  We perform the theoretical study on the unconventional Bose-Einstein
condensations (UBEC) in the high bands of optical lattices observed by
Hemmerich's group. These exotic states are characterized by complex-valued
condensate wavefunctions with nodal points, or real-valued ones with nodal
lines, thus are beyond the {\it "no-node"} paradigm of the conventional BECs. A
quantum phase transition is driven by the competition between the single
particle band and interaction energies. The complex UBECs spontaneously break
time-reversal symmetry, exhibiting a vortex-antivortex lattice structure.
