Zero sound is the name given by Lev Landau in 1957 to the unique quantum vibrations in quantum Fermi liquids.[1] The zero 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 (or largely), zero sound propagates in the direction for the head of Fermi surface with no change of the density of the liquid. Predictions and subsequent experimental observations of zero sound[2][3][4] was one of the key confirmation on the correctness of Landau's Fermi liquid theory.

Derivation from Boltzmann transport equation

edit

The Boltzmann transport equation for general systems in the semiclassical limit gives, for a Fermi liquid,

 ,

where   is the density of quasiparticles (here we ignore spin) with momentum   and position   at time  , and   is the energy of a quasiparticle of momentum   (  and   denote equilibrium distribution and energy in the equilibrium distribution). 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   (equivalently, relaxation time  ), ordinary sound waves ("first sound") propagate with little absorption. But at low temperatures   (where   and   scale as   ), 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  [5] 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   (note that this constraint is stricter than the Pomeranchuk instability) 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   (where   and   are the azimuthal and polar angle of   about the propagation direction  ) and dispersion relation

 .

See also

edit

References

edit
  1. ^ Landau, L. D. (1957). Oscillations in a Fermi liquid. Soviet Physics Jetp-Ussr, 5(1), 101-108.
  2. ^ Keen, B. E., Matthews, P. W., & Wilks, J. (1965). The acoustic impedance of liquid helium-3. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 284(1396), 125-136.
  3. ^ Abel, W. R., Anderson, A. C., & Wheatley, J. C. (1966). Propagation of zero sound in liquid He 3 at low temperatures. Physical Review Letters, 17(2), 74.
  4. ^ Roach, P. R., & Ketterson, J. B. (1976). Observation of Transverse Zero Sound in Normal He 3. Physical Review Letters, 36(13), 736.
  5. ^ Lifshitz, E. M., & Pitaevskii, L. P. (2013). Statistical physics: theory of the condensed state (Vol. 9). Elsevier.

Further reading

edit
  • Piers Coleman (2016). Introduction to Many-Body Physics (1st ed.). Cambridge University Press. ISBN 9780521864886.