Elasticity of an electron liquid

The zero-temperature response of an interacting electron liquid to a time-dependent vector potential of wave vector q and frequency $\omega ,$ such that $q\ll q_F,$ ${\mathrm{qv}}_F\ll \omega \ll E_F/ħ$ (where $q_F,$ $v_F,$ and $E_F$ are the Fermi wave vector, velocity, and energy, respectively), is equivalent to that of a continuous elastic medium with nonvanishing shear modulus $\mu ,$ bulk modulus K, and viscosity coefficients $\eta$ and $\zeta .$ We establish the relationship between the viscoelastic coefficients and the long-wavelength limit of the “dynamical local-field factors” $G_{L\left(T\right)\mathrm{}}(q,\omega ),$ which are widely used to describe exchange-correlation effects in electron liquids. We present several exact results for $\mu ,$ including its expression in terms of Landau parameters, and practical approximate formulas for $\mu ,$ $\eta ,$ and $\zeta$ as functions of density. These are used to discuss the possibility of a transverse collective mode in the electron liquid at sufficiently low density. Finally, we consider impurity scattering and/or quasiparticle collisions at nonzero temperature. Treating these effects in the relaxation-time $\left(\tau \right)$ approximation, explicit expressions are derived for $\mu$ and $\eta$ as functions of frequency. These formulas exhibit a crossover from the collisional regime $(\omega \tau \ll 1),$ where $\mu \sim 0$ and $\eta \sim {\mathrm{nE}}_F\tau ,$ to the collisionless regime $\left(\omega \tau \gg 1\right),$ where $\mu \sim {\mathrm{nE}}_F$ and $\eta \sim 0.$