  We consider the problem of $N$ identical fermions interacting via a
zero-range attractive potential with a lighter atom in one dimension. Using the
few-body approach based on the Skorniakov and Ter-Martirosian equation, we
determine the energies and the critical mass ratios for the emergence of the
tetramer, pentamer, and hexamer. For large $N$, we solve the problem
analytically by using the mean-field theory based on the Thomas-Fermi
approximation. The system becomes bound when the heavy-to-light mass ratio
exceeds a critical value which grows as $N^3$ at large $N$. We also employ a
more sophisticated Hartree-Fock approach, which turns out to be equivalent to
the Thomas-Fermi approximation for determining the energies, but provides a
better description of the microscopic structure of the clusters.
