Tutorial Solutions 3
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Transcript Tutorial Solutions 3
Many-body Green’s Functions
• Exercise (15) Solution: 10 second order diagrams for the Green’s function
Equation of Motion for the Green’s Function
• Exercise (16) Solution: 10 second order diagrams for the Green’s function
• To obtain self energy, cut off Go at either end to leave stubs
• There are six proper self-energy diagrams at second order
Evaluation of the Single Loop Bubble
• Exercise (17) Solution: Same result by closing the contour in the lower half
plane
d
2 iG o (, ) iG o ( q, )
Anti clockwise
Lower half plane
semicircle in lower half plane
2i residues
2i residues
(i ) 2
residue at - i
- i q
i
(i ) 2
2i
d
2 - i q q - i
2
same result as closing in upper half plane
Evaluation of the Single Loop Bubble
• Exercise (18) Solution: continued
d
iG o (, ) iG o ( q, )
• For
2
i
i
residue
i
i
q
i 2
i q
2i
1
q i
d
1
i
2 q i q i
kF
q kF
y
i
kF
x
q i
q kF
d 3
2i
d 3
2i
i o (q, )
2 3 q i 2 3 q i
kF
q kF
kF
q kF