Physics 216: Special topics in many-body physics, Spring 2003:
http://socrates.berkeley.edu/˜ jemoore/phys216.html
Lecture VI
We continue using the assumption that while the interparticle interactions in a Fermi liquid
may be strong, we consider only small perturbations of the quasiparticle occupancies from the
equilibrium distribution. This is somewhat justified because the “most probable” configuration, as
shown last time, still is given by the Fermi-Dirac distribution of quasiparticles, even though the
quasiparticles are no longer free. The assumption that we could keep only the leading term in the
quasiparticle interaction gives
δαβ (p) = Zfαγ,βδ (p,p0)δn(p0)γ δ dp0.(1)
Recall that δαβ (p) was defined as the change in the effective energy of quasiparticles of momentum
pinduced by a given nonequilibrium distribution of the other quasiparticles:
δE
V=Zαβ(p)δnβαdτ . (2)
The total value of ˆis now
ˆ(p)−F=vF(p−pF) + tr 0Zˆ
f(p,p0)δˆn(p0).(3)
In practice we will only be interested in small deviations near the Fermi surface, so the interaction
ˆ
fwill only need to be evaluated when both momenta lie on the Fermi surface. We will assume
spherical symmetry of the Fermi surface, so ˆ
fdepends only on the angle θbetween pand p0.
The final simplifying assumption we will make is that the spin part of the interaction is rota-
tionally symmetric in spin space, so that the Pauli matrices can only appear as a scalar product.
Then we can finally write pFm∗
π2¯
h3ˆ
f(p,p0) = F(θ) + σ·σ0G(θ).(4)
Here Fand Gare two dimensionless scalar functions of the angle θbetween pand p0. The units
work since ˆ
fmultipled by a density gives an energy; the term on the left side is the density of
quasi-particle states per energy on the Fermi surface
ν(F) = 2·4πpF2
(2π¯
h)3
dp
d |pF=pF2
π2¯
h3vF
=pFm∗
π2¯
h3.(5)
Now we will show how the effective mass m∗can be written in terms of F, using Galilean
invariance: this is the first of many examples of how experimental quantities can be related to
Fand G. (We also need to make use of our assumption that the total number of quasiparticles
is equal to the total number of original particles, so that the Fermi surface is unchanged.) The
number flux of quasiparticles is
tr Zˆn∂
∂pdτ. (6)
1
