报告摘要：Feynman's "no-node" theorem states that the conventional many-body ground state wave functions of bosons in the coordinate representation are positive definite. This implies that time-reversal symmetry cannot be spontaneously broken. In this talk, we present 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 metastable states of ultracold 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 with on-site orbital angular momentum, the half-quantum vortex, and the spin texture of skyrmions.