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Physics 216: Special topics in many-body physics, Spring 2003:
http://socrates.berkeley.edu/˜ jemoore/phys216.html
Lecture VIII
In order to understand the significance of the correlation function
Gαβ (t 1 , r 1 , t 2 , r 2 ) =
{ −i〈 0 |Ψα(t 1 , r 1 )Ψ† β (t 2 , r 2 )| 0 〉 if t 2 < t 1 i〈 0 |Ψ† β (t 2 , r 2 )Ψα(t 1 , r 1 )| 0 〉 if t 2 ≥ t 1
we attempt to calculate it for the simple Hamiltonian
H =
∑
kα
kc† kαckα. (2)
Assume that k is a momentum index, so the Hamiltonian is translation-invariant and spin-symmetric: then Gαβ is a function of two variables Gαβ (t = t 1 − t 2 , r = r 1 − r 2 ). Note that this convention of t = t 1 − t 2 is the one used in Landau and Lifshitz volume 9. Some other books use t = t 2 − t 1 , so be alert for this difference in conventions.
We’ll often use the Green’s function in the momentum representation, which is
G(ω, p) =
∫ G(t, r)e−i(p·r−ωt)^ dt d^3 x. (3)
The inverse relation is
G(t, r) =
∫ G(ω, p)ei(p·r−ωt)^
dω d^3 p (2π)^4
For the simple noninteracting Hamiltonian, we actually know the ground state: it is obtained by occupying all the states of momentum less than the Fermi momentum:
| 0 〉 =
∏
|k|<kF
c† k↑c† k↓|vac〉 (5)
where here our notation is | 0 〉 for the many-particle ground state, and |vac〉 for the vacuum (the state of zero particles).
First assume t > 0, so the time-ordering operator doesn’t do anything. Then
Gαβ (t, r) = −i〈 0 |Ψα(0, 0)Ψ† β (−t, −r)| 0 〉 = −i〈 0 |ψα(0)e−iHtψ† β (−r)eiHt| 0 〉 = −ieiE^0 t〈 0 |ψα(0)e−iHtψ† β (−r)| 0 〉
= −i eiE^0 t V
∑
kk′
〈 0 |ckαe−iHteik ′r c† k′β | 0 〉. (6)
From this we see that we must have k′^ = k and α = β. Now a special property of the noninteracting Hamiltonian is that the action of the creation operator in the above will either give 0, if |k| < kF , or another eigenstate of energy E 0 + k, if |k| > kF. So we are left with, for t 1 − t 2 > 0,
G(t, r) = −i
∑
|k|>kF
e−ik^ t+ikr V
Now we perform the same calculation for t < 0.
Gαβ (t, r) = i〈 0 |Ψ† β (0, 0)Ψα(t, r)| 0 〉 = i〈 0 |ψ† β (0)eiHtψα(r)e−iHt| 0 〉 = i〈 0 |ψ† β (0)eiHtψα(r)e−iE^0 t| 0 〉
= i
e−iE^0 t V
∑
kk′
〈 0 |c† k′β eik
′r eiHtckα| 0 〉. (8)
So again α = β and k = k′, and for t < 0,
G(t, r) = i
∑
|k|<kF
e−ik^ t+ikr V
Now let us find what this means in Fourier space. We want to calculate
G(ω, p) =
∫ G(t, r)e−i(p·r−ωt)^ dt d^3 x. (10)
The sum over k in G(t, r) can be replaced by
∫ d^3 k (^) (2Vπ) 3 in our units (recall ¯h = 1). Then the spatial integral gives 3 factors of 2π and a δ-function δ(k − p), leaving only
G(ω, p) =
{ ∫ (^) ∞ 0 −ie −i(p−ω−iη)t (^) dt if |p| > kF ∫ (^0) −∞ ie −i(p−ω+iη)t (^) dt if |p| < kF
=
{ (^) i −i(p−ω−iη) if^ |p|^ > kF i −i(p−ω+iη) if^ |p|^ < kF
Here η is some infinitesimal positive number (η = 0+) introduced to make the integrals convergent. So in general we can write
G(ω, p) =
ω − (p − μ) + iη sign(p − μ)
The meaning of the small imaginary part is that it controls where the pole in the complex ω plane lies, relative to the real axis. Since the small imaginary part only matters at the pole, the above is often written G(ω, p) =
ω − (p − μ) + iη sign ω
For p − μ negative, the pole lies just above the real axis. For p − μ positive, the pole lies just below the real axis. This gives the correct prescription for obtaining G(t, p) for t = 0−, which is related to the occupancy of momentum p:
N (p) = −iG(t = 0−, p) = −i
∫ (^) ∞
−∞
G(ω, p)e−iωt^ dω 2 π
For this slightly negative value of t, we can close the contour in the upper half-plane. Then for |p| < kF , we obtain 1 (since there is a pole of residue 1, and the integral gives 2πi/(2π), while for |p| > kF , there are no poles in the upper half-plane and the integral gives 0. This is as expected since −iG(t = 0−, p) = 〈 0 |c† pcp| 0 〉 = N (p). (15)
operators; also assume spin symmetry. We obtain
G(t, r) = −i
∑ m
〈 0 |Ψ(t, r)|m〉〈m|Ψ†(0, 0)| 0 〉
= −i
∑ m
ei(E^0 −Em)t〈 0 |ψ(0)|m〉〈m|ψ†(r)| 0 〉. (21)
If there is overall translational invariance, then we can choose a basis of eigenstates |m〉 that are also eigenstates of momentum. Then rewriting ψ(r) = V −^1 /^2 ∑ k e ikrck, and similarly ψ†(0),
the two momentum indices must be the same, and we are left with
G(t, r) = −i
∑
k
∑ m
eikr+i(E^0 −Em)t V
〈 0 |ck|m〉〈m|c† k| 0 〉
= −i
∑
k
∑ m
eikr+i(E^0 −Em)t V |〈 0 |ck|m〉|^2. (22)
So we have expressed G(t, r) for positive t 1 − t 2 as a sum over states that have one more particle than the ground state. We can also see from the exponential that G will take a much simpler form as a function of ω and p, which we give below.
For negative t, we obtain
G(t, r) = i
∑
k
∑ m
eikr+i(Em−E^0 )t V 〈 0 |c† k|m〉〈m|ck| 0 〉
= i
∑
k
∑ m
eikr+i(Em−E^0 )t V
|〈 0 |c† k|m〉|^2. (23)
So we have, for negative t 1 − t 2 , a sum over states that have one fewer particle than the ground state. It is easy to substitute in values for the free Fermi gas and show that these reduce to the previous expressions.
In (ω, p) space, the Green’s function becomes
G(ω, p) ∼
( ∑ m
|〈 0 |cp|m〉|^2 ω − (Em − E 0 − μ) + iη
∑ m
|〈 0 |c† p|m〉|^2 ω + (Em − E 0 + μ) − iη
)
. (24)
We can use this Green’s function to define a general electron density of states for a many-body system, that reduces to the ordinary density of states for the noninteracting system:
=G(ω, p) =
{ −πA(ω, p) if ω > 0 πB(ω, p) if ω < 0.
where
A(ω, p) =
∑ m
|〈 0 |cp|m〉|^2 δ(ω+μ−(Em −E 0 )), B(ω, p) =
∑ m
|〈 0 |c† p|m〉|^2 δ(ω−μ−(Em −E 0 )). (26)
Note that these definitions differ by an overall numerical factor from those in Landau and Lifshitz volume 9. These “spectral functions” for addition or subtraction of an electron to a many-body system are directly measured in experiments like photoemission and tunneling.
