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Physics 216: Special topics in many-body physics, Spring 2003:
http://socrates.berkeley.edu/˜ jemoore/phys216.html
Lecture VII
In the previous lecture we developed a phenomenological theory of Fermi liquids, and were able to connect some experimental quantities to the parameters Fl and Gl that describe the interaction between quasiparticles near the Fermi surface. This theory, due to Landau, has been justified by later authors in various ways; probably the most powerful is using the renormalization group (cf. RMP article of R. Shankar). Here we will focus on understanding what the notion of “adiabatic continuity” means in terms of microscopic properties like Green’s functions. Most of the lecture follows Schrieffer chapter 5 except that the notation used is more modern than Schrieffer’s (it is more like Landau and Lifshitz or AGD).
We now introduce a few formal quantities that you have probably seen in a discussion of many- body perturbation theory. We will not be doing too much perturbation theory in this course, as the focus is on nonperturbative phenomena, but there is a philosophical underpinning of many- body perturbation theory that is actually quite important. The idea is to avoid dealing with the enormously complicated wavefunction, which is in any event cannot be measured directly. Instead we focus on correlation functions (“Green’s functions”) which are both easier to calculate and more directly measurable.
This idea is analogous to dealing with reduced one- and two-particle distribution functions f 1 and f 2 in classical statistical mechanics/kinetic theory, rather than the full distribution function g. First let us review some of the main facts from classical many-body physics. Recall the definition of the one-particle distribution function in momentum and position space,
f 1 (t, p, r) ≡ 〈
∑^ N
i=
δ(p − pi)δ(r − ri)〉 = N
∫ g(z 1 ,... , zN ) dz 2... dzN. (1)
Here zi = (pi, ri) and g is normalized to give 1 when integrated over the positions and momenta of all particles. In classical kinetic theory, one writes the (approximate) Boltzmann equation as an evolution equation for the one-particle distribution function f 1 :
∂tf 1 + v · ∇xf 1 =
∫ w′(f 1 ′f 2 ′ − f 1 f 2 )dp 2 dp′ 1 dp′ 2. (2)
This equation comes from truncating the BBGKY hierarchy, which relates the time evolution of fi to that of fi+1:
( (^) ∂ ∂t
) fn(z 1 ,... , zn) = −
∑^ n
i=
∫ dzs+1Ki,s+1 · ∇pi fs+1(z 1 ,... , zs+1), (3)
where hn is a differential operator reflecting flows of and collisions between the first n particles, and the collision term on the right side reflects collisions between one of these particles and the n + 1th particle.
How does the above carry over to quantum mechanics? It will still be useful to define quantum versions of the reduced distribution functions, but for the most part Boltzmann-type time evolution equations will not be used to calculate these functions: instead one usually works in frequency space because the time scales involved are so rapid (an exception is the quantum Boltzmann equation
for liquid He). Instead we will come up with a diagrammatic theory for understanding small perturbations to the free Fermi or Bose gas. Another reason is that quantum systems are often studied under conditions of “linear response”: when driven far from equilibrium, their behavior is not nearly as well understood as in classical systems, which already show surprising behavior (like turbulence) far from equilibrium.
Schrodinger’s equation for the evolution of the wave function is
ih ¯ ∂ψs(t) ∂t
= H(t)ψs(t). (4)
Here the Hamiltonian is time-varying only if some external field applied to the system is time- varying. For an isolated system, H is constant and
ψs(t) = e−iH(t−t^0 )/¯hψs(t 0 ). (5)
We can make a unitary transformation to the Heisenberg picture:
ψh(t) = ψh = eiH(t−t^0 )/¯hψs(t 0 ). (6)
Any operator (not just the Hamiltonian) also must transform between the two pictures:
Oh(t) = eiH(t−t^0 )/ ¯h Os(t)e−iH(t−t^0 )/ h¯ (7)
so that expectation values, which are physical, are unchanged. Hence the time evolution of a Heisenberg operator is given by
i¯h dOh(t) dt
= [Oh(t), H] + i¯h ∂Oh(t) ∂t
Finally, in the “interaction picture” for a Hamilonian divided into H = H 0 + H′, we treat H 0 in the Heisenberg rep. and H′^ in the Schrodinger rep.:
ψi(t) = eiH^0 (t−t^0 )/h¯ψs(t) (9)
and Oi(t) = eiH^0 (t−t^0 )/¯hOs(t)e−iH^0 (t−t^0 )/h¯. (10)
Schrodinger’s equation in the interaction picture becomes
i¯h ∂ψi(t) ∂t
= H i′(t)ψi(t), (11)
where H i′(t) is H i′(t) = eiH^0 (t−t^0 )/ h¯ H s′e−iH^0 (t−t^0 )/ ¯h (12)
Henceforth in this lecture we set ¯h = 1 and write r 1 for r 1.
Let’s define the quantum field operators for a many-particle system to be, in the Schrodinger representation (so the operators are independent of time),
ψα(x) =
∑
k
ukα(x)ckα, ψ α†(x) =
∑
k
ukα(x)∗ckα†^ (13)
essentially determines all one-body observables of the form Fαβ =
∑ i f^ i αβ through the quantum stat. mech. relation 〈 0 |Fαβ | 0 〉 =
∫ (^) ( f (^) αβ^1 ρβα(t, r 1 , r 2 )
) r 1 =r 2 d^3 x 1. (21)
(The reason for allowing r 1 to be different from r 2 for the action of the operator, but not the integral, can be understood by considering the single-particle momentum operator, for example.)
For a translationally invariant system, only the difference r = r 1 − r 2 matters, and with no spin-dependence, we have ραβ = ρδαβ and
ρ(r) = −iG(0−, r). (22)
So the equal-time correlation function has a natural interpretation in terms of the particle density. At r = 0, for a translationally invariant system the density is just the mean density N/V :
N/V = 2ρ(0) = − 2 iG(0−, 0). (23)
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. (24)
The inverse relation is
G(t, r) =
∫ G(ω, p)ei(p·r−ωt)^
dω d^3 p (2π)^4
Next time we will calculate the one-particle correlation function exactly for the noninteracting problem H =
∑
kα
kc† kck (26)
and then show how its behavior in an interacting system can be used to give a more precise definition of a Fermi liquid.
