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 much less than q(F), qvF much less than omega E(F)much less than E-F/(h) over bar (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(T))(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 (tau) approximation, explicit expressions are derived for mu and eta as functions of frequency. These formulas exhibit a crossover from the collisional regime (omega tau much less than 1), where mu similar to 0 and eta similar to nE(F)tau, to the collisionless regime (omega tau much greater than 1), where mu similar to nE(F) and eta similar to 0. [S0163-1829(99)02632-6].