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Renormalization group analysis of $p$-orbital Bose-Einstein condensates in a square optical lattice

We investigate the quantum fluctuation effects in the vicinity of the critical point of a $p$-orbital bosonic system in a square optical lattice using Wilsonian renormalization group, where the $p$-orbital bosons condense at nonzero momenta and display rich phases including both time-reversal symmetry invariant and broken BEC states. The one-loop renormalization group analysis generates corrections to the mean-field phase boundaries. We also show the quantum fluctuations in the $p$-orbital system tend to induce the ordered phase but not destroy it via the the Coleman-Weinberg mechanism, which is qualitative different from the ordinary quantum fluctuation corrections to the mean-field phase boundaries in $s$-orbital systems. Finally we discuss the observation of these phenomena in the realistic experiment.