# Field theories of condensed matter systems by Eduardo Fradkin

By Eduardo Fradkin

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On the other hand, the spin-oﬀ diagonal part, which means less in the case without electron–electron interactions, is responsible for the superconductivity. Up to the previous chapter, the problem had been investigated in the site representation. That is to say, the system is considered to be composed Theory of Superconductivity 25 of N sites. However, the real substance is formed from unit cells, so that the system is a repetition of the unit cell. The usual band theory of polyacene has thus been completed at this stage.

If we put iωn → ω + iη, we can obtain the retarded Green function, whose imaginary part gives the line shape: 1 D(ω) ∼ ω + iη − KI + ΣKM − ΣKM −1 = P 1 ω − −ΣKM −1 KI − iπδ(ω − KI + ΣKM − ΣKM −1 ) i(Im) (at the resonance point). 185) (Re)2 + (Im)2 The line shape is now changed from the δ function type to the Lorentz type, as expected. 1 The extended Nambu spinor Now, we investigate how the line shape obtained above is further modiﬁed in a superconductor. The electron propagators in the previous section are replaced by those in a superconductor.

If we ignore ρ and η, this is reduced to the normal propagator. We have already tried this simple case. 42 Modern Aspects of Superconductivity: Theory of Superconductivity Now we assume the terms responsible for the superconductivity as a perturbation. Namely (S is the electron propagator), S(υn ) = = iυn + + ρΣ+ + ηΣ− (iυn )2 − Ek2 3 kΣ iυn + k Σ3 ρΣ+ + ηΣ− + + ··· . 202) It is helpful to separate this relation into components and to manipulate each one individually. 203) In order to get the self-energy part of the nuclear propagator, we have to evaluate 1 S(υn )S(υn + ω).