Zero sound


Zero sound is the name given by Lev Landau to the unique quantum vibrations in quantum Fermi liquids.
This sound can no longer be thought of as a simple wave of compression and rarefaction, but rather a fluctuation in space and time of the quasiparticles' momentum distribution function.
As the shape of Fermi distribution function changes slightly, zero sound propagates in the direction for the head of Fermi surface with no change of the density of the liquid.

Derivation from Boltzmann transport equation

The Boltzmann transport equation for general systems in the semiclassical limit gives, for a Fermi liquid,
where is the density of quasiparticles with momentum and position at time, and is the energy of a quasiparticle of momentum . The semiclassical limit assumes that fluctuates with angular frequency and wavelength, which are much lower than and much longer than respectively, where and are the Fermi energy and momentum respectively, around which is nontrivial. To first order in fluctuation from equilibrium, the equation becomes
When the quasiparticle's mean free path , ordinary sound waves propagate with little absorption. But at low temperatures , the mean free path exceeds, and as a result the collision functional. Zero sound occurs in this collisionless limit.
In the Fermi liquid theory, the energy of a quasiparticle of momentum is
where is the appropriately normalized Landau parameter, and
The approximated transport equation then has plane wave solutions
with
given by
This functional operator equation gives the dispersion relation for the zero sound waves with frequency and wave vector . The transport equation is valid in the regime where and .
In many systems, only slowly depends on the angle between and. If is an angle-independent constant with then the wave has the form and dispersion relation where is the ratio of zero sound phase velocity to Fermi velocity. If the first two Legendre components of the Landau parameter are significant, and, the system also admits an asymmetric zero sound wave solution and dispersion relation