16 5 19 10 d (i) (ii) 1 Georges[2] Maier [3] 2 10 1 [1] ω = 0 1
[4, 5] Dynamical Mean-Field Theory (DMFT) [2] DMFT I CPA [10] CPA CPA Σ(z) z CPA Σ(z) Σ(z) Σ(z) z - CPA Σ(z) DMFT Σ(z) CPA [6] 3 1960 [7] 3.1 1990 [8] DMFT H = t ij c iσ c jσ + U c i c i c i c i ijσ i ɛ k t ij 1960 [9] t ij III t 1 2
k z z λ(z) Σ(z) (2) G(k, z) = [z ɛ k Σ(z)] 1 λ(z) Σ(z) Σ(z) k DMFT λ(z) DMFT Σ(z) DMFT CPA [11, 2] DMFT Ḡ(z) G(k, z) DMFT N III Ḡ(z) = 1 CPA 1 N z ɛ k k Σ(z) 1980 DMFT ρ(ɛ) = dɛ [4] z Σ(z) ɛ [5] = g(z Σ(z)) (1) NCA [12] extended NCA g(z) U = 0 (XNCA) ρ(ɛ) g(z Σ(z)) XNCA λ(z) z Σ(z) Ḡ(z) 1990 (QMC) U Georges-Kotliar G(z) [13] λ(z) G(z) = [z λ(z)] 1 λ(z) λ(z) = ɛ f + 1 V k 2 (3) N z ɛ c (k) k U Σ(z) ɛ f V k Ḡ(z) (1) ɛ c (k) 1 g(z Σ(z)) = [z λ(z) Σ(z)] 1 k (2) G(k, z) Ḡ(z) CPA Σ(z) [5] [1, 14] 3
(2) Φ F Potthoff Σ t Caffarel-Krauth Σ(z) [17] Σ t [15] Σ t Φ G t Φ [16] G Caffarel-Krauth F Σ t Potthoff Ω [17] Potthoff DMFT Ω t F Ω Ω t + T Tr ln(g 1 Σ t )G t Ω{Σ(t )} (5) t g G t Ω T = β 1 [18] βω{g} = βφ{g} Tr(ΣG) + Tr ln G (4) Ω{Σ(t )}/ t = 0 (6) G Potthoff Self- Φ{G} Energy Functional Theroy (SFT) δφ/δg = Caffarel-Krauth Σ G 1 = g 1 Σ (6) δω{g}/δg = 0 Potthoff DMFT [19] Φ{G} 3.2 DMFT LDA [18] G Σ F {Σ} = Φ T Tr(ΣG) LMTO δf/δσ = G δω{σ}/δσ = 0 4
) H LDA (k) LDA ) Sr 1 x Ca x VO 3 (1) LDA+DMFT Ḡ(z) = 1 [zi H LDA (k) Σ(z)I d ] 1 [21] N k (7) I I d f Ce α-γ d 4f LDA+DMFT LDA+DMFT α- d f γ [22, 23] QMC α-ce La 1 x Sr x TiO f 3 Ti 3d γ-ce 2 2 LDA+DMFT LDA t 2g α-γ 1 d γ-ce 2 1 LDA+DMFT [20] α-ce 1 QMC LDA+DMFT U QMC f 1: La 1 x Sr x TiO 3 LDA, LDA+DMFT [20] GW U [24] 5
DMFT [12] [25] 4 4.1 2: α-ce γ-ce PES BIS LDA+DMFT [23] CPA [10, 26] 3.3 DMFT DMFT DMFT [2] 1 U U N N/N c N c r 1 R U r = R+ r DMFT r R k K k 3 2 N c = 4 = L 2 DMFT DMFT DMFT 6
R r ~ K L 1stBZ 2π/L k ~ 3: exp[ ik ( r i r j )] (10) 1stBZ k = k + K K CPT 4.2 (CPT) (8) Σ c (z) t ij CPT t( r i r j ) = δ ri, r j t c + t ( r i r j ) t c, t Σ(z) z R N c G(k, z) N c G( r i r j, z) DCA CDMFT SFT G( r i r j, z) 1 = δ ri, r j Ḡ(z) 1 t ( r i r j ) Ḡ(z) 1 N c 4.3 (DCA) Ḡ(z) = [z t c Σ c (z)] 1 (8) G( r i r j, z) r i r j G( k, z) 1 = Ḡ 1 (z) t ( k) (9) G(k, z) k CPT G(k, z) = 1 N c CPT [27, 28] Ḡ(K, z) = [z ε K Σ(K, z)] 1 (12) R t ε K k G ij ( k, z) t (K + k) = ɛ K+ k ε K (13) N c i,j=1 G ij ( k, z) t c ε K = (N c /N) k ε K+ k (11) 7
G(K + k, NCA QMC z) [Ḡ 1 k)] DCA N 1 (14) c = 1 DMFT = (K, z) t (K + DMFT DCA DCA [29] N c G(K + k, z) Ḡ(K, z) = (N c /N) k G(K + k, z) (15) 4.4 (CDMFT) CDMFT CPA Molecular CPA Ḡ(K, z) 1 = G(K, z) 1 Σ(K, z) (16) (MCPA) [26] DMFT (1) G(K, z) N c N c (1) k k (i) G(K, z) 0 CDMFT CPT (ii) G c (K, z) (8) Ḡ(K, z) (iii) Σ(K, z) = G(K, z) 1 G c (K, z) 1 Ḡ(z) = (N c /N) k G( k, z) (17) (iv) Σ(K, z) G(K + k, z) Ḡ(K, z) = (N c /N) k G(K + k, z) [30] (v) G(K, z) 1 = Ḡ(K, (15) DCA z) 1 +Σ(K, z) G( k, z) (10) (vi) G(K, z) 1 CDMFT (ii) Ḡ(K, z) 4 CDMFT G c (K, z) (ii) (v) 4.5 (SFT) (ii) N c K Potthoff 8
U/t 4: SFT CDMFT SFT N c 5: n = 1 U/t = N b 4 2 [17] DMFT DCA+QMC) DMFT(N c = 1) N c = 1, N b = SFT N b N c Caffarel-Krauth (ω = 0) CDMFT N c > 1, N b = [31] SFT t 4 SFT U U/t [32, 33, 34] 6 CDMFT SFT 1 µ 5 [16] 5.1 1 µ 1 n = 1 2 N c = 2, N b = 8 DFMT U/t n(µ) DMFT 2 5 5.2 [31] DCA QMC NMR DCA+QMC [35] 9
7: δ 6: U/t = 4 1 U/t = 8 2 n µ DCA+QMC N c = 2, N b = 8 [16] BA N PCDMFT c = 4)[37] CDMFT (i) (ii) (iii) [36] DCA+QMC 7 [37] 5.3 δ = 0.050 1 δ = 0.200 2 2 Jarrel (i),(ii) DMFT (RVB) DCA+QMC N c T N N c = 40 N c = 1 7 [39] J 4t 2 /U J [3, 38] CPT 1 [34] T = 0 10
CPT Potthoff h 1 h = 0 2 h 0 [40] 1 h = 0 0 5.4 0 s p d DMFT [41] s d 4 f 2 0 RKKY 2 1 8 2 N c = 4 DCA+QMC [3] 2 d DCA+NCA 6 8: 2 [3] U/t = 8 [25] DMFT N c = 4 DCA 1 DMFT f 1 RKKY 11
RKKY RKKY 1 0.4 7 9 DMFT ɛ f ɛ f = 2, U = 4, V = 1 V 1 K = (0, 0) Γ ρ f (Kω) ρ f (Kω) 0.4 0.3 0.2 0.1 (a) K=(0,0) K=(π,0), (0,π) K=(π,π) 0-6 -4-2 0 2 4 6 ω (b) 0.4 0.3 0.2 0.1 0-6 -4-2 0 2 4 6 ω 2 [1] : 29 (1994) 9: DCA+NCA 777. [25] T = 0.2 T = 0.02 [2] A. Georges, G. Kotliar, W. Krauth, and M. Rozenberg: Rev. Mod. Phys. 68 (1996) 13. [3] T. Maier et al: cond-mat/0404055. DMFT K = (π, π) [4] Y. Kuramoto: Theory of Heavy Fermions and Valence Fluctuations, 12
eds. T. Kasuya and T. Saso (Springer Verlag, 1985) p.152. [5] Y. Kuramoto and T. Watanabe: Physica 148B (1987) 80. [6] L. Onsager: J. Am. Chem. Soc. 58 (1936) 1486. [7] R. Brout: Phys. Rev. 122 (1960) 469. [8] Y. Kuramoto and N. Fukushima: J. Phys. Soc. Jpn. 67 (1998) 583. [9] J. Hubbard: Proc. Royal. Soc. London 276 (1963) 238. [10] R.J. Elliott, J.A. Krumhansl and P.A. Leath: Rev. Mod. Phys. 46 (1974) 465. [11] E. Müller-Hartmann: Z. Phys. B74 (1989) 507. [12] C.-I. Kim, Y. Kuramoto and T. Kasuya: J. Phys. Soc. Jpn. 59 (1990) 2414. [13] A. Georges and G. Kotliar: Phys. Rev. B 45 (1992) 6479. [14] O. Sakai and Y. Kuramoto: Solid State Commun. 89 (1994) 307. [15] M. Caffarel and W. Krauth: Phys. Rev. Lett. 72 (1994) 1545. [16] M. Capone et al.: cond-mat/0401060. [17] M. Potthoff, M. Aichhorn and C. Dahnken: Phys. Rev. Lett. 91 (2003) 206402. [18] G. Baym: Phys. Rev. 127 (1962) 835. [19] M. Potthoff: Eur. Phys. J. B36 (2003) 335. [20] I. A. Nekrasov et al: Euro. Phys. J. B. 18 (2000) 55. [21] S.-K. Mo et al.: Phys. Rev. Lett. 90 (2003) 186403. [22] M. B. Zölfl et al.: Phys. Rev. Lett. 87 (2001) 276403. [23] A. K. McMahan, K. Held and R. T. Scalettar: Phys. Rev. B 67 (2003) 75108. [24] S. Biermann, F. Aryasetiawan and A. Georges: cond-mat/0401653. [25] Y. Shimizu: J. Phys. Soc. Jpn. 71 (2002) 1166. [26] M. Tsukada: J. Phys. Soc. Jpn. 26 (1969) 684. [27] C. Gros and R. Valenti: Annalen der Phys. 3 (1994) 460. [28] D. Sénéchal, D. Perez and M. Pioro- Ladriére: Phys. Rev. Lett. 84 (2000) 522. [29] M.H. Hettler et al.: Phys. Rev. B58 (1998) R7475. [30] G. Kotliar et al.: Phys. Rev. Lett. 87 (2001) 186401. [31] S. Moukouri and M. Jarrell: Phys. Rev. Lett. 87 (2001) 167010. [32] Y. Imai and N. Kawakami: Phys. Rev. B65 (2002) 233103. [33] O. Parcollet, G. Biroli and G. Kotliar: cond-mat/0308577. [34] D. Sénéchal and A.-M. Tremblay: condmat/0308625. [35] H. Yasuoka, T. Imai, T. Shimizu: Strong Correlation and Superconductivity, (Springer Verlag, Berlin, 1989) p. 254. 13
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1: NCA (QMC) 2: CPT CDMFT DCA SFT DCA CDMFT 15