Probing holographic semi-local quantum liquids with D-branes

We study dynamics of probe D-branes in $(d+2)$-dimensional background with general semi-locality. The background is characterized by a parameter $\eta$ and is conformal to $AdS_{2}\times\mathbb{R}^{d}$. We discuss thermodynamics of the probe D-branes and find that the entropy density is vanishing in the extremal limit, which indicates that the background may correspond to the true ground state of the system. We also clarify the conditions under which the specific heat matches to the behavior of a Fermi liquid or a Bose liquid. We calculate the current-current and density-density retarded Green's functions, from which we can obtain the AC conductivity and the zero sound mode. The AC conductivity scales as $\omega^{-1}$ when $d/\eta<2$ and $\omega^{-2\eta/d}$ when $d/\eta>2$, while it contains a logarithmic term in $\omega$ when $d/\eta=2$. We also observe that there is no quasi-particle excitation when $d/\eta\geq2$ and classify the conditions under which the quasi-particle description is valid when $d/\eta<2$.