The Elasticity of an Electron Liquid
Abstract
It is shown that, at zero temperature, the response of an interacting electron liquid to a time-dependent potential of wave vector q and frequency omega such that q << q_F, qv_F << omega << E_F/hbar (where q_F, v_F and E_F are the Fermi wavevector, velocity and energy respectively), is equivalent to that of a continuous elastic medium with nonvanishing {it shear modulus} mu, bulk modulus K, and viscosity coefficients eta and zeta. Whereas K - related to the thermodynamic compressibility - is well known from Monte Carlo calculations in cases of practical interest, little attention has been paid so far to mu, eta and zeta. Here we present several exact results for mu, and practical formulas for both mu, eta and zeta. These are used to discuss the possibility of a {it transverse} collective mode in the electron liquid at sufficiently low density. We also show that the "dynamical local field factor" G(q,omega), widely used to describe exchange-correlation effects in electron liquids, is {it singular} for q,omega -> 0: in particular, lim_{omega -> 0} lim_{q -> 0}G(q,omega)v(q) ne lim_{q -> 0}lim_{omega -> 0}G(q,omega)v(q) (where v(q) is the Fourier transform of the interaction) in contrast to what has been generally assumed thus far. Finally, we consider impurity scattering and thermal relaxation effects, which introduce another frequency scale hbar/ tau << E_F, where tau is the relaxation time. Treating these effects in the relaxation time approximation, explicit expressions are derived for mu and eta as functions of frequency. These formulas exhibit a crossover from the hydrodynamic regime (omega tau << 1), where mu \sim 0 and eta \sim n E_F tau, to the dynamic regime (omega tau >> 1), where mu \sim n E_F and eta \sim 0.Keywords
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