  We study the Nambu--Jona-Lasinio model at finite temperature and finite
density by using the functional renormalization group. The RG flows of the
four-fermi coupling constant in the NJL model are investigated. We obtain the
chiral phase boundary in cases of the large-$N$ leading approximation and an
improved approximation. The large-$N$ non-leading term at the vanishing
temperature has a singularity at the fermi surface. We show that the quantum
corrections by the large-$N$ non-leading term largely influence the phase
boundary at the low temperature and high density region.
