  According to the "no-node" theorem, many-body ground state wavefunctions of
conventional Bose-Einstein condensations (BEC) are positive-definite, thus
time-reversal symmetry cannot be spontaneously broken. We find that
multi-component bosons with spin-orbit coupling provide an unconventional type
of BECs beyond this paradigm. We focus on the subtle case of isotropic Rashba
spin-orbit coupling and the spin-independent interaction. In the limit of the
weak confining potential, the condensate wavefunctions are frustrated at the
Hartree-Fock level due to the degeneracy of the Rashba ring. Quantum zero-point
energy selects the spin-spiral type condensate through the
"order-from-disorder" mechanism. In a strong harmonic confining trap, the
condensate spontaneously generates a half-quantum vortex combined with the
skyrmion type of spin texture. In both cases, time-reversal symmetry is
spontaneously broken. These phenomena can be realized in both cold atom systems
with artificial spin-orbit couplings generated from atom-laser interactions and
exciton condensates in semi-conductor systems.
