  We study a microscopic model for four spinless fermions on the square lattice
which exhibits a quartet bound state in the strong coupling regime. The
four-particle quantum states are analyzed using symmetry arguments and by
introducing a zoo of relevant lattice animals. These considerations, as well as
variational and exact diagonalization calculations demonstrate the existence of
a narrow quartet band at small hopping and a first order transition to
delocalized fermions at a critical hopping parameter, in qualitative contrast
to, e. g., the BCS-BEC crossover in the attractive Hubbard model. In the case
of pure attraction, an intermediate phase is found, in which a more extended
and presumably more mobile hybrid quartet dominates the ground state. We
comment on the relevance of the spin degree of freedom and on the reasons why
electron quartetting is rarely observed in real materials.
