Set of supplementary notes.

152 CHAPTER 10. FERMI LIQUID THEORY
will evolve in time following the Schrödinger prescription
ψ k (r, t) = ψ k (r)e −iɛ kt/h¯
. (10.3)
Here ψ k is the Bloch wavefunction satisfying the time-independent Schrödinger equation,
and the time-dependent solution oscillates in time with a single frequency, given by h¯ω = ɛ k ,
the band energy.
If we look at this in Fourier space, we should say that
ψ k (r, ω) = 2πψ k (r)δ(ω − ɛ k ) (10.4)
so that the wavefunction has spectral weight only at ω = ɛ k /h¯. 1
We can say that the probability amplitude of finding an electronic state with energy ω and
momentum k is just
A(k, ω) = δ(ω − ɛ k ) , (10.5)
where the quantity A(k, ω) is usually called the electron spectral function.
What about an interacting system In this case, (i) interactions modify the precise form
of the dispersion relation, so we should replace ɛ k by a renormalised ˜ɛ k . This latter is often
referred to as a mass renormalisation: m ∗ /m = ɛ k / ˜ɛ k . Moreover, (ii), if we add a quasiparticle,
it will scatter from other quasiparticles or by creating particle-hole pairs. Because of this, the
probability amplitude of finding a quasiparticle of momentum k at time t will decay exponentially
at a rate given by the quasiparticle scattering rate Γ k . These two effects change the
time-dependence of the state to e (iɛ k/h¯−Γ k )t and after a Fourier transform lead to an ansatz for
the spectral function in an interacting system of the form
A(k, ω) = − 1 [
]
π I 1
(10.6)
ω − ˜ɛ k /h¯ + iΓ(k)
Notice that if the inverse lifetime Γ k → 0, the spectral function reduces to the noninteracting
(10.5). (10.6) describes quasiparticles with a dispersion curve ˜ɛ k and a decay rate (inverse
lifetime) Γ k .
We must be careful about the chemical potential. If we are in equilibrium (and at T=0), we
cannot add fermionic excitation at an energy ω < µ. So we shall infer that for ω > µ, (10.6) is
the spectral function for particle-like excitations, whereas for ω < µ it is the spectral function
for holes.
If Γ k is small, then the quasiparticles are long-lived and have some real meaning. More
precisely: a quasiparticle resonance at energy h¯ω = ˜ɛ k can be resolved, if the peak width in
the spectral function is less than the peak’s centre position, or, borrowing the terminology
developed for damped harmonic oscillators, if the quality factor (ratio of peak position over
peak width) is larger than 1. For an interacting system (right panel in Fig. 10.7), A has width
= 2 × Γ k . The phase-space argument made in section 10.3.1 showed that Γ k ∝ ɛ 2 . The quality
factor ˜ɛ k /Γ ∝ ˜ɛ k /˜ɛ 2 k therefore diverges for ɛ → 0. We find that quasiparticles are overdamped
at high energies, but well-defined for E → E F .
1 Getting the sign here requires one to adopt a sign convention for Fourier transforms that is opposite to the
one often used in maths books. We have also set h¯ = 1.