  We investigate the first emergence of the so-called cold accretion, the
accretion flows deeply penetrating a halo, in the early universe with
cosmological N-body/SPH simulations. We study the structure of the accretion
flow and its evolution within small halos with $\lesssim 10^8~{\rm M}_\odot$
with sufficiently high spatial resolutions down to $\sim 1 \ {\rm pc}$ scale.
While previous studies only follow the evolution for a short period after the
primordial cloud collapse, we follow the long-term evolution until the cold
accretion first appears, employing the sink particle method. We show that the
cold accretion emerges when the halo mass exceeds $\sim 2.2\times 10^7 \ {\rm
M}_\odot\left\{\left(1+z\right)/15 \right\}^{-3/2}$, the ${\it minimum}$ halo
masses above which the accretion flow penetrates halos. We further continue
simulations to study whether the cold accretion provides the dense shock waves,
which have been proposed to give birth to supermassive stars (SMSs). We find
that the accretion flow eventually hits a compact disc near the halo centre,
creating dense shocks over a wide area of the disc surface. The resulting
post-shock gas becomes dense and hot enough with its mass comparable to the
Jeans mass $M_{\rm J}\sim 10^{4-5} \ {\rm M}_\odot$, a sufficient amount to
induce the gravitational collapse, leading to the SMS formation.
