  We use unbiased computational methods to elucidate the onset and properties
of pair superfluidity in two-species fermionic and bosonic systems with onsite
interspecies attraction loaded in one-dimensional optical lattice. We compare
results from quantum Monte Carlo (QMC) and density matrix renormalization group
(DMRG), emphasizing the one-to-one correspondence between the Drude weight
tensor, calculated with DMRG, and the various winding numbers extracted from
the QMC. Our results show that, for any nonvanishing attractive interaction,
pairs form and are the sole contributors to superfluidity, there are no
individual contributions due to the separate species. For weak attraction, the
pair size diverges exponentially, i.e. Bardeen-Cooper-Schrieffer (BCS) pairing
requiring huge systems to bring out the pair-only nature of the superfluid.
This crucial property is largely overlooked in many studies, thereby
misinterpreting the origin and nature of the superfluid. We compare and
contrast this with the repulsive case and show that the behavior is very
different, contradicting previous claims about drag superfluidity and the
symmetry of properties for attractive and repulsive interactions. Finally, our
results show that the situation is similar for soft core bosons: superfluidity
is due only to pairs, even for the smallest attractive interaction strength
compatible with the largest system sizes that we could attain.
