Outline PGeneral Many body 2Pconsistent Conclusions Basics
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4 Ĥ = kσ ϵ(k)c kσ c kσ + iσ V i n iσ + U i n i n i { Ĥ ext = d1d2 η σ(1, 2)c σ(1)c σ (2)... σ ] + [η (1, 2)c (1)c (2) + η (1, 2)c (2)c (1)... + ] [ ξ σ(1, 2)c σ (1)c σ (2) + ξ σ(1, 2)c σ(1)c σ(2)... σ ] } + [ ξ (1, 2)c (1)c (2) + ξ (1, 2)c (2)c (1)...
5 [ { )}] Ω[G (0) 1, H] = β 1 log exp β (Ĥ + Ĥ ext µ N G (0) 1 = [iω n ϵ(k) µ] G σσ (k, k ; τ, τ ) = 1 ħ { ρ H T { } { } ρ H = exp βĥ / exp βĥ [ ]} c kσ (τ)c k σ (τ ) = G 2 (12, 34) = 1 [ { ρ ħ 2 H T ψ(1) ψ(3) ψ(4) ψ(2) ]} δω[g (0) 1, H] δg (0) 1 (k, τ; k, τ ) 1 = (R 1, τ 1 )
6 ( ) Φ[G, H] = Ω[G (0) 1, H] d 1 G (0) 1 (1, 1) G 1 (1, 1) G( 1, 1 ) G α δφ[g, H] (12) =, Σ α δφ[g, 0] (12) = δhᾱ(2, 1) δgᾱ(2, 1) H=0 G (2)α δ 2 Φ[G, H] (13, 24) =, Λ α δ 2 Φ[G, 0] (13, 24) = δh α (4, 3)δHᾱ(2, 1) δg α (4, 3)δGᾱ(2, 1) H=0
7 G (0) 1 (1, 2) G 1 (1, 2) = Σ α δφ[g, H] (12) = δg α (2, 1) H=0 [ ] Σ σ (k) = U G σ (k ) 1 1 Γ σ σ (k, k ; q)g σ (k + q)g σ (k + q) βn βn k q Γ(k, k ; q) = Λ α (k, k ; q) [Λ α GG Γ] (k, k ; q) Λ α (13, 24) = δσα (1, 2) δg α (4, 3)
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9 σk σk σk Γσσ = Λσσ Λσσ Γσσ σ k +q σ k +q σ k +q k k Γ σσ (k, k ; q) = Λ eh σσ (k, k ; q) 1 βn q Λ eh σσ (k, k ; q ) G σ (k + q )G σ (k + q )Γ σσ (k + q, k + q ; q q ) Γ σσ = Λ eh σσ + Keh σσ
10 σk σk σk Γσσ = κσσ κσσ Γσσ σ k +q σ k +q σ k +k +q k k + k + q Γ σσ (k, k ; q) = Λ ee σσ (k, k ; q) 1 βn q Λ ee σσ (k, k + q ; q q ) G σ (k + q q )G σ (k + q )Γ σσ (k + q q, k ; q ) Γ σσ = Λ ee σσ + Kee σσ
11 Γ σσ = Λ ee σσ + Kee σσ = Λeh σσ + Keh σσ I = Λ eh Λ ee K ee K eh = Γ = I K ee K eh = Λ eh Λ ee \ I = I + K eh + K ee = Λ ee + Λ eh I Γ I G σ Λ eh Λ ee Σ Λ α Γ
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13 σk Σ σ = σk σk σk σk σk σk +q σk +q σk Γ σ σ σk σk k +q σk Σ σ = Λ eh σσ G σ k +q σk σk Σ σ (k, k ) = Σ σ (k) Σ σ (k ) G σ (k, k ) = G σ (k) G σ (k )
14 Γ (k, k ; q) = Λ U 1 Λ U χ (k k )Λ U χ (k k ) χ σσ (q) = (βn) 1 k G σ(k)g σ (k + q) Λ U χ (0) = 1
15 Λ eh Λ U eh Σ (η ) = Λ U η eh η Σ ( η ) = Λ eh
16 Λ ee ee Σ ( ξ ) = Λ ee ξ ee ξ Σ ( ξ ) = Λ ee
17 Σ = Λ U G Σ = Σ (k) = 1 βn Σ (k) = 1 βn Λ eh G Λ U (k, k; q)g (k + q) q k Λ eh (k, k ; 0)G (k )
18 M k,k = δ k,k + Λ (k, k ; 0)G (k )G (k ) Σ Γ σσ G σ [ Σ (k) = U G (k ) 1 1 G (k )G (k + k k ) βn βn k k ] Γ (k, k ; k k )
19 M k,k = δ k,k + Λ (k, k ; 0)G (k )G (k ) Σ Γ σσ G σ [ Σ (k) = U G (k ) 1 1 G (k )G (k + k k ) βn βn k k ] Γ (k, k ; k k )
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21 Ĥ AD = <ij> t ij c i c j + i V i c i c i X(V i) av = dvρ(v)x(v) ρ(v) = cδ(v ) + (1 c)δ(v + ) { } F av = k B T ln exp βĥad(t ij, V i ) av
22 G(E ±, k) = 1 N Nδ k,k 1 E ± i0 + ϵ(k ) Σ(E ± i0 +, k) k G(E +, k)g(e, q + k) G(E +, k)g(e, q k)
23 eh Γ kk (E +, E ; q) = Λ eh kk (E +, E ; q) + 1 N k Λ eh kk (E +, E ; q) G(E +, k )G(E, k + q)γ k k (E +, E ; q) ee Γ kk (E +, E ; q) = Λ ee kk (E +, E ; q)+ 1 N k Λ ee kk (E +, E ; q+k k ) G(E +, k )G(E, Q k )Γ k k (E +, E ; q + k k ) Q = q + k + k Γ kk (E +, E ; q) = Λ eh kk (E +, E ; q)+λ ee kk (E +, E ; q) I kk (E +, E ; q) I
24 G(k, z) = G( k, z) Γ kk (z +, z ; q) = Γ kk (z +, z ; Q) = Γ k k(z +, z ; Q) Q = q + k + k k+q Γ k k +q k = = Γ k k k k = k+q Γ k k +q k Λ ee kk (z +, z ; q) = Λ eh kk (z +, z ; Q) = Λ eh k k(z +, z ; Q)
25 k ± = k ± q/2 Σ(z +, k + ) Σ(z, k ) = 1 [ Λ kk (z +, z ; q) G(z+, k N +) G(z, k ) ] k G (2) = GG + ΛGG G (2) ω = 0 q = 0 Λ kk (z +, z ; q) Λ kk (z +, z ; q)
26 E ± = E ± i0 + IΣ(E +, k) = 1 Λ kk (E +, 0)IG(E +, k ) N k RΣ(E +, k) = Σ + P dω π IΣ(ω +, k) ω E Λ eh ee Λ Λ kk (q) = Λ kk (q) + 1 Λ kk (q) [G + (k ) G + G + G (k ) N k G + G ] Λ k k (q) G ± = N 1 k G(E ±, k)
27 G k (ω, q) = G(E +, k + ) G(E, k ) Σ k (ω, q) = Σ k+ (E +, k + ) Σ(E, k ) R k (ω, q) = 1 Λ kk (ω, q) G k (ω, q) Σ k (ω, q) N k G(ω, q) 2 = 1 G k (ω, q) 2 N k E ± = E ± ω/2 ± i0 + k ± = k ± q/2 Γ 1 N { [ δ k,k k Λ kk G kr k G 2 R k G k G 2 + R G G k G k G 2 2 G k + G k } Γ k k = Λ kk G kr k G 2 R k G k G 2 + R G G k G k G 2 2 ]
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