  In the context of Gross-Pitaevskii theory, we investigate the unconventional
Bose-Einstein condensations in the two-species mixture with $p$-wave symmetry
in the second band of a bipartite optical lattice. A new imaginary-time
propagation method is developed to numerically determine the $p$-orbital
condensation. Different from the single-species case, the two-species boson
mixture exhibits two non-equivalent complex condensates in the
intraspecies-interaction-dominating regime, exhibiting the vortex-antivortex
lattice configuration in the charge and spin channels, respectively. When the
interspecies interaction is tuned across the SU(2) invariant point, the system
undergoes a quantum phase transition toward a checkerboard-like spin density
wave state with a real-valued condensate wavefunction. The influence of lattice
asymmetry on the quantum phase transition is addressed. Finally, we present a
phase-sensitive measurement scheme for experimentally detecting the UBEC in our
model.
