  We predict the existence of a pair-liquid phase in lattice fermion systems
with finite-range attractive interactions. This exotic state competes on one
side with a normal Fermi liquid of unpaired fermions and on the other side with
a phase-separated state where all fermions are coupled into macroscopic
clusters. We show that such a phase is absent in bosonic systems and therefore
is protected by the exclusion principle. In contrast with zero-range attractive
systems where clustering of more than two fermions is directly prohibited, here
quantum statistics acts dynamically and in a more subtle way. Since a
many-fermion wave function must have nodes, the cluster formation threshold is
larger than the pair formation threshold. By directly solving a four-body
Schroedinger equation on one- and two-dimensional lattices, we map the
boundaries of pair stability. The pair liquid phase should be observable in
cold atom systems. We also discuss implications for the preformed-pair
mechanism of high-temperature superconductivity.
