  We study the critical behavior of Bose-Einstein condensation in the second
band of a bipartite optical square lattice in a renormalization group framework
at one-loop order. Within our field theoretical representation of the system,
we approximate the system as a two-component Bose gas in three dimensions. We
demonstrate that the system is in a different universality class than the
previously studied condensation in a frustrated triangular lattice due to an
additional Umklapp scattering term, which stabilizes the chiral superfluid
order at low temperatures. We derive the renormalization group flow of the
system and show that this order persists in the low energy limit. Furthermore,
the renormalization flow suggests that the phase transition from the thermal
phase to the chiral superfluid state is first order.
