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 ω, such that qqF, qvFωEF/ (where qF, vF, and EF are the Fermi wave vector, velocity, and energy, respectively), is equivalent to that of a continuous elastic medium with nonvanishing shear modulus μ, bulk modulus K, and viscosity coefficients η and ζ. We establish the relationship between the viscoelastic coefficients and the long-wavelength limit of the ``dynamical local-field factors'' GL(T)(q,ω), which are widely used to describe exchange-correlation effects in electron liquids. We present several exact results for μ, including its expression in terms of Landau parameters, and practical approximate formulas for μ, η, and ζ 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 (τ) approximation, explicit expressions are derived for μ and η as functions of frequency. These formulas exhibit a crossover from the collisional regime (ωτ1), where μ~0 and η~nEFτ, to the collisionless regime (ωτ1), where μ~nEF and η~0.