$p_{x,y}$-orbital counterpart of graphene: Cold atoms in the honeycomb optical lattice

We study the ground-state properties of the interacting spinless fermions in the $p_{x,y}$-orbital bands in the two-dimensional honeycomb optical lattice, which exhibit different features from those in the $p_z$-orbital system of graphene. In addition to two dispersive bands with Dirac cones, the tight-binding band structure exhibits another two completely flat bands over the entire Brillouin zone. With the realistic sinusoidal optical potential, the flat bands acquire a finite but much smaller bandwidth compared to the dispersive bands. The band flatness dramatically enhanced interaction effects giving rise to various charge and bond ordered states at commensurate fillings of $n=\frac{i}{6}\left(i=1–6\right)$. At $n=\frac{1}{6}$, the many-body ground states can be exactly solved as the close-packed hexagon states which can be stabilized even in the weakly interacting regime. The dimerization of bonding strength occurs at both $n=\frac{1}{2}$ and $\frac{5}{6}$, and the latter case is accompanied with the charge-density wave of holes. The trimerization of bonding strength and charge inhomogeneity appear at $n=\frac{1}{3},\frac{2}{3}$. These crystalline orders exhibit themselves in the noise correlations of the time-of-flight spectra.

Figure 1

(Color online) (a) The contour plot of the optical potential of the 2D honeycomb lattice described by Eq. (1). (b) The optical potential distribution around the potential maximum in one unit cell.

Figure 2

(a) The two sublattice structure ($A$ and $B$) of the honeycomb lattice. (b) The hexagon Brillouin zone with edge length $4\pi /\left(3\sqrt{3}a\right)$.

Figure 3

(Color online) Dispersion of the two-lowest $p_{x,y}$-orbital bands $E_{1,2}$. The band $E_1$ is completely flat, while $E_2$ exhibits Dirac points at $K_{1,2}=\left(\pm \frac{4\pi }{3\sqrt{3}a},0\right)$. The other two bands are symmetric with respect to $E=0$ [from Wu et al. (Ref. 19)].

Figure 4

(Color online) The Wannier-type localized eigenstate for the lowest band. The orbital configuration at each site is oriented along a direction tangential to the closed loop on which the particle is delocalized. The absence of the $\pi$ hopping and the destructive interference together ensure such a state as an eigenstate.

Figure 5

(Color online) The orbital configurations of eigenstates at $K_1$, which are of $p_x\pm i{p}_y$ type. The phase of each lobe is presented $\left(\omega =e^{i\left(2\pi /3\right)}\right)$. (a) ${\psi }_1\left({\vec{K}}_1\right)$ $\left(E_1=-\frac{3}{2}{t}_{\parallel }\right)$. (b) ${\psi }_2\left({\vec{K}}_1\right)$ with ${\alpha }_k=0$ $\left(E_2=0\right)$.

Figure 6

(Color online) Dispersion of the two-lowest $p_{x,y}$-orbital bands $E_{1,2}$ in the presence of the $\pi$-bonding $t_{\perp }$. The bottom band is no longer rigorously flat but acquires a narrow width of $t_{\perp }$.

Figure 7

(Color online) Dispersion of the two-lowest $p_{x,y}$-orbital bands $E_{1,2}$ in the presence of the on-site potential.

Figure 8

Band structures for the realistic optical potential in Eq. (1) along the path from $O\to K_1\to M\to K_2$ in the Brillouin zone with (a) $V/E_r=5$ and (b) $V/E_r=10$.

Figure 9

Width of the beginning six bands of the optical potential in Eq. (1). Bands 1 and 2 are the $s$-orbital bands. Moreover, bands 3–6 are the $p_{x,y}$-orbital bands, where those of 3 and 6 are flat as discussed before.

Figure 10

(Color online) The configuration of the close-packed Wigner-crystal state for both bosons and fermions at $n=\frac{1}{6}$. Each thickened plaquette has the same configuration as in Fig. 4 [from Wu et al. (Ref. 19)].

Figure 11

The phase boundary of the incompressible plaquette Wigner-crystal state of spinless fermions at $⟨n⟩=\frac{1}{6}$ [from Wu et al. (Ref. 19)].

Figure 12

The filling $⟨n⟩$ vs the chemical potential $\mu$ for spinless fermions for (a) weak and (b) strong interactions. Due to the particle-hole symmetry, only the part with $\mu$ from the band bottom $-\frac{3}{2}{t}_{\parallel }$ to $U/2$ is shown. Only one plateau appears in (a) at $n=\frac{1}{6}$, while a series of plateaus appears in (b) at $n=\frac{1}{6}$, $\frac{1}{3}$, $\frac{1}{2}$, $\frac{2}{3}$, $\frac{5}{6}$, and 1 [from Wu et al. (Ref. 19)].

Figure 13

(Color online) Bonding strength dimerization can occur at both $⟨n⟩=\frac{1}{3}$ in the weak-coupling regime and $⟨n⟩=\frac{1}{2}$ in the strong-coupling regime as depicted by the thickened (red) bonds. The orbital orientation in the dimer is along the bond direction. In the weak-coupling case $\left(⟨n⟩=\frac{1}{3}\right)$, the thickened bonds correspond to the shared edges of two neighboring plaquette states in the flat band. In the strong-coupling case $\left(⟨n⟩=\frac{1}{2}\right)$, each dimer contains one particle as an entangled state of occupied and empty sites.

Figure 14

(Color online) The trimerized states at fillings $⟨n⟩=\frac{1}{3},\frac{2}{3}$ in the strong-coupling regime as described by thickened bonds. Each trimer contains one particle at $⟨n⟩=\frac{1}{3}$ and two particles at $⟨n⟩=\frac{2}{3}$.

Figure 15

(Color online) The orbital configuration at the filling $⟨n⟩=\frac{5}{6}$ exhibits the dimerized state of holes in the strong-coupling regime. The holes mainly distribute on the position of sites 1 and 4 in each unit cell with a large bonding strength.