Unusual States of Vortex Matter in Mixtures of Bose-Einstein Condensates on Rotating Optical Lattices

In a single-component superfluid under rotation a broken symmetry in the order parameter space results in a broken translational symmetry in real space: a vortex lattice. If translational symmetry is restored, the phase of the order parameter disorders and thus the broken symmetry in the order parameter space is also restored. We show that for Bose-Einstein condensate mixtures in optical lattices with negative dissipationless drag, a new situation arises. This state is a modulated vortex liquid which breaks translational symmetry in the direction transverse to the rotation vector.

Figure 1

Vortex orderings for the parameters $n_2/n_1=4$, $n_d/n_1=-5.0$, and $m_2/m_1=2$. Panel (a) shows the low-temperature phase ($T=6.90$), where vorticity averages [shown in gold (light gray) for component 2 and green (dark gray) for component 1] produce a hexagonal vortex lattice in one component and a honeycomb lattice in the other component. Panel (b) shows the situation occurring at a higher temperature ($T=9.52$), where a hexagonal sublattice of the honeycomb lattice melts, the number of vorticity peaks in real-space averages diminishes by a factor of 2, and the remaining hexagonal lattice is cocentered with a vortex lattice of component 1. The insets in the top panels in parts (a) and (b) show the structure factor evolution. The lower panels show a typical 3D vortex configuration in a $24\times 24\times 24$ segment of the system.

Figure 2

Vortex orderings for the parameters $n_2/n_1=16$, $n_d/n_1=-2.5$, and $m_2/m_1=2$. The $\mathit{k}$-space inset is the full $\mathit{k}$-space structure factor for component 1, in both panels (a) and (b). The left-hand side in panels (a) and (b) is a real-space average of component 2 [gold (light gray)], while the right-hand side is component 1 [green (dark gray)]. Column (a) corresponds to the ordering in low-temperature ($T=11.0$). Column (b) shows the situation taking place at higher temperature ($T=13.3$) where the green (dark gray) vortices (component 1) are in the phase corresponding to a dual superfluid bosonic density wave. It is far from obvious from the 3D picture in the panel (b) that component 1 breaks translational symmetry. However, when averaged over many configurations, we see that component 1 indeed breaks translational symmetry.

Figure 3

Plot of the evolution of the structure factor for component 1 when with increasing temperature the system undergoes a sequence of the phase transitions from the state with $\mathrm{U}(1)\times \mathrm{U}(1)$ broken symmetry to the state with U(1) broken symmetry and finally to the fully symmetric state (denoted as “Sym”). The squares represent the structure function of the square lattice in component 1, $S_{\mathrm{sq}}^{(1)}$, at the Bragg peaks at ${\mathit{k}}_{\perp }=(0.7854,0.00)\approx (\pi /4,0)$. The crosses represent the structure function of the triangular lattice in component 1, $S_{\mathrm{tri}}^{(1)}$, at the Bragg-peak at ${\mathit{k}}_{\perp }=(1.1781,0.0982)\approx (3\pi /8,\pi /64)$. Furthermore, the circles represent the helicity modulus in the $z$ direction for component 1. It vanishes at the same temperature as the square lattice ordering ceases. However, at the same temperature there appears a nonzero structure factor for a triangular lattice $S_{\mathrm{tri}}^{(1)}$. This is associated with the vortex state dual to bosonic superfluid density wave. The parameters are $n_2/n_1=16$, $n_d/n_1=-2.5$ and $m_2/m_1=2$. The system size is $L\times L\times L$, with $L=64$. Periodic boundary conditions are used in all directions, ${10}^5$ sweeps are used for thermalization and ${10}^6$ sweeps for collecting average values with sampling every 100th sweep.