  Motivated by the fundamental question of the fate of interacting bosons in
flat bands, we consider a two-dimensional Bose gas at zero temperature with an
underlying quartic single-particle dispersion in one spatial direction. This
type of band structure can be realized using the NIST scheme of spin-orbit
coupling [Y.-J. Lin, K. Jim\'{e}nez-Garcia, and I. B. Spielman, Nature
$\textbf{471}$, 83 (2011)], in the regime where the lower band dispersion has
the form $\varepsilon_{\textbf{k}} \sim k_{x}^{4}/4+k_{y}^{2}+\ldots$, or using
the shaken lattice scheme of Parker $\textit{et al.}$ [C. V. Parker, L.-C. Ha
and C. Chin, Nature Physics $\textbf{9}$, 769 (2013)]. We numerically compare
the ground state energies of the mean-field Bose-Einstein condensate (BEC) and
various trial wave-functions, where bosons avoid each other at short distances.
We discover that, at low densities, several types of strongly correlated states
have an energy per particle ($\epsilon$), which scales with density ($n$) as
$\epsilon \sim n^{4/3}$, in contrast to $\epsilon \sim n$ for the weakly
interacting Bose gas. These competing states include a Wigner crystal,
quasi-condensates described in terms properly symmetrized fermionic states, and
variational wave-functions of Jastrow type. We find that one of the latter has
the lowest energy among the states we consider. This Jastrow-type state has a
strongly reduced, but finite condensate fraction, and true off-diagonal long
range order, which suggests that the ground state of interacting bosons with
quartic dispersion is a strongly-correlated condensate reminiscent of
superfluid Helium-4. Our results show that even for weakly-interacting bosons
in higher dimensions, one can explore the crossover from a weakly-coupled BEC
to a strongly-correlated condensate by simply tuning the single particle
dispersion or density.
