Quantum Stripe Ordering in Optical Lattices

We predict the robust existence of a novel quantum orbital stripe order in the $p$-band Bose-Hubbard model of two-dimensional triangular optical lattices with cold bosonic atoms. An orbital angular momentum moment is formed on each site exhibiting a stripe order both in the superfluid and Mott-insulating phases. The stripe order spontaneously breaks time-reversal, lattice translation, and rotation symmetries. In addition, it induces staggered plaquette bond currents in the superfluid phase. Possible signatures of this stripe order in the time of flight experiment are discussed.

Figure 1

The condensate configuration in the real space described by Eq. (3) with TR, rotation, and $U(1)$ symmetry breaking. The unit cell contains 4 sites as marked with thick lines. The OAM moments (dashed arrowed circles) form the stripe order, and induce bond currents exhibiting staggered plaquette moments (solid arrowed circles). $\alpha$ in Eq. (3) depends on the interaction strength with $\alpha =\frac{\pi }{6}(\frac{\pi }{4})$ in the weak (strong) coupling limit, respectively. Currents vanish in the weak-coupling limit, and exist on tilted bonds at finite interaction strength with directions specified by arrows.

Figure 2

(a) Phase diagram based on the GMF theory in the $2\times 2$ unit cell (see Fig. 1). Large scale GMF calculations in a $30\times 30$ lattice are performed to confirm the stripe-ordered superfluid (SF) phase at points 1, 2, and 3 with $(t/U,\mu /U)=(0.02,2)$, (0.03, 1.5), and (0.038, 2.2), respectively. (b) The flux $\Phi$ around a rhombic plaquette vs $t/(nU)$. It decays from $\frac{1}{3}$ in the strong coupling limit to 0 in the noninteraction limit. The solid line is the GMF result at $n=3$, while the dashed line is based on the energy function Eq. (5) of the trial condensate.

Figure 3

Predicted TOF image for the stripe-ordered superfluid phase with the condensation wave vectors $K_{2,3}$. Note that the locations of the highest peaks depend on the size $l_{x,y}$ of the $p$-orbital Wannier function. Parameters are $\alpha =\pi /6$; $l_{x,y}/a=0.1$; the $\delta$ function is replaced by a Lorentzian line for display.