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Strong correlation effects in a two-dimensional Bose gas with quartic dispersion

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. Jiménez-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 ($ε$), which scales with density ($n$) as $ε\sim n^{4/3}$, in contrast to $ε\sim n$ 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.