5) G (p) ε p = p 2 /2m µ µ 1/(iω p ε p ) (5) Ĩ Σ Σ input Γ input Σ input Γ input Γ input 4.3 (5) Ĩ Γ ν(p + q, p) Γ (p + q, p) R(p + q, p) R(p + q, p) Γ (p + q, p) ν=x,y,z q ν γ ν (p + q, p)/ ν=x,y,z q ν Γ ν (p + q, p) (6) R(p + q, p) (6) (5) Γ (p + q, p) Γ (p + q, p) = G (p + q) 1 Σ(p + q) G (p) 1 + Σ(p) iω q (ε p+q ε p )/R(p + q, p) (7) Γ ν (p + q, p) ν=x,y,z q ν q Γ ν(p + q, p) = 1 G (p + q) 1 Σ(p + q) G (p) 1 + Σ(p) q 1 + R(p + q, p)iω q /(ε p+q ε p ) (8) (7) (8) R(p + q, p) 8
4.4 Σ input [G] Σ[G] R(p + q, p) [4] 1 ω p F 2 q lim lim R(p + q, p) p =pf = 1 (9) ω q q lim lim R(p + q, p) p =pf = κ (1) q ω q κ κ κ 4.5 (4) Ĩ(p + q, p; p + q, p ) p p q Ĩ(p + q, p; p + q, p ) Ĩ(q) p Ĩ(q) p p Ĩ(p + q, p; p + q, p )G(p )G(p + q)γ (p + q, p ) p G(p )G(p + q)γ (p + q, p ) (4) i = i = ν Ĩ(p+q, p; p +q, p ) Ĩ(q) p ν=x,y,z (11) Γ (p + q, p) = 1 Ĩ(q) pπ(q) (12) q ν Γ ν (p + q, p) = ε p+q ε p iω q Ĩ(q) pπ(q) (13) ν=x,y,z q ν γ ν (p + q, p) = ε p+q ε p (12) (13) R(p + q, p) R(p + q, p) = 1 Ĩ(q) pπ(q) 1 Ĩ(q) pπ(q)iω q /(ε p+q ε p ) 9 (14)
(14) (9) (1) [ lim 1 Ĩ(q) p Π(q) ] p =pf,ωq = κ (15) q κ Ĩ(q) p 4.6 Γ (14) (7) Γ Γ (p + q, p) = [ 1 Ĩ(q) pπ(q) ] G(p + q) 1 G(p) 1 G (p + q) 1 G (p) 1 (16) Γ = Γ (a) Γ (b) Γ (a) 1 Ĩ(q) pπ(q) Γ (b) [G(p + q) 1 G(p) 1 ]/[G (p + q) 1 G (p) 1 ] Γ (12) Γ = Γ (a) Γ = Γ (a) Γ (b) Γ (b) Γ = Γ (b) GISC(Gauge-Invariant Self-Consistent ) [5] Γ (a) Ĩ(q) p p Ĩ(q) (16) Π(q) Π(q) = p G(p + q)g(p)γ (p + q, p) = Π (b) (q) [ 1 Ĩ(q) Π(q)] (17) Π (b) (q) Π (b) (q) p G(p + q)g(p)γ (b) (p + q, p) = Π (q) 2 p [ ] G (p)σ(p)g(p) Re (18) iω q ε p+q + ε p Π (q) [ p G (p + q)g (p)] (17) ϵ(q) u(q) 4πe 2 /q 2 Π (b) (q) ϵ(q) 1 + u(q)π(q) = 1 + u(q) 1 + Ĩ(q) Π(b) (q) (19) 1
G + (q) Π (q) ϵ(q) = 1 + u(q) 1 G + (q)u(q)π (q) (2) [6] (19) Π (q) Π (b) (q) G + (q)u(q) Ĩ(q) Ĩ(q) Ĩ(q) G +(q)u(q) f xc (q) [7] (16) 4.7 GWΓ (16) Γ input F Σ 3 W = Σ = GWΓ W u 1+uΠ Σ Π Γ Π= GGΓ G 1 = -1 G Σ G -1 ~ Γ = (1 I Π) G G -1 G G -1-1 3: GWΓ Γ = 1 GW GWΓ 11
5 GWΓ r s p F = 1/αr s a B α = (4/9π) 1/3.521 a B GWΓ Ĩ(q) f xc(q) f xc (q) [8] G(p) G(p) n(p) = lim η + T ω p G(p)e iω pη (21) n(p) z F 4 (a) n(p) (b) z F r s T E F n(p) 1..5 (a) : EPX : GWΓ..5 1. 1.5 2. p (units of p ) 1. (b).9 GWΓ.8 F z.7.6 : EPX : GW : FHNC.5 1 2 3 4 5 rs 4: (a) (b) 12
n(p) z F ( EPX Effective Potential Expansion [9] GW [1] FHNC Fermi Hypernetted Chain [11] GWΓ A(p, ω) G(p) ω A(p, ω) = ImG(p, ω)/π Padé 5 r s = 1 µ c (.136Ry) ω = ω ω + iγ ω = γ =.1πT =.1πE F 18 16 r =1 s 14 Dispersion of a free electron A(p,ω) (units of E ) -1 F 12 1 8 p=2.4p F p=2.2p F p=2.p F p=1.8p F p=1.6p F p=1.4p F 6 4 2 p=1.2p F p=.4p F p=.2p F p=1.p F p=.8p F p=.6p F p= -3-2 -1 1 2 3 4 5 6 ω (units of E F) 5: r s = 1 5 A(p, ω) p ω ω = E p ω = ε p E p ε p [12] 13
( ω p p < p F p > p F ω = E p ω p ω = E p + ω p 6 RPA Π (q) GW G W RPA GWΓ 6 (b) GWΓ ω = E p ω p ω p A (p,ω) (units of E ) 3-1 F 2 1 (a) p = p F r s = 4 E F = 3.1eV ω p= 1.9E F : GWΓ : GW : RPA : Noninteracting -6-4 -2 2 4 6 ω (units of E F) 3 A (p,ω) (units of E ) -1 F 2 1 (b) p = E p ωp -6-5 -4-3 -2-1 ω (units of E F) 6: r s = 4 5 A(p, ω) γ p p F ε p p 1.6p F p 2p F 14
p G Π [13] 7 Q ρρ (q, ω) ω = [14] RPA -1 -Q ρρ (q,)/n [units of (27.2eV) ] 15 1 5 r = 5 s : Monte Carlo : GWΓ : RPA 1 2 3 4 5 q (units of p ) 7: Q ρρ (q, ) S(q, ω) 8 RPA r s = 5 S(q, ω) r s S(q, ω) q q c.9p F q RPA Π q q c q b q a 15
-1 S(q,ω)/N [units of (27.2eV) ] 11 1 9 8 7 6 5 4 3 2 1 a a a a b a b a a ω p b b b q=.8p F q=.6p F b q=1.p F b b q=1.4p F q=1.2p F q=2.4p F q=2.2p F q=2.p F q=1.8p F q=1.6p F Electron Gas at r s= 5 : GWΓ : RPA 1 2 3 4 5 6 7 8 9 1 ω (units of E F) 8: S(q, ω) q RPA r s r s = 5.25 r s [15] S(q, ω) [13] 6 Σ Π GW GWΓ Ĩ(q) 16
f xc (q) Σ Π GWΓ Ĩ(q) (11) ( (11) Padé Σ G GWΓ [16] [1] Y. Takada, Phys. Rev. B 52 (1995) 1278. [2] P. Hohenberg and W. Kohn, Phys. Rev. 136 (1964) 864; W. Kohn and L. J. Sham, Phys. Rev. 14 (1965) A1133. [3] Y. Takada, Phys. Rev. Lett. 87 (21) 22642. [4] P. Nozières, Theory of interacting Fermi Systems (Benjamin, New York, 1964), Chap. 6. [5] Y. Takada, J. Phys. Chem. Solids 54 (1993) 1779. [6] C. A. Kukkonen and A. W. Overhauser, Phys. Rev. B 2 (1979) 55. [7] E. K. U. Gross, J. F. Dobson, and M. Petersilka, Density Functional Theory II, edited by R. F. Nalewajski (Springer, Berlin, 1996), Chap. 2, p. 81. [8] Y. Takada, Int. J. Mod. Phys. B 15 (21) 2595. [9] Y. Takada and H. Yasuhara, Phys. Rev. B 44 (1991) 7879. [1] L. Hedin, Phys. Rev. 139 (1965) A796. [11] L. J. Lantto, Phys. Rev. B 22 (198) 138. [12] H. Yasuhara, S. Yoshinaga, and M. Higuchi, Phys. Rev. Lett. 83 (1999) 325. [13] Y. Takada and H. Yasuhara, Phys. Rev. Lett. 89 (22) 21642. [14] S. Moroni, D. M. Ceperley, and G. Senatore, Phys. Rev. Lett. 75 (1995) 689. [15] Y. Takada, to appear in J. Superconductivity, 18 (25). [16] F. Bruneval, F. Sottile, V. Olevano, R. Del Sole, and L. Reining, Phys. Rev. Lett. 94 (25) 18642. 17