  The pseudogap state of cuprate high-temperature superconductors has been
often viewed as either a yet unknown competing order or a precursor state to
superconductivity. While awaiting the resolution of the pseudogap problem in
cuprates, we demonstrate that local pairing fluctuations, vortex liquid
dynamics and other precursor phenomena can emerge quite generally whenever
fermionic excitations remain gapped across the superconducting transition,
regardless of the gap origin. Our choice of a tractable model is a lattice band
insulator with short-range attractive interactions between fermions in the
s-wave channel. An effective crossover between Bardeen-Cooper-Schrieffer (BCS)
and Bose-Einstein condensate (BEC) regimes can be identified in any band
insulator above two dimensions, while in two dimensions only the BEC regime
exists. The superconducting transition is "unconventional" (non-pair-breaking)
in the BEC regime, identified by either the bosonic mean-field or XY
universality class. The insulator adjacent to the superconductor in the BEC
regime is a bosonic Mott insulator of Cooper pairs, which may be susceptible to
charge density wave ordering. We construct a function of the many-body
excitation spectrum whose non-analytic changes define a sharp distinction
between band and Mott insulators. The corresponding "second order transition"
can be observed out of equilibrium by driving a Cooper pair laser in the Mott
insulator. We explicitly show that the gap for charged bosonic excitations lies
below the threshold for Cooper pair breakup in any BEC regime, despite quantum
fluctuations. Our discussion ends with a view of possible consequences for
cuprates, where antinodal pair dynamics has certain features in common with our
simple s-wave picture.
