Time-Reversal Symmetry Breaking of $p$-Orbital Bosons in a One-Dimensional Optical Lattice

We study bosons loaded in a one-dimensional optical lattice of twofold $p$-orbital degeneracy at each site. Our numerical simulations find an anti-ferro-orbital $p_x+i{p}_y$, a homogeneous $p_x$ Mott-insulator phase, and two kinds of superfluid phases distinguished by the orbital order (anti-ferro-orbital and paraorbital). The anti-ferro-orbital order breaks time-reversal symmetry. Experimentally observable evidence is predicted for the phase transition between the two different superfluid phases. We also discover that the quantum noise measurement is able to provide a concrete evidence of time-reversal symmetry breaking in the first Mott phase.

Figure 1

Phase diagram of a one-dimensional lattice Bose gas with $p_x$ and $p_y$ orbital degrees of freedom. The upper panel shows the sketch of the experimental setup we proposed. The (green) circles are used to denote the requirement of the approximate local isotropy of the lattice potential at each site. The lowest Mott lobe (with filling $\nu =1$) is dominated by $p_x$ bosons. The Mott state (with $\nu >1$) has an AFO order (see the text). We do not claim another phase for the tiny tip of the second Mott lobe beyond the AFO SF to PO SF transition line (red line) because of numerical errors. For sufficiently large hopping $t_x$ or for low filling, the Bose gas has a crossover from PO SF to a $p_x$ SF phase, which will not be discussed in this Letter.

Figure 2

Momentum distributions $\tilde{n}(\mathbf{k})$. (a) shows the 1D momentum distribution ${\tilde{n}}_{1\mathrm{D}}(k_x)$ in a geometrically 2D experimental setting (see the paragraph labeled “Experimental proposal”) for the PO SF phase. $p_x$ peaks at $k_x{a}_x=(2j+1)\pi$ ($j$ is some integer) are sharp, and $p_y$ peaks at $k_x{a}_x=2j\pi$ are broad. The insets show that the double logarithmic plot of ${\tilde{n}}_{1\mathrm{D}}(k_x)$ near the sharp (broad) peaks is linear (nonlinear). (b) shows the sketch of 2D momentum distributions [$\tilde{n}(\mathbf{k})$] in different phases (PO SF, AFO SF, and AFO Mott, from right to left). In three subgraphs, the horizontal (vertical) axis is $k_y{a}_y/\pi$ ($k_x{a}_x/\pi$). The (purple) wiggles along each subgraph show ${\tilde{n}}_{1\mathrm{D}}(k_x)$. In the AFO SF phase, the $p_y$ peaks, which are broad in PO SF, are replaced by sharp peaks. In the AFO Mott phase, there are no sharp peaks.

Figure 3

Properties of the AFO-to-PO phase transitions with total filling $\nu =1.5$. (a) shows the $Z_2$ order parameter ${\tilde{L}}_z$. Our results indicate a continuous phase transition of the AFO order. (b) shows the filling of the $p_x$ and $p_y$ bosons. The $p_y$ component does not vanish across the phase transition. (c) shows the ratio $K_x/K_y$ in the PO SF phase.