  The neutron-drip-line nucleus $^{23}$N is investigated in a three-body model
consisting of a $^{21}$N core and two valence neutrons. By solving the Faddeev
equations with the realistic neutron-neutron potentials and the neutron-core
potentials, we calculate the ground state properties of $^{23}$N and also find
that there is a new excited state with two-neutron separation energy at about
0.18 MeV. The properties, such as the two-neutron separation energies, are
obtained with a good agreement with experiments. By calculating the
root-mean-square matter radii, the average distances between the valence
neutrons, and the average distances between the core and the center-of-mass of
the neutron pair, we show that the excited state of $^{23}$N has a clear halo
structure. The correlation density distributions of the three-body system are
also calculated to analyze its geometric configuration. At last, we find that
the excited state of $^{23}$N has a very small binding energy, a large radius
and distribution, and a triangular shape similar with the ground state. These
roperties of the excited state indicate that it can be an Efimov state.
