  Recent experiments on ultracold atoms in optical lattices have synthesized a
variety of tunable bands with degenerate double-well structures in momentum
space. Such degeneracies in the single particle spectrum strongly enhance
quantum fluctuations, and may lead to exotic many-body ground states. Here we
consider weakly interacting spinor Bose gases in such bands, and discover a
universal quantum "order by disorder" phenomenon which selects a novel chiral
spin superfluid with remarkable properties such as spontaneous anomalous spin
Hall effect and momentum space antiferromagnetism. For bosons in the excited
Dirac band of a hexagonal lattice, such a state supports staggered spin loop
currents in real space. We show that Bloch oscillations provide a powerful
dynamical route to quantum state preparation of such a chiral spin superfluid.
Our predictions can be readily tested in spin resolved time-of-flight
experiments.
