  The atomic Bose gas is studied across a Feshbach resonance, mapping out its
phase diagram, and computing its thermodynamics and excitation spectra. It is
shown that such a degenerate gas admits two distinct atomic and molecular
superfluid phases, with the latter distinguished by the absence of atomic
off-diagonal long-range order, gapped atomic excitations, and deconfined atomic
pi-vortices. The properties of the molecular superfluid are explored, and it is
shown that across a Feshbach resonance it undergoes a quantum Ising transition
to the atomic superfluid, where both atoms and molecules are condensed. In
addition to its distinct thermodynamic signatures and deconfined half-vortices,
in a trap a molecular superfluid should be identifiable by the absence of an
atomic condensate peak and the presence of a molecular one.
