One way to model the strange metal phase of certain materials is via a
holographic description in terms of probe D-branes in a Lifshitz spacetime,
characterised by a dynamical exponent z. The background geometry is dual to a
strongly-interacting quantum critical theory while the probe D-branes are dual
to a finite density of charge carriers that can exhibit the characteristic
properties of strange metals. We compute holographically the low-frequency and
low-momentum form of the charge density and current retarded Green's functions
in these systems for massless charge carriers. The results reveal a
quasi-particle excitation when z<2, which in analogy with Landau Fermi liquids
we call zero sound. The real part of the dispersion relation depends on
momentum k linearly, while the imaginary part goes as k^2/z. When z is greater
than or equal to 2 the zero sound is not a well-defined quasi-particle. We also
compute the frequency-dependent conductivity in arbitrary spacetime dimensions.
Using that as a measure of the charge current spectral function, we find that
the zero sound appears only when the spectral function consists of a single
delta function at zero frequency.Comment: 20 pages, v2 minor corrections, extended discussion in sections 5 and
6, added one footnote and four references, version published in JHE
