Zero sound explained
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
The Boltzmann transport equation for general systems in the semiclassical limit gives, for a Fermi liquid,
}\cdot\frac-\frac\cdot \frac = \text[f] ,
where
f(\vec{p},\vec{x},t)=f0(\vec{p})+\deltaf(\vec{p},\vec{x},t)
is the density of quasiparticles (here we ignore
spin) with momentum
and position
at time
, and
E(\vec{p},\vec{x},t)=E0(\vec{p})+\deltaE(\vec{p},\vec{x},t)
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
| \partial\deltaf | + |
\partialt |
}\cdot\frac-\frac\cdot \frac = \text[f] .
(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
E\rm+v\rm(|\vec{p}|-p\rm)+\int
p\rmm*
} F(p, p') \delta f(p'),where
is the appropriately normalized Landau parameter, and
f0(\vec{p})=\Theta(p\rm-|\vec{p}|)
.The approximated transport equation then has plane wave solutions
\deltaf(\vec{p},\vec{x},t)=\delta(E(\vec{p})-E\rm)ei(\vec{k ⋅ \vec{r}-\omegat)}\nu(\hat{p})
,with
[5] given by
(\omega-v\rm\hat{p} ⋅ \hat{k})\nu(\hat{p})=v\rm\hat{p} ⋅ \hat{k}\intd2
F(\hat{p},\hat{p}')\nu(\hat{p}')
.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
\nu(\hat{p})\propto({\omega}/({v\rm\hat{p} ⋅ \vec{k}})-1)-1
and dispersion relation
} - 1 = 1/F_0 where
} is the ratio of zero sound phase velocity to Fermi velocity. If the first two Legendre components of the Landau parameter are significant,
F(\hat{p},\hat{p}')=F0+F1\hat{p} ⋅ \hat{p}'
and
, the system also admits an asymmetric zero sound wave solution
\nu(\hat{p})\propto{\sin(2\theta)}/({s-\cos{\theta}})ei\phi
(where
and
are the azimuthal and polar angle of
about the propagation direction
) and dispersion relation
| \sin3\theta\cos\theta |
s-\cos\theta |
d\theta=
.
See also
Further reading
- Book: Piers Coleman. Introduction to Many-Body Physics. 9780521864886. 1st. Cambridge University Press. 2016.
Notes and References
- Landau, L. D. (1957). Oscillations in a Fermi liquid. Soviet Physics Jetp-Ussr, 5(1), 101-108.
- 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.
- 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.
- Roach, P. R., & Ketterson, J. B. (1976). Observation of Transverse Zero Sound in Normal He 3. Physical Review Letters, 36(13), 736.
- Lifshitz, E. M., & Pitaevskii, L. P. (2013). Statistical physics: theory of the condensed state (Vol. 9). Elsevier.