Zero sound

Derivation from Boltzmann transport equation

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

${\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }t}+\frac{\mathrm{\partial }E}{\mathrm{\partial }\overrightarrow{p}}\cdot \frac{\mathrm{\partial }f}{\mathrm{\partial }\overrightarrow{x}}-\frac{\mathrm{\partial }E}{\mathrm{\partial }\overrightarrow{x}}\cdot \frac{\mathrm{\partial }f}{\mathrm{\partial }\overrightarrow{p}}=\text{St}[f]}$,

where ${\displaystyle f(\overrightarrow{p},\overrightarrow{x},t)=f_0(\overrightarrow{p})+\delta f(\overrightarrow{p},\overrightarrow{x},t)}$ is the density of quasiparticles (here we ignore spin) with momentum ${\displaystyle \overrightarrow{p}}$ and position ${\displaystyle \overrightarrow{x}}$ at time ${\displaystyle t}$, and ${\displaystyle E(\overrightarrow{p},\overrightarrow{x},t)=E_0(\overrightarrow{p})+\delta E(\overrightarrow{p},\overrightarrow{x},t)}$ is the energy of a quasiparticle of momentum ${\displaystyle \overrightarrow{p}}$ (${\displaystyle f_0}$ and ${\displaystyle E_0}$ denote equilibrium distribution and energy in the equilibrium distribution). The semiclassical limit assumes that ${\displaystyle f}$ fluctuates with angular frequency ${\displaystyle \omega }$ and wavelength ${\displaystyle \lambda =2\pi /k}$, which are much lower than ${\displaystyle E_{\mathrm{F}}/\hslash }$ and much longer than ${\displaystyle \hslash /p_{\mathrm{F}}}$ respectively, where ${\displaystyle E_{\mathrm{F}}}$ and ${\displaystyle p_{\mathrm{F}}}$ are the Fermi energy and momentum respectively, around which ${\displaystyle f}$ is nontrivial. To first order in fluctuation from equilibrium, the equation becomes

${\displaystyle \frac{\mathrm{\partial }\delta f}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{E}_0}{\mathrm{\partial }\overrightarrow{p}}\cdot \frac{\mathrm{\partial }\delta f}{\mathrm{\partial }\overrightarrow{x}}-\frac{\mathrm{\partial }\delta E}{\mathrm{\partial }\overrightarrow{x}}\cdot \frac{\mathrm{\partial }{f}_0}{\mathrm{\partial }\overrightarrow{p}}=\text{St}[f]}$.

When the quasiparticle's mean free path ${\displaystyle \ell \ll \lambda }$ (equivalently, relaxation time ${\displaystyle \tau \ll 1/\omega }$), ordinary sound waves ("first sound") propagate with little absorption. But at low temperatures ${\displaystyle T}$ (where ${\displaystyle \tau }$ and ${\displaystyle \ell }$ scale as ${\displaystyle T^{-2}}$ ), the mean free path exceeds ${\displaystyle \lambda }$, and as a result the collision functional ${\displaystyle \text{St}[f]\approx 0}$. Zero sound occurs in this collisionless limit.

In the Fermi liquid theory, the energy of a quasiparticle of momentum ${\displaystyle \overrightarrow{p}}$ is

${\displaystyle E_{\mathrm{F}}+v_{\mathrm{F}}(|\overrightarrow{p}|-p_{\mathrm{F}})+\int \frac{d^3{\overrightarrow{p}}^{\prime }}{4\pi p_{\mathrm{F}}{m}^{\ast }}F(p,p^{\prime })\delta f(p^{\prime })}$,

where ${\displaystyle F}$ is the appropriately normalized Landau parameter, and

${\displaystyle f_0(\overrightarrow{p})=\mathrm{\Theta }(p_{\mathrm{F}}-|\overrightarrow{p}|)}$.

The approximated transport equation then has plane wave solutions

${\displaystyle \delta f(\overrightarrow{p},\overrightarrow{x},t)=\delta (E(\overrightarrow{p})-E_{\mathrm{F}})e^{i(\overrightarrow{k}\cdot \overrightarrow{r}-\omega t)}\nu (\hat{p})}$,

with ${\displaystyle \nu (\hat{p})}$ given by

${\displaystyle (\omega -v_{\mathrm{F}}\hat{p}\cdot \hat{k})\nu (\hat{p})=v_{\mathrm{F}}\hat{p}\cdot \hat{k}\int d^2\frac{{\hat{p}}^{\prime }}{4\pi }F(\hat{p},{\hat{p}}^{\prime })\nu ({\hat{p}}^{\prime })}$.

This functional operator equation gives the dispersion relation for the zero sound waves with frequency ${\displaystyle \omega }$ and wave vector ${\displaystyle \overrightarrow{k}}$ . The transport equation is valid in the regime where ${\displaystyle \hslash \omega \ll E_{\mathrm{F}}}$ and ${\displaystyle \hslash |\overrightarrow{k}|\ll p_{\mathrm{F}}}$.

In many systems, ${\displaystyle F(\hat{p},{\hat{p}}^{\prime })}$ only slowly depends on the angle between ${\displaystyle \hat{p}}$ and ${\displaystyle {\hat{p}}^{\prime }}$. If ${\displaystyle F}$ is an angle-independent constant ${\displaystyle F_0}$ with ${\displaystyle F_0>0}$ (note that this constraint is stricter than the Pomeranchuk instability) then the wave has the form ${\displaystyle \nu (\hat{p})\propto (\omega /(v_{\mathrm{F}}\hat{p}\cdot \overrightarrow{k})-1{)}^{-1}}$ and dispersion relation ${\displaystyle \frac{s}{2}\log \frac{s+1}{s-1}-1=1/F_0}$ where ${\displaystyle s=\omega /k{v}_{\mathrm{F}}}$ is the ratio of zero sound phase velocity to Fermi velocity. If the first two Legendre components of the Landau parameter are significant, ${\displaystyle F(\hat{p},{\hat{p}}^{\prime })=F_0+F_1\hat{p}\cdot {\hat{p}}^{\prime }}$ and ${\displaystyle F_1>6}$, the system also admits an asymmetric zero sound wave solution ${\displaystyle \nu (\hat{p})\propto \sin (2\theta )/(s-\cos \theta )e^{i\varphi }}$ (where ${\displaystyle \varphi }$ and ${\displaystyle \theta }$ are the azimuthal and polar angle of ${\displaystyle \hat{p}}$ about the propagation direction ${\displaystyle \hat{k}}$) and dispersion relation

${\displaystyle {\int }_0^{\pi }\frac{{\sin }^3\theta \cos \theta }{s-\cos \theta }d\theta =\frac{4}{F_1}}$.