Orbital Ordering and Frustration of $p$-Band Mott Insulators

We investigate the general structure of orbital exchange physics in Mott-insulating states of $p$-orbital systems in optical lattices. Orbital orders occur in both the triangular and kagome lattices. In contrast, orbital exchange in the honeycomb lattice is frustrated as described by a novel quantum 120° model. Its classical ground states are mapped into configurations of the fully packed loop model with an extra $U(1)$ rotation degree of freedom. Quantum orbital fluctuations select a six-site plaquette ground state ordering pattern in the semiclassical limit from the “order from disorder” mechanism. This effect arises from the appearance of a zero energy flat band of orbital excitations.

Figure 1

The $\tau$-vector (a), (c) and the corresponding orbital (b), (d) configurations. The $p$-orbitals in the triangular lattices form 45° and 135° angles with the $x$-axis, and those in the kagome lattices are 15°, 75°, and 135°.

Figure 2

The fully packed oriented loop configurations in which $\tau$-vectors lie in directions of $\phi =\pm 30°$, $\pm 90°$, $\pm 150°$. (a) The closest packed loop configuration with all the loops in the same chirality. (b) The $p$-orbital configuration for one closed loop in (a). The azimuthal angles of the $p$-orbitals are 45°, 105°, 165°, 225°, 285°, and 345°.

Figure 3

The fully packed unoriented loop configurations in which $\tau$-vectors lie along the bond directions. (a), (c) are the $\tau$-vector configurations with the closest packed loops and the ferromagnetic state, respectively. (b), (d) are their corresponding $p$-orbital configurations.

Figure 4

The “orbital wave” correction to the ground state energy per site for two manifolds of classic ground states depicted in Fig. 3a (the solid line) and 3c (the dashed line). $\theta$ is the rotation angle. The selected ground states correspond to that depicted in Fig. 3a and its six symmetry related counterparts in which $\theta =n\frac{\pi }{3}$ ($\theta =n\times 60°$).