  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 $d/\eta<2$.
