I am trying to clarify a conceptual issue about phenomenological Fermi liquid theory. My confusion can be explained using the following two sentences from Dupuis's many body theory notes, but the same sentiment is present in many other sources as well. The sentences are:
According to the adiabatic continuity assumption, as the interaction is slowly turned on we generate an (excited) eigenstate of the interacting system. However, because of the interactions the state under study is damped and acquires a finite lifetime.
There seems to be two competing concepts here, both of which seem central to FLT. On one hand,
- Quasiparticles correspond to excited energy eigenstates of the interacting Fermi liquid. These eigenstates can be obtained by starting with a corresponding excited state of the free Fermi gas and adiabatically switching on the interactions. Landau's theory postulates adiabatic continuity, so that the interacting eigenstates stand in 1-1 correspondence with the free eigenstates and can therefore be labeled by the same quantum numbers.
On the other hand,
- Whereas free particles do not interact, and an excited state of the free theory will persist indefinitely, quasiparticles of the interacting theory do interact with each other. While a quasiparticle will then in general decay, its lifetime will diverge as it approaches the Fermi surface.
These two notions of quasiparticles seem contradictory to me. If the quasiparticles are eigenstates of the interacting theory, then they should not decay. Conversely, if the quasiparticles do interact and decay, then how are they related to the free particle excitations, and how should I understand that the quasiparticles carry the same quantum numbers as the free particles?
EDIT: After talking to a friend, I think the answer lies in the fact that the adiabatic theorem does not hold for an eigenstate without a gap. If the system were gapped, and none of the energies crossed as the interaction were turned on, then eigenstates would necessarily evolve into eigenstates. But since the Fermi system is gapless, there's no reason that the adiabatically evolved eigenstates remain eigenstates. But it would be nice to have confirmation from someone more knowledgeable, and it's strange that this point is not discussed in any of the sources I've checked.
EDIT 2: Apologies for the multiple edits. After doing some research, I think that my previous edit was incorrect. As far as I can tell, the Gell-Mann and Low theorem guarantees that an infinitely slow adiabatic turning on of the interaction evolves eigenstates of the free theory into eigenstates of the interacting theory. The application to FLT seems immediate to me: if we start with an excited free particle state, and turn on the interactions infinitely slowly, we expect the state we obtain to be an eigenstate. But clearly this cannot be what we are actually doing in FLT, since the quasiparticles are not eigenstates. So how should I make sense of this?