Intertwined Superfluid and Density Wave Order in a $p$-Orbital Bose Condensate

We study a continuum model of the weakly interacting Bose gas in the presence of an external field with minima forming a triangular lattice. The second lowest band of the single-particle spectrum ($p$-band) has three minima at non-zero momenta. We consider a metastable Bose condensate at these momenta and find that, in the presence of interactions that vary slowly over the lattice spacing, the order parameter space is isomorphic to $S^{5}$. We show that the enlarged symmetry leads to the loss of topologically stable vortices, as well as two extra gapless modes with quadratic dispersion. The former feature implies that this non-Abelian condensate is a "failed superfluid" that does not undergo a Berezinskii-Kosterlitz-Thouless (BKT) transition. Order-by-disorder splitting appears suppressed, implying that signatures of the $S^5$ manifold ought to be observable at low temperatures.