P-orbital Condensations of Two-species Bose Mixture in Optical Lattice

We investigate the $p$-orbital Bose-Einstein condensations~(BECs) of two-species mixture in a bipartite optical lattice. A new imaginary-time propagation method is developed to numerically solve the Gross-Pitaevskii equation with truncating states below the $p$ bands, which can be applicable to even higher orbital bands. Our study confirms that the intra-species interactions favor complex time-reversal broken BECs with staggered orbital currents. However, when the inter-species interaction increases, the complex condensate state undergoes a quantum phase transition toward a real-valued TR invariant condensate with staggered spin density structure. We discuss the origin and properties of such phase transition and its implication in the experimental measurement.