Fermi Liquid Theory: Principles

Landau developed the idea of quasiparticle excitations in the context of interacting Fermi systems. His theory is known as Fermi liquid theory. He introduced the idea phenomenologically, and later Abrikosov and Kalatnikov gave a formal derivation using diagrammatic perturbation theory to all orders. Landau suggested describing the excited states of the interacting system as in one-to-one correspondence with the excited states of the noninteracting system, through “switching on” the pair interactions. The interactions conserve the total particle number, spin, and momentum. Starting with a noninteracting system with one particle added in state p, σ to the ground state Fermi sea, and switching on the interactions, so that the particle becomes “dressed” by its interaction with the other particles, gives a state with characteristics of a particle in an excited state with definite momentum p, spin state σ , and adding one to the particle count. The energy of course is not preserved because the Hamiltonian is changed. In addition the state given by this switch-on process will eventually decay into a collection of more complicated states (e.g. by exciting particle-hole pairs out of the Fermi sea) so that there is a finite lifetime. Thus the process gives a state with simple quantum numbers p, σ,N, and counting, because of the one-to-one correspondence with the noninteracting system, but it is not a true eigenstate of the interacting Hamiltonian. It is called a quasiparticle or quasi-excitation. In the noninteracting system particles can only be added for p > pF , and so this gives quasiparticle excitation with p > pF . (Remember, pF is not changed by interactions.) For p < p, no particles can be added to the noninteracting system, but a particle can be removed from p, σ to form an excited state (of the N − 1 particle system). Switching on the interaction now gives a quasihole state with momentum −p,−σ . We can account for both types of excitations in terms of a change in occupation number δnp,σ which is+1 for the added particle/quasiparticle for p > pF , and −1 for the removed particle or hole/quasihole for p < pF . In this notation we are using the filled Fermi sea as a reference for the quasiparticles. The idea of switching on the interaction to define the quasi-excitations only makes sense if an appropriate switching rate τ−1 s can be found. This has to be slow enough that perturbations in the energy ∼ h̄/τs are small compared to the energy scale of interest. This is of order p2/2m− εF ∼ vF (p−pF )with vF ∼ pF/m the Fermi velocity. On the other hand τs must be shorter than the lifetime of the quasiparticle, otherwise it will decay away during its birth. The lowest order decay process is scattering a particle out of the Fermi sea. Applying the Fermi Golden rule shows that the decay rate of a quasiparticle of momentum p will vary proportionate to (p−pF )2 for p near pF , since by energy conservation and the Pauli exclusion principle the two particles must scatter into a narrow band of states of width about (p−pF ) near the Fermi surface. Thus the quasiparticle is well defined only for p→ pF—typically we might guess for |p − pF | pF , although the estimate must also depend on the strength of the interactions. For a small number of quasiexcitations the energy relative to the ground state is given by superposition