1 1.1 hν A(k,ε)[ k ρ(ω)] [1] A(k,ε) ε k μ f(ε) 1/[1 + exp( ε μ k B T )] A(k,ε)f(ε) ρ(ε)f(ε) A(k,ε)(1 f(ε)) ρ(ε)(1 f(ε)) A(k,ε) σ(ω) χ(q,ω) k B T ev k

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1 ARPES ARPES ARPES - ARPES BCS 2.4

2 1 1.1 hν A(k,ε)[ k ρ(ω)] [1] A(k,ε) ε k μ f(ε) 1/[1 + exp( ε μ k B T )] A(k,ε)f(ε) ρ(ε)f(ε) A(k,ε)(1 f(ε)) ρ(ε)(1 f(ε)) A(k,ε) σ(ω) χ(q,ω) k B T ev k B T ev ev ε d ε p... t U T K J k B T 1.2 N Ψ N = ψ 1 ψ 2...ψ N N ψ i (i =1,..., N) ε i (i =1,..., N) 1(a) ε i hν ε kin ε kin = ε i + hν V 0 (1) V 0 : μ E B ( hν ε kin + V 0 + μ = ε i + μ) 1(b) ε kin ε f

3 hν ε kin = ε f + hν V 0 (2) 1: [2] ψ k ARPES ε k hk hk (1) k- hk hk hk = hk ε = ε k ε kin = h2 K 2 2m e m e : h k = h K = 2m e ε kin cosθ θ: K k [2] A(k,ε) A(k,ε)=δ(ε ε k ) (3) ε = ε kin + V 0 hν A(k,ε) ε<μ ε >μ [2] ε k = μ k k ρ(ε)( k A(k,ε))

4 ρ(ε) = k δ(ε ε k) 1.3 2: [3] k =(0, 0) (π, 0) (π, π) k (a1) Cu t [4] q (0,π/2) (a2) t t q (π, π) (b) [5] (a1) q (0,π/2) ARPES 2 A(k,ε) k =(π, 0) k ε = ε k 2 k =(π, 0) k Cu 3d x 2 y 2 2p x, 2p y 4 ev Cu 3d x 2 y 2 ε k = ε 0 2t(cos k x a + cos k y a) 4t cos k x a cos k y a (4)

5 t, t 2 Cu p k =(π, π) (1 + p) 1 2(a1) k =(0, 0) Γ 1 p 2(a2) T c [6] Cu 3d x 2 y 2 2p 2p x,y 2p z Cu 3d 3z 2 r2 Cu 4s Cu t Cu t t 2(a1) k =(π, 0) q (0,π/2) t 2(a2) (π, π) q 1/8 [7] 1.4 ψ k k k Block k + q k q 3 Ψ N = ψ k ψ k ψ k... ψ k+q ψ k qψ k... q electron correlation 3 - Feynman N q=1 hk (q) N hk hk - hk hk hk - - ε k ε = ε k - τ k h/τ k - ε = ε k ΔE k 4 1

6 3: - t (k) (k ) (k + q) (k q) (k) (k ) (k + q) (k q) - (k) (k + q) (k ) +(k q) - (k) (k + q) +(k ) (k q) k k 4: ε = ε k ReΣ(k,ε) ImΣ(k,ε) self-energy ε hk Σ(k,ε) ε = ε k +ReΣ(k,ε) (5) ε = ε k ImΣ(k,ε k ) (5) 5 N N 1 N +1 Ψ N g N-

7 5: ε = ε k + ReΣ(k,ε) (5) A(k,ε)= i + i Ψi N 1 c k Ψ N g 2 δ(ε + Ei N 1 Eg N ) Ψ N+1 i c k ΨN g 2 δ(ε E N+1 i + E N g ), (6) [8] Ψ N 1 i N 1- Ψ N+1 i N +1- Eg N N- EN 1 i Ei N+1 N 1- N +1- c k c k k ε ε kin + V 0 hν < μ ε >μ E B μ E B = ε + μ = Ei N 1 Eg N + μ (6) : G(k,ε) = i 0 dt Ψ N g {c k (t),c k } Ψ N g e iεt 0+ t = Ψ N g c 1 k ε + i0 + + H Eg N = i + i Ψ N 1 i c k Ψ N g 2 [ ε + Ei N 1 Eg N i c k ΨN g 2 P [ ε Ei N +1 + Eg N Ψ N+1 c k Ψ N g + Ψ N 1 g c k ε + i0 + H + Eg N P iπδ(ε + E N 1 i E N g )] iπδ(ε E N+1 i + E N g )]. c k ΨN g (7) A(k,ε)= 1 ImG(k,ε). (8) π H P 2 G 0 (k,ε)=1/[ε ε k + i0 + ] (7) 2 G(k,ε) G(k,t) iθ(t) Ψ N g {c k (t),c k } ΨN g {A, B} AB + BA A(t) e iht Ae iht

8 (8) A(k,ε)=δ(ε ε k ) (3) ε = ε k ε = ε k Σ(k,ε) 1 G(k,ε)= (9) ε ε k Σ(k,ε) A(k,ε) = 1 π ImG(k,ε) = 1 π ImΣ(k,ε) [ε ε k ReΣ(k,ε)] 2 + [ImΣ(k,ε)] 2. (10) (5) ε = ε k A(k,ε) ε = ε k ReΣ(k,ε) ε = ε k (10) A(k,ε) z k(ε k ) π z k (ε k )ImΣ(k,ε k ) [ε ε k ]2 +[z k (ε, (11) k )ImΣ(k,ε)]2 z k (ε) [1 ReΣ(k,ε)/ ε] 1 (< 1) (11) z k (ε k )(< 1) 2z k (ε k )ImΣ(k,ε k ) - ε - ReΣ(k,ε) ReΣ(k,μ)= 1 π P ImΣ(k,ε ) ε ε dε ImΣ(k,ε)= 1 π P ReΣ(k,ε ) ReΣ(k,μ) ε ε dε ε μ e h 6: μ e - h - 6

9 - - ε - (ε μ) 2 μ (ε μ) 2 ImΣ(k,ε) h/τ (ε μ) 2 - ReΣ(k,ε) μ ε μ 3 α k β k Σ(k,ε) α k (ε μ) iβ k (ε μ) 2. (12) k k=k F A k 7: (12) A(k,ε) (11) A(k,ε) 1 β k (ε μ) 2 π [ε ε k + α k (ε μ)] 2 + βk 2 (ε μ)4 = z zβ k (ε μ) 2 π [ε μ z k (ε k μ)] 2 +[z k β k (ε μ) 2 ] 2 (13) z k 1/(1+α k )(< 1) 7 ε = ε k h/τ k 2z k ImΣ(k,ε k ) 7 3 ReΣ(k,μ)=0

10 μ ε μ 2z k ImΣ(k,ε)=2z k β k (ε μ) 2 < ε μ ε μ < 1/2z k β k ε QP 1/2z k β k (13) z k 1 z k 7 k k k F μ A(k,ε) β k π (ε μ) 2 (ε k μ) 2 ImΣ(k,ε) k = k F z k δ(ε μ) 1 z k (13) ε = μ A(k F,ε)= z k F π z kf β kf 1+zk 2 F βk 2 F (ε μ) 2 z2 k F β kf π (ε μ), ε = μ A(k,ε) k-ε 8 ε = ε k μ ε k = μ k k F k k F A(k,ε) n(k) 8 k F n(k) k (2π) 3 n N/V 8 ε = ε k = z k(ε k μ)+μ ε = ε k μ z k < 1 m m b z 1 > 1 k v F (1/ h)dε k /dk z k z k

11 8: ε = ε k n(k) μ = ε k k F v F E F [9] (a) (b) z A(k,ε) [ z k 1 ReΣ(k,ε) ] 1 ε k=kf,ε=μ z k â k,â k z k Ψ N 1,g â k Ψ N,g 2 = Ψ N+1,g â k Ψ N,g 2 z k n(k) 8 k z k (< 1) (12) Σ(k,ε) ε = μ - (12) ε μ - ε μ Σ(k,ε)=g k (ε μ + iγ k ) 2 (14)

12 9 (14) ε = μ Σ(k,ε) g k Γ 2 (ε μ) 2i g k k Γ 3 (ε μ) 2 (15) k (12) (12) α k β k α k = g k /Γ 2 k, β k =2g k /Γ 3 k =2α k /Γ k z k 1 α k =1+1/z k 1/z k, β k =2α k /Γ k 2/z k Γ k z k 1 A(k, ) k Re (k, ) Im (k, ) k ev 9: Σ(k,ε) A(k,ε) Σ(k,ε)= 10i(ε μ)/(ε μ + i) 2 ε k μ = 2 ev ε = ε k + ReΣ(k,ε) (5) ε = ε k A(k,ε) [1] (14) Γ k Γ k z k Σ(k,ε) Σ max g k /Γ k = α k Γ k Γ k /z k Γ k z k α k 1/z k, β k 2/z 2 kσ max β k α k ε QP ε QP 1/2z k β k =(1/2z k )(z 2 kσ max /2) = z k Σ max /4

13 ε QP Σ max - (14) 4 ε = ε k +ReΣ(k,ε) (5) ε = ε k ε = ReΣ(k,ε) 9 μ ε = ε k ImΣ(k,ε) ε ε k + ReΣ(k,ε) A(k,ε) U = 1 U = 2 N( ) U = 2.7 U = 3 U = : n(ε)[10] U U =0 k Σ(ε) Σ(k,ε) z k k z k z z k z 1 z n(μ) n 0 (μ) k 10 [10] U/t z [11]

14 - μ v F - λ ε = ε k ε = ε k /(1 + λ) 11 ω D μ ε = ε k μ 11 Mo 110 [12] k =(π, π) [13] - SrVO 3 - [14] 11: - ω D Mo [12] 1.5 μ μ n μ μ/ n χ c μ/ n = χ 1 c χ c - μ/ n =(1+Fs 0 )/n (μ) [15] 12

15 講義ノート n (μ) m Fs 0 > 0 ε n ε n+δn s δμ= (1+F )δn/n 0 (μ) μ μ+δμ δn/n (μ) δn 12: δ n δμ n (ε) F s 0 n (ε) - 1 ev/ δ 0 m [16] La 1 x Sr x TiO 3 0 x 1 δ = x 0 γ m [17] La 1 x Sr x TiO 3 x =0 n =1 LaTiO 3 La 3+ Sr 2+ x Ti 3d n =1 x La 2 x Sr x CuO 4 δ = x 0 m [18] μ/ n 1/m t J d p

16 ε d ε p... U t t U ε d ε p 13: [20] Δ >U Δ <U p d (b) Δ U Zaanen-Sawatzky-Allen [21] CI [2] d p d p d d p p d p d U d -p t pd d p Δ ε d ε p ε d d ε p p Δ U 13(a) p Δ >U d U W W Δ <U p

17 14: [20] p [10] 10 Δ W Δ U 13(b) [21] Zaanen-Sawatzky-Allen [22] Δ >U Δ <U U W Δ W U Δ W t n U W Δ W 13(a) - n - - Hubbard 14 U/t [23] Brinkman Rice U/t m m [24] (1970) [25, 26] Brinkman-Rice Hubbard Kotliar

18 [10] 9 CuO 2 Zhang-Rice [27] La 2 x Sr x CuO 4 0 x< 0.35 La 2 CuO 4 n =9 La 3+ Sr 2+ CuO 2 X δ = x Zhang-Rice [28] x >0.25 Zhang-Rice Mn Fe Zhang-Rice ARPES [29] SrVO 3 SrRuO 3 La 1 x Sr x MnO 3 [30] d 2.2 σ = σ = α β a 1 =( 2a, 2a) a 2 =( 2a, 2a) α β s Cu 3d x 2 y 2 ψ k = c α k ψ kα + c β k ψ kβ h MF ψ = εψ h MF

19 ( ψkα h MF 0 ψ kα Δ ψ kα h MF 0 ψ kβ ψ kβ h MF 0 ψ kα ψ kβ h MF 0 ψ kβ +Δ )( c α c β ) ( c α = ε c β ) (16) h MF 0 h MF α Δ Δ > 0 β +Δ t t t 0,t = 0,... ψ kα h MF 0 ψ kβ = ε k ψ kα h MF ψ kα 0 ( )( ) ( ) Δ εk c α c α Δ c β = ε c β (17) ε k ε k± = ± ε 2 k +Δ2, (18) n =1 2Δ Fermi ε = μ 2Δ Fermi 4 t 2Δ Fermi Q Q =(π, π) σ = ±Δ hk hk + hq = c kψ k + c k+q ψ k+q ψ AF k ( εk Δ Δ ε k+q )( ck c k+q ) = ε ( ck c k+q ) (19) ε k± = ε k + ε k+q 2 (εk ) ε 2 k+q ± +Δ 2 2, (20) ψ k ψ k+q Σ(k,ε)= Δ 2 ε ε k+q + i0 + (21) 4 k

20 Q =(π, π) 15(b) [31] 2.3 BCS BCS μ k ε = ε k ε =2μ ε k Δ 2 k Σ(k,ε)= ε + ε k 2μ + i0 + (22) Δ k s d d x 2 y2 Δ(k) =Δ 0 (cos k x cos k y ) δ 0 ε = ε k +ReΣ(k,ε) (5) ε k = μ ± (ε k μ) 2 +Δ 2 k (23) μ 2 Δ k μ μ - k k F - (18) - n =1 n μ - ARPES μ μ ARPES μ μ ARPES

21 15: [3] k (a1) Cu t [4] (a2) t (b) k =(π, 0) d Δ(k) = Δ 0 (cos k x cos k y ) k x = ±k y T c 15(a1)(a2) T δ 16 ARPES k [32] Δ Δ sc T <x< 1 La 1 x Sr x TiO 3 La 1 x Sr x TiO 3 e/(1 x) d 1 x La 2 x Sr x CuO 4 x <0.2 +e/x x - Imada (1) m (2) 0

22 16: [9] Oda [34] (π, 0) Δ T <T c T c <T <T k T <T Δ sc L a Δ 0 d v 2 Δ 0 v 2 Δ 0 =( 2/a)v 2 17: La 2 x Sr x CuO 4 LSCO [35] Bi 2 Sr 2 CaCu 2 O 8+δ Bi2212 [36, 37] Δ T Δ sc T c Δ 0 [9]

23 [33] La 2 x Sr x CuO 4 n(ε) [28] La 2 x Sr x CuO 4 [3] - ARPES [38] [39] [40] [1] p.321. [2] II 1996 p.149. [3] 51, No.11 ( ), (2016). [4] T. Yoshida et al. : Phys. Rev. B 74, (2006). [5] N. P. Armitage et al. : Phys. Rev. Lett. 88, (2002). [6] E. Pavarini, I. Dasgupta, T. Saha-Dasgupta, O. Jepsen, and O. K. Andersen, Phys. Rev. Lett. 87, (2001). [7] T. Yoshida et al : Condens. Matter 19, (2007). [8] 2005 [9] 62, 815 (2007). [10] X. Y. Zhang, M. J. Rozenberg, and G. Kotliar: Phys. Rev. Lett. 70, 1666 (1993). [11] p [12] T. Valla, A. V. Fedorov, P. D. Johnson, and S. L. Hulbert: Phys. Rev. Lett. 83, 2085 (1999). [13] A. Lanzara et al. : Nature 412, 510 (2001). [14] S. Aizaki et al. : Phys. Rev. Lett. 109, (2012).

24 [15] N. Furukawa and M. Imada: J. Phys. Soc. Jpn. 61, 3331 (1992). [16] T. Yoshida, A. Ino, T. Mizokawa, A. Fujimori. Y. Taguchi, T. Katsufuji and Y. Tokura: Europhys. Lett. 59, 258 (2002). [17] Y. Tokura, Y. Taguchi, Y. Okada, Y. Fujishima, T. Arima,K. Kumagai and Y. Iye: Phys. Rev. Lett. 70, 2126 (1993). [18] N. Momono, M. Ido, T. Nakano, M. Oda, Y. Okajima, and K. Yamaya: Physica C 233, 395 (1994). [19] A. Ino, T. Mizokawa, A. Fujimori, K. Tamasaku, S. Uchida, T. Kimura, T. Sasagawa and K. Kishio: Phys. Rev. Lett. 79, 2101 (1997). [20] 54, (1999). [21] A. E. Bocquet, T. Mizokawa, K. Morikawa, A. Fujimori, K. B. Maiti, S. R. Barman, D. D. Sarma, Y. Tokura and M. Onoda: Phys. Rev. B 53, 1161 (1996). [22] J. Zaanen, G. A. Sawatzky and J. W. Allen: Phys. Rev. Lett. 55, 418 (1985). [23] N. F. Mott: Metal Insulator Transitions (Taylor & Francis, 1974) p [24] W. F. Brinkman and T. M. Rice: Phys. Rev. B 2, 4302 (1970). [25] I. H. Inoue, I. Hase, Y. Aiura, A. Fujimori, Y. Haryuama, T. Maruyama and Y. Nishihara: Phys. Rev. Lett. 74, 2539 (1995). [26] T. Yoshida, M. Hashimoto, T. Takizawa, A. Fujimori, M. Kubota, K. Ono, and H. Eisaki: Phys. Rev. B 82, (2010). [27] F. C. Zhang and T. M. Rice: Rhys. Rev. B 37 (1988) [28] A. Ino, T. Mizokawa, K. Kobayashi, A. Fujimori, T. Sasagawa, T. Kimura, K. Kishio, K. Tamasaku, H. Eisaki, and S. Uchida: Phys. Rev. Lett. 81 (1998) [29] M. Kobayashi, I. Muneta, Y. Takeda, Y. Harada, A. Fujimori, J. Krempasky, T. Schmitt, S. Ohya, M. Tanaka, M. Oshima, and V. N. Strocov: Phys. Rev. B 89, (2014). [30] K. Yoshimatsu, T. Okabe, H. Kumigashira, S. Okamoto, S. Aizaki, A. Fujimori, and M. Oshima: Phys. Rev. Lett. 104, (2010). [31] M. Horio et al. : Nat. Commun. 7, (2016). [32] T. Yoshida, M. Hashimoto, I. M. Vishik, Z.-X. Shen, and A. Fujimori: J. Phys. Soc. Jpn. 81, (2012). [33] M. Imada: J. Phys. Soc. Jpn. 62, 1105 (1993).

25 [34] M. Oda, R.M. Dipasupil, N. Momono and M. Ido: J. Phys. Soc. Jpn. 69, 983 (2000). [35] M. Hashimoto, T. Yoshida, K. Tanaka, A. Fujimori, M. Okusawa, S. Wakimoto, K. Yamada, T. Kakeshita, H. Eisaki and S. Uchida: Phys. Rev. B 75, (2007) [36] A. Kanigel et al. : Nature Phys. 2, 447 (2006). [37] K. Tanaka, W.S. Lee, D.H. Lu, A. Fujimori, T. Fujii, Risdiana, I. Terasaki, D. J. Scalapino, T.P. Devereaux, Z. Hussain and Z.-X. Shen: Science 314, 1910 (2006). [38] M. Hashimoto et al.: Nat. Phys. 6, 414 (2010). [39] J. Chang et al. : Nat. Phys. 8, 871 (2012). [40] O. Cyr-Choiniere, G. Grissonnanche, S. Badoux, J. Day, D. A. Bonn, W. N. Hardy, R. Liang, N. Doiron-Leyraud, and L. Taillefer: Phys. Rev. B 92, (2015).

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lect.dvi 2001 7 3{5 1 2 1.1 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 1.2 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 1.3 : : : : : : : :