1. Many-body Green’s Functions
Propagating electron or hole interacts with other e-/h+
Interactions modify (renormalize) electron or hole energies
Interactions produce finite lifetimes for electrons/holes (quasi-particles)
Spectral function consists of quasi-particle peaks plus ‘background’
Quasi-particles well defined close to Fermi energy
MBGF defined by

2. Many-body Green’s Functions
Space-time interpretation of Green’s function
(x,y) are space-time coordinates for the endpoints of the Green’s function
Green’s function drawn as a solid, directed line from y to x
Non-interacting Green’s function Go represented by a single line
Interacting Green’s Function G represented by a double or thick single line y x
time
Add particle
Remove particle
t > t’ t’
t’ > t t
Go(x,y)
x,t
y,t’
G(x,y)
x,t
y,t’ x y

3. Many-body Green’s Functions
Lehmann Representation (F 72 M 372) physical significance of G

4. Many-body Green’s Functions
Lehmann Representation (physical significance of G)

5. Many-body Green’s Functions
Lehmann Representation (physical significance of G)

6. Many-body Green’s Functions
Lehmann Representation (physical significance of G)
Poles occur at exact N+1 and N-1 particle energies
Ionisation potentials and electron affinities of the N particle system
Plus excitation energies of N+1 and N-1 particle systems
Connection to single-particle Green’s function

7. Many-body Green’s Functions
Gell-Mann and Low Theorem (F 61, 83)
Expectation value of Heisenberg operator over exact ground state expressed in terms of evolution operators and the operator in question in interaction picture and ground state of non-interacting system

8. Many-body Green’s Functions
Perturbative Expansion of Green’s Function (F 83)
Expansion of the numerator and denominator carried out separately
Each is evaluated using Wick’s Theorem
Denominator is a factor of the numerator
Only certain classes of (connected) contractions of the numerator survive
Overall sign of contraction determined by number of neighbour permutations
n = 0 term is just Go(x,y)
x, y are compound space and time coordinates i.e. x ≡ (x, y, z, tx)

9. Many-body Green’s Functions
Fetter and Walecka notation for field operators (F 88)

10. Many-body Green’s Functions
Nonzero contractions in numerator of MBGF
(6) ) ( ψ ˆ '
(5)
(4)
(3)
(2)
(1) y x r +
(-1)3 (i)3v(r,r’)Go(r’,r) Go(r,r’) Go(x,y)
(-1)4(i)3v(r,r’)Go(r,r) Go(r’,r’) Go(x,y)
(-1)5(i)3v(r,r’)Go(x,r) Go(r’,r’) Go(r,y)
(-1)4(i)3v(r,r’)Go(r’,r) Go(x,r’) Go(r,y)
(-1)6(i)3v(r,r’)Go(x,r) Go(r,r’) Go(r’,y)
(-1)7(i)3v(r,r’)Go(r,r) Go(x,r’) Go(r’,y)

11. Many-body Green’s Functionsx r r’
(1)
(2)
(3)
(4)
(5)
(6)
Nonzero contractions
-(i)3v(r,r’)Go(r’,r) Go(r,r’) Go(x,y) (1)
+(i)3v(r,r’)Go(r,r) Go(r’,r’) Go(x,y) (2)
-(i)3v(r,r’)Go(x,r) Go(r’,r’) Go(r,y) (3)
+(i)3v(r,r’)Go(r’,r) Go(x,r’) Go(r,y) (4)
+(i)3v(r,r’)Go(x,r) Go(r,r’) Go(r’,y) (5)
-(i)3v(r,r’)Go(r,r) Go(x,r’) Go(r’,y) (6)

12. Many-body Green’s Functions
Nonzero contractions in denominator of MBGF
Disconnected diagrams are common factor in numerator and denominator
(-1)3(i)2v(r,r’)Go(r’,r) Go(r,r’)
(-1)4(i)2v(r,r’)Go(r,r) Go(r’,r’) r r’
(7) r r’
(8)
Numerator = [ … ] x [ … ]
Denominator = …

13. Many-body Green’s Functions
Expansion in connected diagrams
Some diagrams differ in interchange of dummy variables
These appear m! ways so m! term cancels
Terms with simple closed loop contain time ordered product with equal times
These arise from contraction of Hamiltonian where adjoint operator is on left
Terms interpreted as
iG(x, y) = …

14. Many-body Green’s Functions
Rules for generating Feynman diagrams in real space and time (F 97)
(a) Draw all topologically distinct connected diagrams with m interaction lines and 2m+1 directed Green’s functions. Fermion lines run continuously from y to x or close on themselves (Fermion loops)
(b) Label each vertex with a space-time point x = (r,t)
(c) Each line represents a Green’s function, Go(x,y), running from y to x
(d) Each wavy line represents an unretarded Coulomb interaction
(e) Integrate internal variables over all space and time
(f) Overall sign determined as (-1)F where F is the number of Fermion loops
(g) Assign a factor (i)m to each mth order term
(h) Green’s functions with equal time arguments should be interpreted as G(r,r’,t,t+) where t+ is infinitesimally ahead of t
Exercise: Find the 10 second order diagrams using these rules

15. Many-body Green’s Functions
Feynman diagrams in reciprocal space
For periodic systems it is convenient to work in momentum space
Choose a translationally invariant system (homogeneous electron gas)
Green’s function depends on x-y, not x,y
G(x,y) and the Coulomb potential, V, are written as Fourier transforms
4-momentum is conserved at vertices q1 q2 q3
Fourier Transforms
4-momentum Conservation

16. Many-body Green’s Functions
Rules for generating Feynman diagrams in reciprocal space
(a) Draw all topologically distinct connected diagrams with m interaction lines and 2m+1 directed Green’s functions. Fermion lines run continuously from y to x or close on themselves (Fermion loops)
(b) Assign a direction to each interaction
(c) Assign a directed 4-momentum to each line
(d) Conserve 4-momentum at each vertex
(e) Each interaction corresponds to a factor v(q)
(f) Integrate over the m internal 4-momenta
(g) Affix a factor (i)m/(2p)4m(-1)F
(h) A closed loop or a line that is linked by a single interaction is assigned a factor eied Go(k,e)

17. Equation of Motion for the Green’s Function
Equation of Motion for Field Operators (from Lecture 2)

18. Equation of Motion for the Green’s Function
Equation of Motion for Field Operators

19. Equation of Motion for the Green’s Function
Differentiate G wrt first time argument

20. Equation of Motion for the Green’s Function
Differentiate G wrt first time argument

21. Equation of Motion for the Green’s Function
Evaluate the T product using Wick’s Theorem
Lowest order terms
Diagram (9) is the Hartree-Fock exchange potential x Go(r1,y)
Diagram (10) is the Hartree potential x Go(x,y)
Diagram (9) is conventionally the first term in the self-energy
Diagram (10) is included in Ho in condensed matter physics
(9) y r1 x
(i)2v(x,r1)Go(x,r1) Go(r1,y) x y r1
(10)
(i)2v(x,r1)Go(r1,r1) Go(x,y)

22. Equation of Motion for the Green’s Function
One of the next order terms in the T product
The full expansion of the T product can be written exactly as
(i)3v(1,2) v(x,r1)Go(1,x) Go(r1,2) Go(2,r1) Go(1,y)
(11)
Go(1,y) y 1 x
S(x,1) 2 r1

23. Equation of Motion for the Green’s Function
The proper self-energy S* (F 105, M 181)
The self-energy has two arguments and hence two ‘external ends’
All other arguments are integrated out
Proper self-energy terms cannot be cut in two by cutting a single Go
First order proper self-energy terms S*(1)
Hartree-Fock exchange term Hartree (Coulomb) term
Exercise: Find all proper self-energy terms at second order S*(2)
(9) x’ x x x’
(10) r1

24. Equation of Motion for the Green’s Function
Equation of Motion for G and the Self Energy

25. Equation of Motion for the Green’s Function
Dyson’s Equation and the Self Energy

26. Equation of Motion for the Green’s Function
Integral Equation for the Self Energy

27. Equation of Motion for the Green’s Function
Dyson’s Equation (F 106)
In general, S* is energy-dependent and non-Hermitian
Both first order terms in S are energy-independent
Quantum Chemistry: first order self energy terms included in Ho
Condensed matter physics: only ‘direct’ first order term is in Ho
Single-particle band gap in solids strongly dependent on ‘exchange’ term
G(x,y) = = …
S(x’,x’’)= …

28. Evaluation of the Single Loop Bubble
One of the 10 second order diagrams for the self energy
The first energy dependent term in the self-energy
Evaluate for homogeneous electron gas (M 170)
a+b, ℓ+q
b, ℓ
w-a, k-q
a, q
a+b, ℓ+q
b, ℓ

29. Evaluation of the Single Loop Bubble
Polarisation bubble: frequency integral over b
Integrand has poles at b = e ℓ - id and b = -a + e ℓ+q + id
The polarisation bubble depends on q and a
There are four possibilities for ℓ and q x y

30. Evaluation of the Single Loop Bubble
Integral may be evaluated in either half of complex plane x y

31. Evaluation of the Single Loop Bubble
From Residue Theorem
Exercise: Obtain this result by closing the contour in the lower half plane

32. Evaluation of the Single Loop Bubble
Polarisation bubble: continued
For
Both poles in same half plane
Close contour in other half plane to obtain zero in each case
Exercise: For
Show that
And that

33. Evaluation of the Single Loop Bubble
w-a, k-q
a, q
a+b, ℓ+q
Self Energy

34. Evaluation of the Single Loop Bubble
Self Energy: continued

35. Evaluation of the Single Loop Bubble
Real and Imaginary Parts
Quasiparticle lifetime t diverges as energies approach the Fermi surface