Figure 1

The density wave formed in a honeycomb lattice for the $m_F=1$ atoms. The points represent lattice sites. Larger points indicate a site filled with more atoms. This pattern is periodically repeated. A complementary density wave is formed by $m_F=-1$ atoms. This density wave does not lead to a sixfold symmetry breaking in time-of-flight unless additional phases appear on the sites.Reuse & Permissions

Figure 2

Schematic of the time-of-flight pattern for a superfluid in a 2D honeycomb lattice. Larger darker dots correspond to more particles with a given momentum. The complex numbers $\left|t\right|$ and $\left|z\right|$ represent the amplitudes of the Fourier transform of the condensate wave function at $k=(\frac{\pi }{a},0)$ and $k=(\frac{\sqrt{3}\pi }{2a},\frac{\pi }{2a})$ (see text). The twisted superfluid is described by $\left|t\right|\ne \left|z\right|$.Reuse & Permissions

Figure 3

Minimum eigenvalue of the Hessian, ${\lambda }_0$, in the normal superfluid phase plotted against the lattice depth $V_{\mathrm{lat}}$ (in units of $E_{\mathrm{R}}$) when $U=0.05E_{\mathrm{R}}$ and five particles (of each species) are present per unit cell. All the eigenvalues of the Hessian are positive, thereby showing the stability of the normal phase. We conclude that there is no twisted superfluid state for these potential depths. This result is illustrative of all parameter ranges we explored.Reuse & Permissions

Figure 4

Slice through the energy landscape at $V_{\mathrm{lat}}=1.8E_{\mathrm{R}}$ and $U=0.05E_{\mathrm{R}}$ and five particles (of each species) per unit cell. Dotted curve: The ratio Re[$z$]/Re[$t$]is varied and the energy is found by minimizing with respect to the other variational parameters. Solid curve: Same, but with varying Im[$z$]/Im[$t$]. We find that the overall energy minimum occurs when $\text{Re}\left[z\right]=\text{Re}\left[t\right]$ and $\text{Im}\left[z\right]=\text{Im}\left[t\right]$.Reuse & Permissions