Exotic phases of interacting $p$-band bosons

We study a model of interacting bosons that occupy the first excited $p$-band states of a two-dimensional optical lattice. In contrast to the much studied single band Bose-Hubbard Hamiltonian, this more complex model allows for nontrivial superfluid phases associated with condensation at nonzero momentum and staggered order of the orbital angular momentum in addition to the superfluid-Mott insulator transition. More specifically, we observe staggered orbital angular momentum order in the Mott phase at commensurate filling and superfluidity at all densities. We also observe a transition between the staggered angular momentum superfluid phase and a striped superfluid, with an alternation of the phase of the superfluid along one direction. The transition between these two phases was observed in a recent experiment, which is then qualitatively well described by our model.

Figure 1

Hopping parameters on the square lattice for $p_x$ and $p_y$ orbitals. Because $p_{x,\left(y\right)}$ orbitals are parity odd, the overlap of orbitals changes sign depending on the direction. Along the parallel direction ($x$ for $p_x$ orbitals, $y$ for $p_y$ orbitals), the overlap is negative, which gives a positive $+t_{\parallel }$ hopping parameter. On the other hand, along the perpendicular direction, there is a conventional negative hopping parameter $-t_{\perp }$.

Figure 2

Staggered orbital angular momentum order. A configuration having states proportional to $|L_z=\pm 1\rangle$ on each site maximizes $L_z^2$, minimizing the interaction energy. Concentrating on the $p_x$ orbital, there is a phase alternation along the $x$ direction and a phase coherence along the $y$ direction, which minimizes the kinetic energy and corresponds to the condensation with wave vector $\mathbf{k}=(\pi ,0)$. The same phenomenon is observed for the $p_y$ orbitals, with reversed axes and $\mathbf{k}=(0,\pi )$. This gives an alternation of sites with $L_z=\pm 1$ and thus a staggered order for the angular momentum along $z$.

Figure 3

Stripe phase. Due to the different signs of the hopping parameters, when the system is composed of just one species ($p_y$ in this figure), it will have phase alternation along the parallel ($y$) direction and phase coherence along the transverse ($x$) direction to minimize the kinetic energy. This forms stripes aligned with the transverse direction and striped superfluid phases can be observed.

Figure 4

Phase diagram for $\Delta /t=0.5,2.0,3.5,$ and 5.0. We observe two Mott phases and a superfluid phase. The extents of the Mott phases are not varying a lot with $\Delta$, especially for the $\rho =1$ Mott phase and large values of $\Delta$.

Figure 5

Phase diagram for $\Delta /t=3.5$ and different sizes. The finite size effects are negligible.

Figure 6

Cut in the phase diagram at $\rho =1$ and $\Delta =0.5$. The system goes from a Bose condensed phase at low interaction to a Mott phase as ${\rho }_s$ goes to zero with increasing interaction $U$. There is an intermediate SAM-Bose condensed phase.

Figure 7

Cuts in the phase diagram at $\rho =1$ as a function of $\Delta /t$. At $U/t=10$ (top), the system is always in the superfluid phase (${\rho }_s\ne 0$) and we observe a staggered angular momentum order when $\Delta <1$. For $\Delta >1$, the superfluid is composed of only one dominant species. The same happens in the Mott phase for large interaction $U/t=40$ (bottom) with a smaller value ($\Delta \simeq 0.15$) for the disappearance of the SAM order.

Figure 8

Phase diagram for $\rho =1$ as a function of $U/t$ and $\Delta /t$. There are four different phases: a Mott phase, a Mott phase with staggered orbital momentum (SAM) order, a superfluid phase with SAM order, and a striped superfluid.

Figure 9

Cut in the phase diagram at $\rho =2$ as a function of $U/t$, the quantities and the observed behavior are similar to the $\rho =1$ case (see Fig. 6) but the $S_{\mathrm{SAM}}$ order is more robust as it is still nonzero in the Mott phase, despite the much larger value of $\Delta$ ($\Delta /t=3$ in this case compared to $\Delta /t=0.5$ in Fig. 6).

Figure 10

Cut in the phase diagram at $\rho =2$ as a function of $\Delta /t$. The quantities and the observed behavior are similar to the $\rho =1$ case (see Fig. 7) with a more robust SAM order persisting up to $\Delta /t\simeq 3$.

Figure 11

Phase diagram for $\rho =2$ as a function of $U/t$ and $\Delta /t$. The SAM phases are more robust than in the $\rho =1$ case (Fig. 8) but the same four phases are observed.

Figure 12

Cuts as functions of $\Delta$ for $U=20$ and different integer and noninteger densities. For these regimes, the system is always superfluid, even for integer densities. We observe that, as the density is increased, the structure factor $S_{\mathrm{SAM}}$ becomes larger and larger and persists for larger values of $\Delta$. The SAM order is found for all densities, provided $\Delta$ is small enough.