Complex and real unconventional Bose-Einstein condensations in high orbital bands

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.