(a) Calculate the tadpole diagram for a zero-temperature Fermi gas using the Feynman method (treating the vacuum as the ground state of the Fermi sea) and show that this gives the same answer as the T = 0 limit of the Matsubara calculation given here. You will need to add a similar exponential factor to have the integral over energy for the single-propagator loop give a meaningful answer.

(b) What Matsubara self-energy diagram (and self-energy diagram for a T = 0 Fermi sea) corresponds to the exchange self-energy diagram of Figure 8.27(b)? Show that this diagram gives an energy of opposite sign to the direct Coulomb energy diagram.

Transcribed Image Text:

Fig.8.27 戸 k (b) P k