Quantum Quasicrystals of Spin-Orbit-Coupled Dipolar Bosons

We study quasi-two-dimensional dipolar Bose gases in which the bosons experience a Rashba spin-orbit coupling. We show that the degenerate dispersion minimum due to the spin-orbit coupling, combined with the long-range dipolar interaction, can stabilize a number of quantum crystalline and quasicrystalline ground states. Coupling the bosons to a fermionic species can further stabilize these phases. We estimate that the crystalline and quasicrystalline phases should be detectable in realistic dipolar condensates, e.g., dysprosium, and discuss their symmetries and excitations.

Figure 1

Ground-state phase diagram of two-dimensional Rashba-coupled dipolar bosons. The horizontal axis is the parameter $R$—defined in the text—which measures the ratio of the dipolar interaction to the contact interaction. The vertical axis is $k_0{d}_z$, where $k_0$ is the Rashba coupling strength and $d_z$ is the transverse confinement. In the region marked $Q{C}^{\prime }$, the system exhibits quasicrystalline phases involving $\ge 7$ pairs of momenta.

Figure 2

(a) Interaction potential $U(q)$ as a function of momentum $q$ for $R=0.6$ (dashed line) and $R=1$ (thick line). The inset shows the interaction potential in momentum space due to a single condensate at $k_0\widehat{\mathbf{y}}$ in the $R=1$ case; the color coding matches that in the main panel and indicates that the interaction energy is attractive for a condensate sufficiently far away in momentum space. (b) Ten-component (pentagonal) quasicrystal pictured in momentum space; dashed squares show typical momentum combinations that are coupled by interactions.

Figure 3

Real-space spin and total density plots for various ordered states. (a) Density of the spin-up component of the “square-lattice” phase. (b) Total density profile of the square-lattice phase. (c) Total density profile for the quasicrystalline phase; pentagons are guides to the eye. (d)–(f) Total density profile of the hexagonal phase, as a function of the relative phase ${\Xi }_{\mathbf{k}}$ [Eq. (10)] of any one of the three condensate pairs. Varying this phase smoothly from (d) 0 through (f) $\pi$ corresponds to an overall translation plus a phason.