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








 


semicircle in lower half plane
 2i  residues
 2i  residues
(i ) 2
residue at     - i 
  - i       q



i
(i ) 2
2i
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
2i 
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