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1 RKKY s-d QCD
2 a b c d φ XY Ward-Takahashi BCS Nambu-Jona-Lasinio
3
4 Euler Principia motus fluidorum 1864 Maxwell Electromagnetic field 1900 Planck Lorentz Lorentz 1905 Einstein Special Relativity 1908 Minkowski Raum und Zeit 1911 Kamerlingh Onnes Hg T c = 4.2K 1915 Einstein Allgemeine Relativitästheorie 1925 Heisenberg 1926 Schrödinger Wellenmechanik 1826 Dirac Dirac 1927 Dirac Maxwell field 1927 Bloch 1928 Jordan-Wigner 1929 Heisenberg-Pauli C. D. Anderson 1933 Meissner-Ochsenfeld Meissner 1933 de Haas, de Boer, van den Berg 1934 Pauli-Weisscopf 1934 Fermi Kapitsa T c = 2.17K 1940 Pauli 1943 Heisenberg S ESR NMR 1949 Tomonaga, Schwinger, Feynman, Dyson 1954 Yang-Mills 4
5 1955 S 1957 BCS 1958 Landau 1960 Josephson Gell-Mann, Zwig K. G. Wilson 1972 Osheroff, Richardson, Lee T c = 2.6mK 1973 Kosterlitz-Thouless 1974 Gross, WIlczek, Politzer 1979 Anderson STM Scanning Tunneling Microscopy 1986 Bednorz, Muller Rb K 6 Li 100nK 5
6 1.2 L L L h2 2m 2 ψ = ɛψ. (1) ψ = 1 V e ik r (2) ψ(x + L, y, z) = ψ(x, y + L, z) = ψ(x, y, z + L) = ψ(x, y, z) (3) k x = 2π L n x, k y = 2π L n y, k z = 2π L n z. (4) n x n y n z C T C = 2 π2 π 2 k 3 k2 B 2 Bρ(ɛ F )T = N e h 2 mt (5) kf R/
7 k k k P.W. Anderson Basic Notions of Condensed Matter Physics 1. H H = kσ ɛ k c kσ c kσ g kk c k c k c k c k, (6) 7
8 BCS BCS ψ = k (u k + v k c k c k ) 0 (7) -BCS 11(1976) QED ev 8
9 ɛ F ev (8) e2 d ev (9) - W = 1/τ T 2 1 τ (k BT ) 2 k B T, (10) ɛ F Fermi liquid theory 1/τ H = kσ ɛ k c kσ c kσ + U 1 N c k q c k +q c k c k. (11) kk q ImΣ R k (ɛ) = U 2 q ImG R k q(ɛ ) k dɛ ( coth ɛ ɛ 2π 2k B T ( dx 2π ) ɛ tanh 2k B T tanh x k B T tanhx + ɛ ɛ 2k B T ImG R k (x)imgr k +q(x + ɛ ɛ ) (12) ) 9
10 Table 1: Green z ρ ImΣ ReΣ z ρ d = 1 T or ɛ ɛ ln ɛ z = 0 T d = 2 T 2 ln T or ɛ 2 ln ɛ ɛ z 0 T 2 ln T d = 3 T 2 or ɛ 2 ɛ z 0 T 2 G R k (ɛ) Green ɛ = 0 dɛ ( coth ɛ 2k B T ) ɛ tanh F (ɛ ) = F (0)(πk B T ) 2 (13) 2k B T ImΣ R k (0) = 1 2π U 2 (πk B T ) 2 1 dx 2k B T cosh 2 (x/2k B T ) ImG R k q(0)img R k (x)imgr k +q(x) (14) k q 3 ImΣ R k (0) T 2 (15) ImΣ R k (ɛ) ɛ 2 (T = 0) (16) T I,II. 2. II. 3. A. A. Abrikosov, L. P. Gorkov, I. Ye Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics. 4. K. Nishijima, Fields and Particles (1969). 5. N. N. Bogoliubov, D. V. Shirkov, Introduction to the Theory of Quantized 10
11 Fields C. Hodges, H. Smith and J. W. Wilkins, Phys. Rev. B34, 302 (1971). 8. P. Bloom, Phys. Rev. B12, 125 (1975). 11
Einstein 1905 Lorentz Maxwell c E p E 2 (pc) 2 = m 2 c 4 (7.1) m E ( ) E p µ =(p 0,p 1,p 2,p 3 )=(p 0, p )= c, p (7.2) x µ =(x 0,x 1,x 2,x
7 7.1 7.1.1 Einstein 1905 Lorentz Maxwell c E p E 2 (pc) 2 = m 2 c 4 (7.1) m E ( ) E p µ =(p 0,p 1,p 2,p 3 )=(p 0, p )= c, p (7.2) x µ =(x 0,x 1,x 2,x 3 )=(x 0, x )=(ct, x ) (7.3) E/c ct K = E mc 2 (7.4)
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59 7 7.1 σ-ω σ-ω σ ω σ = σ(r), ω µ = δ µ,0 ω(r) (6-4) (iγ µ µ m U(r) γ 0 V (r))ψ(x) = 0 (7-1) U(r) = g σ σ(r), V (r) = g ω ω(r) σ(r) ω(r) (6-3) ( 2 + m 2 σ)σ(r) = g σ ψψ (7-2) ( 2 + m 2 ω)ω(r) = g ω ψγ
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2 2 T c µ T c 1 1.1 1911 Kammerlingh Onnes 77K ρ 5.8µΩcm 4.2K ρ 1 4 µωcm σ 77K ρ 4.2K σ σ = ne 2 τ/m τ 77K τ 4.2K σ 58 213 email:takada@issp.u-tokyo.ac.jp 1933 Meissner Ochsenfeld λ = 1 5 cm B = χ B =
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1 13 6 8 3.6.3 - Aharonov-BohmAB) S 1/ 1/ S t = 1/ 1/ 1/ 1/, 1.1) 1/ 1/ *1 AB ) e iθ AB S AB = e iθ, AB θ π ϕ = e ϕ ϕ ) ϕ 1.) S S ) e iθ S w = e iθ 1.3) θ θ AB??) S t = 4 sin θ 1 + e iθ AB e iθ AB + e
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