  Motivated by the recent experiment on p-orbital band bosons in optical
lattices, we study theoretically the quantum phases of Mott insulator and
superfluidity in two-dimensions. The system features a novel superfluid phase
with transversely staggered orbital current at weak interaction, and a Mott
insulator phase with antiferro-orbital order at strong coupling and
commensurate filling. We go beyond mean field theory and derive from a
microscopic model an effective action that is capable of describing both the
p-band Mott insulating and superfluid phases in strong coupling. We further
calculate the excitation spectra near the quantum critical point and find two
gapless modes away from the tip of the Mott lobe but four gapless modes at the
tip. Our effective theory reveals how the phase coherence peak builds up in the
Mott regime when approaching the critical point. We also discuss the finite
temperature phase transition of p-band superfluidity.
