  By analyzing simple models of fermions in lattice potentials we argue that
the zero-temperature pairing instability of any ideal band-insulator occurs at
a finite momentum. The resulting supersolid state is known as "pair density
wave". The pairing momentum at the onset of instability is generally
incommensurate as a result of phase-space restrictions and relative strengths
of interband and intraband pairing. However, commensurate pairing occurs in the
strong-coupling limit and becomes a Cooper-channel analogue of the
Halperin-Rice exciton condensation instability in indirect bandgap
semiconductors. The exceptional sensitivity of incommensurate pairing to
quantum fluctuations can lead to a strongly-correlated insulating regime and a
non-BCS transition, even in the case of weak coupling as shown by an exact
renormalization group analysis.
