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1 ,,,.,, 19,,,,,

2 2 7,.,,., DFT), LDA) GGA GW, La 2 CuO 4 La 2 CuO 4 LDA 3d. 2eV

3 ,.,, E F,. E F = 2 kf 2 /2m k F m, m kg, Jsec J 10 ev., r s, ke 2 /r r, k 9

4 Nm 2 C 2 e = C J 10 ev.,., r s, r s, r s, r s,. r s 1-10,..,,.,,,,.,,.,,. DFT,,, DFT. d E F f p E F, d p f (

5 κ ET. 3 2 Energy (ev) Γ Y Μ Z Γ Μ 7. 1 κ-(et) 2 Cu 2 (CN) 3. E =0, ( 2 ) [K. Nakamura, Y. Yoshimoto, T. Kosugi, R. Arita, and M. Imada. (2009): J. Phys. Soc. Jpn. 78, (2009).].. /

6 6 7 LDA 20 E F E F HOMO LUMO 7. 3,..,. σ.

7 N s N b jk Φ = k N N s [ k=k 1 j=1 b jk c j ] c j, c j, j,. N s N b jk... Φ Ξ = exp[ c i A ijc j ] ij Ξ Φ = k { n } (e A b) nk c n N Ξ,. 1 4 i μ iμ, i j iμ jν, iμ jν 0.,,,,. H K, iμ ( 2 2 ) 2m r 2 jν = t ijμν 7. 3

8 8 7 t ijμν, H K K K i,j,μ,ν iμ H K jν = t ijμν + μδ ij iμ jν = δ ijμν, μ H K H K = i,j μ,νσ c iμσ K i,j,μ,νc jνσ 7. 5 tight binding model. (7. 5) H K, (7. 2), Φ. e τhk Φ = N s k N { [exp( Kτ)b] nk c n} 0 n 7. 6 H = ij c i K ijc j 7. 7 Tr Z =Tre βh =Trρ 7. 8 (??) Z =Tr M e Δ βh m(τ m) m= Q = e Δ βh m(τ m) Q m

9 Tr M m=1 e Δ βh m(τ m) =[det(i + Q)] I e Δ βh m(τ m) = B m B m det(i + Q) = det(i + B 1 B 2 B M ) S =dets I +B M B M 1 I B M 2 I (7. 12) det 1 ψ k N s N s N s (7. 14) S (MN s ) (MN s ) Z =dets τ c (τ) c(τ) (7. 11) Δ β 1 B m =1 Δ β H m (τ m ) (7. 14) N s M S Δ β 1 S = Δ β M m [ i ] c i (τ m)(c i (τ m ) c i (τ m 1 )) + H m (τ m ), Δ β H m (τ m ) c i (τ m)k ij c j (τ m 1 ) ij Δ β 0 (7. 17)

10 10 7 S = β 0 dτ [ i c i (τ) ] τ c i(τ)+h(τ) ,.,., 7.3, (7. 5), H U = U 2 iσ,σ n iσ n iσ 7. 19,. n iσ c iσ c iσ n iσ n iσ = n iσ σ = σ (7. 19) σ σ. (7. 19) H U = U i (n i 1 2 )(n i 1 ) H U (7. 19) N s /4+( i n i + n i )/2, H K μ U/2, H K.. H = H K + H U 7. 21,., 2. 1,

11 ,. U, U,,, 2, U. 1,.,. La 2 CuO 4,. n iσn jσ ij σσ ,., e τh Φ (. e τh Φ, (Stratonovich- Hubbard).. (7. 21). (??), (??), ρ(i, j; β) = i e βh j = i 1 i ρ(δ β ) i 1 i 1 ρ(δ β ) i 2 i 2 i M 1 i M 1 ρ(δ β ) j 7. 22

12 12 7 ρ(δ β ) e HΔ β e HKΔβ/2 e HU Δ β e HKΔβ/2 + O(Δ 7. 3 β) 23.Δ β 0 ρ(δ β ) H U H K. e HKΔ β c c (7. 6). e Δ βh U = i e Δ βu(n i 1/2)(n i 1/2) due α 2 (u O)2 = 2π α O O = n n, 0 1 n 2 σ = n σ e α(n 1/2)(n α 1/2) = du exp[α{ u2 2π 2 + u(n n )}] , α = Δ β U., e α(n 1/2)(n 1/2) = 1 2 s=±1 exp[2a U s(n n )] a U =th th( 1 α ) , 1 0, 1 4. u s ( ). e αn n (7. 26) (7. 27) 2 n n. (7. 26) (7. 27),, (7. 6)

13 s u (. (7. 10), z z,., (7. 22), M. (7. 15) (7. 18), s i (τ j ) Z = Tre βh 1 = 2 M s i(τ j)=±1 dets({s i (τ j )}) (7. 17) (7. 18) H H(τ) = c iσ (τ)h ijσσ (τ)c jσ (τ) ijσσ H ijσσ (τ) = K ij δ σσ +2a U s i (τ)(δ σ δ σ δ σ δ σ )δ ij C T τ, (??) (?? ) Z = Tre βh = e C Tr [e βhk T τ e β dτ(hu (τ) C/β)] 0 = β β ( ) n ( C dτ 1 dτ n Tr [e (β τn)hk 1 βh ) U n 0 0 τ n 1 β C ( e (τ2 τ1)hk 1 βh ) ] U e τ1hk C, 1 βh U C = 1 2 cosh γ = 1+ βu 2C e γs(n n ) s=±

14 14 7, Z = β β dτ 1 dτ n ( C 0 τ n 1 2β )n Z n ({s i,τ i }) n 0 s i ±1,1 i n Z n ({s i,τ i })=Tr 1 i=n e ΔτiHK e siγ(n n ) i n, Δτ i = τ i+1 τ i, Δτ n = β τ n + τ 1.(7. 36) (7. 29), dets ,,.,.,,,,, , Z = Tr exp[ βh] Tr., A A = Tr exp[ βh]a/z A i exp[ βe i ]A i / i exp[ βe i ] 7. 39

15 ,,. i E i, i.,. exp[ βh],,.... exp[ βh]. importance sampling. exp[ βe i ], A i A i / ,,,,.,,,.,. E j <E i W (i j) =1, E i <E j i j. E j E i W (i j) = exp[β(e i E j )] 0 1, R W j, W, i,., j = i j W (i j) = exp[ βe j ]/(exp[ βe i ] + exp[ βe j ]).,,, exp[ βe i ], i.

16 , 7.6.,,, (7. 29) (7. 35).,, (N s ) (M) ((7. 29) (7. 35) s i (τ)). (7. 29) (7. 35) dets),.,, ( 2. (auxiliary-field quantum Monte Carlo). (7. 35), τ i,. τ(, Δ β..., dets. dets, (7. 41)

17 A i sign(dets i )A i / i sign(dets i ) ,,. (path-integral renormalization group, PIRG) 3., Φ initial, Ψ g Ψ g = lim τ e τh Φ initial , H H K H V, τ = MΔ β Δ β M, exp[ τh] [exp[ Δ β H K ]exp[ Δ β H V ]] M. Φ initial, exp[ Δ β H K ] (7. 2),. exp[ Δ β H V ], -,. exp[ Δ β H V ],,,., L Ψ (L) = L α=1, c α Φ (L) α c α Φ (L) α, ,,. L,.,,

18 18 7,,,.,,. Ψ (L),.,,.,, 4. L T (n) w n : L φ = n w n T (n) φ = n w n φ (n) φ φ (n). L, n., T (n) R n. k, w n = exp(ik R n ) (7. 44) n., S.,,,, 4.., 5,, CI) L.

19 (variational Monte Carlo, VMC), Φ Φ HΦdR E =, Φ 2 dr,. (7. 45) R,, 6. E g, Φ H Φ E g Φ Φ. α Φ(α), α,.,.,,,,..,. A A = Φ A Φ Φ Φ = s = s Φ A R s R s Φ s R s Φ 2 Φ A R s Φ R s 1 W s W s s 7. 47

20 20 7. R s, W s W s = R s Φ (7. 47) W s,, (7. 47) A = Φ A R s R s Φ 1 s N sample N sample, s. Φ = PL φ, φ,. L, 4, P ( 7 P L φ P L. φ φ pair = [ Ns i,j=1 f ij c i c j ] N/ f ij, Ns 2.,, f ij,. f ij, N s. 1.

21 N/2, 2., BCS f ij.,,, 1. P exp[ v i,j n i n j ], v i,j i,j i j., P exp[ g n i n i ], g i, (7. 20). ( ) ( ).,., R s Φ.,., (steepest descent method) (Newton method). α δα, α (, δα ( 7, 8 )., δα,,. (Sorella) (stochastic reconfiguration method),, α,. 7, 8, 9.,.

22 E/N s. L 1/L 3. N s = L L(L =4, 6, 8, 10) t =1, U =4,. HF PIRG, VMC [D. Tahara and M. Imada, J. Phys. Soc. Jpn. 77, (2008)] Exact L =4. AFQMC,. 7. 2,,., ( (π, π) ) (AFQMC) (VMC) 7. 3.,.

23 ( U t U/t =4) S(q peak ) (δ). [N. Furukawa and M. Imada, J. Phys. Soc. Jpn. 61, 3331 (1992)], 8 8, (VMC) [D. Tahara and M. Imada, J. Phys. Soc. Jpn. 77, (2008)]...,.,,.,,,,

24 24 7., (dynamical mean field theory, DMFT) 10, 11. DMFT H K.. G 1, 1 Z Tr exp[ S] (7. 18), S = β 0 dτ{ c i (τ) G 1 (τ)c i (τ)+h Ui } H Ui i, G 1 (. G ii (τ) G Σ G 1 ii (τ) G 1 (τ)+σ(τ) (7. 52) 1,. (7. 52) ω n G ii (ω n ) G ii (ω n ) = k 1 iω n ε 0 (k) Σ(ω n ) (7. 52) 7. 54,. ε 0 (k). J S, J Γ = F[J] Tr(JG ii ) Γ/ J =0. F. DMFT. (7. 53), (7. 54) Σ,,.,,., DMFT

25 ,, (7. 52), DMFT 12, 13.., ,. 2, 3,. 11, 14.,,., U/t, U/t,.,,.., U/t, ,..,,,..,,.

26 U/t t / t 7. 4 ( U t t ). [T. Mizusaki and M. Imada, Phys. Rev. B 74, (2006)].., 2,, (RVB) , ,,., 2,.

27 U/t A(ω) ω = 0 [D. Vollhardt, K. Held, G. Keller, R. Bulla, Th. Pruschke, I.A. Nekrasov, and V.I. Anisimov, J. Phys. Soc. Jpn. 74, 136 (2005)]. W ( )ω. U/W ,,., 15, (??), (??) ( 6, 16 ).., 7.5.2,

28 28 7,.,,. (??),,.,., 17,,,.. (Multi-scale Ab initio scheme for Correlated Electrons (MACE),,,. LDA LDA E F ( ev)

29 LDA LDA LDA E F,, E F.

30 LDA GW, 2 E F ( 3,. (??), (H e 2 ), (H e + H en )., (??), H[c,c] = μ,ν h K (μ, ν)c μc ν + μ.μ,ν,ν h V (μ, μ,ν,ν )c μc μ c νc ν ,. μ, ν,,,., LDA, μ LDA,.

31 S Z = Tr exp[s]. S[c,c] = Ldτ, L = ν c ν τ c ν + H[c,c] 7. 57, L. ν r, σ, τ.,.,,. H = H L + H H + H HL H,. L. H L H HL., Z = Tr L Tr H exp[s] ,, S L [c L,c L] = logtr H exp[s] L L S L / τ. H L [c L,c L] L L c τ cdr 7. 61,, H L H L., H L

32 32 7 τ.,. τ ( ), HL., H H L H = h LK (μ, ν)c μc ν μ,ν + μ.μ,ν,ν h LV (μ, μ,ν,ν )c μc μ c νc ν ,.τ,,,.., (7. 55)..., H HL,. h LV (ω) h LV (τ) τ t, ω.,.,,.. (1) h LV (ω) h LV (ω =0), (2) Σ ReΣ(ω =0)+ωdReΣ/dω ω=0,

33 ,.,, ε(k) 1/(1 dreσ/dω ω=0 ) ,.,., SrVO 3, 7. 6.,,..,.,. Marzari, Souza, Vanderbilt 18..,. {ψ nk }., ϕ nr (r) = V (2π) 3 e ik R ψ nk (r)d 3 k k.. n R

34 E (ev) R X M 7. 6 SrVO 3 ( ).,.. t 2g 8. ϕ nr (r) = V (2π) 3 e ik R ψ (w) nk (r)d3 k 7. 64, ψ (w) nk, ψ (w) nk (r) = U mn (k)ψ mk (r) m U mn (k) Ω = [ ϕ n0 r 2 ϕ n0 ϕ n0 r ϕ n0 2 ] n ψ nk (r), ψ (w) nk (r). k ,, H mn (R) = ϕ m0 H ϕ nr 7. 67

35 (a). G 0, V,, (Fig.7. 8) W. (b). (b) ( 7. 10) m = n, R = 0,..,. 3d, 20 ev, ev.

36 RPA. 7. 7(a) Γ ,, (7. 59)., 7. 7(a).,, (b) , 7. 9, RPA. RPA 3d 1eV,.. RPA. RPA (H ) (L ), 7. 9., H. RPA.,,, (H, H), (H, L), (L, H), (L, L). (L, L). RPA RPA (constrained RPA, crpa). RPA, 19. P T P T = P H + P L. P L L, P H P T P L. P H, W H.

37 W H (k) = ε 1 H (V,k)V (k), ε α (w, k) = 1 P α (k)w(k) V, k, α L, H, T = L + H., P H G. G. G(r, r ; ω) = occ n ψ n (r)ψn(r ) unocc ω ε n i0 + + n ψ n (r)ψ n(r ) ω ε n + i n L G L, H G H, G. P H G(k) = G L (k)+g H (k) P H (k) = P T (k) P L (k), P T (k) = dk G(k )G(k + k ), P L (k) = dk G L (k )G L (k + k ) P L L, P H H L H. W H, 19. W (k) = ε 1 T (V,k)V (k) =ε 1 L (W H,k)W H (k) Chain rule). V P H W H, L 2, V P T W., W H. RPA, (7. 62) h LV W H. W H, W.

38 ,., (7. 68) W H.,, d ev W H. P L,,,.,, U(r) = lim W H (r, ω). ω 0 (7. 62) H RPA- GW., W H P L, L., GW Σ(k) = dk G(k )W (k + k ) W, L Σ L. U L. W

39 W (k) =ε 1 L (U, k)u(k), W (7. 75) W, W H (ω) U = W H (ω = 0). ω =0 W W.,, W, W. ω, WH V, W U, V. ΣL W Σ L (k) = dk G L (k ) W (k + k ) , H, (7. 76) (7. 78) ΔΣ(k) = dk [G L (k )(W (k + k ) W (k + k )) + G H (k)w (k + k )] V XC V XC,LDA V XC. G L (W W ) W H., G L > G H G H W., W H,. LDA H LK ω = ε 0 (k), ω = ε (k) =ε 0 (k) +ΔΣ(k)., ΔΣ(k) Σ 0 + Σ 1 ω ω, ω = ε (k) =(ε 0 (k)+σ 0 )/(1 Σ 1 ).,., LDA H. 3d, 10-20%., ΔΣ., (double-counting term). LDA

40 40 7..,,., LDA. H,.,, W H (ω), Σ(ω) Σ(ω =0)+ωdReΣ/dω ω=0 ), RPA(cRPA), H = H K + H U, H K = c Rn t Rn,R n c R n, Rn,R n H U = 1 c Rn 2 c Rn U nn R,mm R c R m c R m R,nn,mm H U,. n, n,m,m, R.,.

41 ,,.,.,,.,,,.,, U,.,,,,.,,.,,.., , GaAs,,,

42 42 7,, , 56K, LaFeAsO, LaFaPO, BaFe 2 As 2, LeFeAs, FeSe, 7. 11, 2, 2.,, LaFeAsO LaFeAsO [Y. Kamihara, T. Watanabe, M. Hirano and H. Hosono, J. Am. Chem. Soc. 130, 3296 (2008)].,. LDA GGA 7. 12, 3d, 1 5 3d 10 (

43 Energy (ev) Γ X M Γ Z LaFeAsO [K. Nakamura, R. Arita and M. Imada, J. Phys. Soc. Jpn. 77, (2008), T. Miyake, K. Nakamura, R. Arita and M. Imada, J. Phys. Soc. Jpn. 79, (2010)]. 2 2eV 10 3d, 0,. 5 2eV p,.

44 ). 6 d 6 LaFeAsO. M,,.. LDA LSDA 2μ B. LaFeAsO μ B, LSDA,,.,,. LaFeAsO FeSe FeTe d 3d x 2 y 2 [T. Miyake, K. Nakamura, R. Arita and M. Imada, J. Phys. Soc. Jpn. 79, (2010)]. 3d Se Te As P p, 8.

45 , (7. 67), RPA. 50%,, LaFePO, LaFeAsO, BaFe 2 As 2,FeSe., 2, p 3d,.,, ( FeTe BaFe 2 As 2 LaFeAsO LaFePO LaFeAsO 1. [T. Misawa, K. Nakamura and M. Imada, J. Phys. Soc. Jpn. 80, (2011)]., LaFePO, LaFeAsO, BaFe2As2, FeTe

46 (d,δ=-1) 1.2(d,δ=0) [T. Misawa, K. Nakamura and M. Imada, Phys. Rev. Lett. 108, (2012)].,,. LaFeAsO,.,,.,, d 6 d 5 3d, , d 6 d 5,.,,,.

47 47 1 (2004).. 2 M. Imada and Y. Hatsugai. (1989): J. Phys. Soc. Jpn. 58, T.Kashima and M. Imada. (2001): J. Phys. Soc. Jpn. 70, T. Mizusaki and M. Imada. (2004): Phys. Rev. B 69, H. Fukutome. (1988): Prog. Theor. Phys. 80, Monte Carlo Methods in Statistical Mechanics (1979) ed. by K. Binder. Springer-Verlag, Berlin. 7 D. Tahara and M. Imada (2008): J. Phys. Soc. Jpn. 77, M. Imada and T. Miyake: (2010) J. Phys. Soc. Jpn. 79, S. Sorella (2001): Phys. Rev. B 64, ; (2005) Phys. Rev. B, 71, W. Metzner and D. Vollhardt (1989): Phys. Rev. Lett. 62, A. Georges, G. Kotliar, W. Krauth, and M. J. Rosenberg (1996): Rev. Mod. Phys. 68, G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet and C. A. Marianetti. (2006): Rev. Mod. Phys. 78, T. Maier, M. Jarrell, T. Pruschke and M. H. Hettler (2006) : Rev. Mod. Phys. 77, M. Imada, A. Fujimori and Y. Tokura (1998): Rev. Mod. Phys. 70, W. M. C. Foulkes, L. Mitas, R. J. Needs and G. Rajagopal. (2001): Rev. Mod. Phys. 73, M. Takahashi and M. Imada. (1984): J. Phys. Soc. Jpn. 53, 963; 53, M. Imada and M. Takahashi. (1984): J. Phys. Soc. Jpn. 53, A. Becke. (1993): J. Chem. Phys ; 98, I. Souza, N. Marzari and D. Vanderbilt.. (2001): Phys. Rev. B F. Aryasetiawan, M. Imada, A. Georges, G. Kotliar, S. Biermann, and A. I. Lichtenstein (2004) : Phys. Rev. B 70, H. Hosono, Y. Nakai and K. Ishida. (2009): J. Phys. Soc. Jpn. 78,

d (i) (ii) 1 Georges[2] Maier [3] [1] ω = 0 1

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