Fermi-Liquid Response and Zero Sound

Landau interactions renormalize static compressibility and spin susceptibility and, in the collisionless regime, can create a collective zero-sound pole. Static response takes $\omega\to0$ before $q\to0$; zero sound takes $q,\omega\to0$ at fixed $s=\omega/(qv_F)$ with $\omega\tau\gg1$. Interchanging these limits changes the observable.

Required background. Use Landau theory and the general Kubo-response framework. Helpful background. Ward-consistent response and diffusion, conductivity, and susceptibility clarify the two limits.

Static Fermi-liquid response

A uniform density change shifts the quasiparticle energy by both the chemical potential and the $\ell=0$ forward interaction. In the total-DOS convention of this chapter,

$$\kappa=\frac{N(0)}{n^2(1+F_0^s)}, \qquad \chi_s=\frac{\mu_*^2N(0)}{1+F_0^a}.$$

These are equilibrium limits. Their positivity is the $\ell=0$ Landau stability condition. Long-range Coulomb interactions modify the charge channel and can lift density motion to a plasmon; the neutral short-range case is assumed below.

Collisionless kinetic equation

Let $\delta n_{\mathbf p}=-(\partial n_0/\partial\epsilon)\nu(\hat{\mathbf p})e^{i(\mathbf q\cdot\mathbf x-\omega t)}$. Neglecting collisions when $\omega\tau\gg1$, the linearized Landau equation is

$$(\omega-\mathbf q\cdot\mathbf v_F)\nu(\hat{\mathbf p}) =\mathbf q\cdot\mathbf v_F \int\frac{\mathrm d\Omega_{\mathbf p'}}{S_{d-1}} F^s(\hat{\mathbf p}\cdot\hat{\mathbf p}')\nu(\hat{\mathbf p}').$$

For a three-dimensional isotropic liquid with only $F_0^s$, an undamped mode with $s=\omega/(qv_F)>1$ obeys

$$1=F_0^s\left[ \frac{s}{2}\ln\left(\frac{s+1}{s-1}\right)-1 \right].$$

For $F_0^s>0$ this equation has a root above the particle–hole continuum, whose small-$q$ upper edge is $\omega=qv_F$. If the root lies inside the continuum, the pole acquires Landau damping. Its residue follows from the derivative of the response denominator, not from the dispersion equation alone. The derivation and angular generalizations are given in Baym and Pethick 1991, chs. 2 and 4.

Zero sound versus first sound

Zero sound is collisionless: the Fermi surface deforms coherently before collisions establish local equilibrium. First sound is hydrodynamic and requires $\omega\tau\ll1$; its speed is fixed by the adiabatic equation of state and its attenuation by transport coefficients. Cooling a Fermi liquid can increase $\tau$ and drive a crossover between the two for a fixed probe frequency.

Exercises

Show that for large positive $F_0^s$, the zero-sound root satisfies $s^2\simeq F_0^s/3$.

Solution

For $s\gg1$, $\frac{s}{2}\ln[(s+1)/(s-1)]-1=1/(3s^2)+O(s^{-4})$. The dispersion equation becomes $1\simeq F_0^s/(3s^2)$, hence $s^2\simeq F_0^s/3$.

References

• Gordon Baym and Christopher Pethick, Landau Fermi-Liquid Theory: Concepts and Applications, Wiley-VCH (1991), chs. 2 and 4, doi:10.1002/9783527617159.