Frustrated Bose-Einstein condensates with noncollinear orbital ordering

We investigate the unconventional Bose-Einstein condensations with the orbital degree of freedom in the three-dimensional cubic optical lattice, which gives rise to various exotic features absent in conventional scalar and spinor Bose-Einstein condensations. Orbital angular momentum moments are formed on lattice sites breaking time-reversal symmetry spontaneously. Furthermore, they exhibit orbital frustrations and develop a chiral ordering selected by the “order-from-disorder” mechanism.

Figure 1

(a) The SO(3) manifold of the 3D $p$-orbital BEC order parameter in terms of Euler angles, where $\mathbf{l}$ is the direction of the orbital angular momentum. (b) Sketch of the noncollinear orderings of orbital angular momenta in real space from Eq. (4).

Figure 2

(a) 2D orbital BEC in the square lattice without frustration. (b) A typical configuration of 3D orbital BEC in a cubic lattice with frustration. The thick blue bonds minimize both the transverse and parallel hopping energy, while the thin red bonds only minimize the transverse hopping energy.

Figure 3

The energy difference between the orbital BEC with OAM towards $z$ direction and body-diagonal direction $\Delta E=E_z-E_{\mathrm{diag}}$ with $t_{\parallel }=t$, $t_{\perp }=0.05t$, and the filling factor $n=2$.

Figure 4

Ordering patterns in a plaquette for (a) $|Q_{\mathrm{diag}}^1\rangle$ and (b) $|Q_{\mathrm{diag}}^2\rangle$ with opposite chirality. (c) Definition of the chirality in a plaquette.