Seeding the second star -- II. CEMP star formation enriched from faint
  supernovae
  Carbon-enhanced metal-poor (CEMP) stars are the living fossils holding
records of chemical enrichment from early generations of stars. In this work,
we perform a set of numerical simulations of the enrichment from a supernova
(SN) of a first generation of metal-free (Pop III) star and the gravitational
collapse of the enriched cloud, considering all relevant cooling/heating
processes and chemical reactions as well as the growth of dust grains. We adopt
faint SN models for the first time with progenitor masses $M_{\rm PopIII} =
13$--$80 \ {\rm M}_{\bigodot}$, which yield C-enhanced abundance patterns
(${\rm [C/Fe]} = 4.57$--$4.75$) through mixing and fallback of innermost layers
of the ejecta. This model also considers the formation and destruction of dust
grains. We find that the metals ejected by the SN can be partly re-accreted by
the same dark matter minihalo, and carbon abundance of the enriched cloud
$A({\rm C}) = 3.80$--$5.06$ is lower than the abundance range of observed CEMP
stars ($A({\rm C}) \gtrsim 6$) because the mass of the metals ejected by faint
SNe is smaller than normal core-collapse SNe due to extensive fallback. We also
find that cloud fragmentation is induced by gas cooling from carbonaceous
grains for $M_{\rm PopIII} = 13 \ {\rm M}_{\bigodot}$ even with the lowest iron
abundance ${\rm [Fe/H]} \sim -9$. This leads to the formation of low-mass
stars, and these ``giga metal-poor'' stars can survive until the present-day
Universe and may be found by future observations.
