  Feynman's "no-node" theorem states that the conventional many-body
ground-state wavefunctions of bosons in the coordinate representation is
positive-definite. This implies that time-reversal symmetry cannot be
spontaneously broken. In this article, we review our progress in studying a
class of new states of unconventional Bose-Einstein condensations beyond this
paradigm. These states can either be the long-lived metal-stable states of
ultra-cold bosons in high orbital bands in optical lattices as a result of the
"orbital-Hund's rule" interaction, or the ground states of spinful bosons with
spin-orbit coupling linearly dependent on momentum. In both cases, Feynman's
argument does not apply. The resultant many-body wavefunctions are
complex-valued and thus break time-reversal symmetry spontaneously. Exotic
phenomena in these states include the Bose-Einstein condensation at non-zero
momentum, the ordering of orbital angular momentum moments, the half-quantum
vortex, and the spin texture of skyrmions.
