Bosonic and Fermionic Dipoles on a Ring

We show that dipolar bosons and fermions confined in a quasi-one-dimensional ring trap exhibit a rich variety of states because their interaction is inhomogeneous. For purely repulsive interactions, with increasing strength of the dipolar coupling there is a crossover from a gaslike state to an inhomogeneous crystal-like one. For small enough angles between the dipoles and the plane of the ring, there are regions with attractive interactions, and clustered states can form.

Figure 1

Sketch of the ring-shaped trap in the $xy$ plane. The dipole moments $\mathbf{d}=d(\sin \alpha ,0,\cos \alpha )$ are aligned by an external field $\mathbf{E}$ in the $xz$ plane.

Figure 2

Left: Coupling strength $V_{\mathrm{c}.\mathrm{m}.}(\Theta )$ for $\alpha =0$ (dashed line), $0.19\pi \approx {\alpha }_c$ (dotted line) and $\pi /2$ (solid line). Right: Relative-coordinate potential $V_{\mathrm{rel}}(\vartheta )$ for $R/a_{\perp }=10$, including the long-range asymptotics (dashed line).

Figure 3

Homogeneous system ($\alpha =0$): Ground-state energy $E(n{r}_d)$ for $N=4$ dipoles. For comparison, the plot includes the Lieb-Liniger results for the Bose gas, Hartree-Fock for fermions, and the classical limit $E_C$.

Figure 4

Density profiles $\rho (\theta )$ for $N=4$ bosons (left) and $N=3$ fermions (right) for $\alpha =0.19\pi$ (uppermost row) and $\alpha =0.22\pi$ (middle row) for different coupling strengths. Bottom: Schematic overview of the different ground states for bosons (for fermions, see text). The dashed lines indicate the boundaries between these approximate regions. On the curved dotted line, the ground-state energy is zero.

Figure 5

Ground-state energy $E(n{r}_d)$ for $N=3$ fermions (solid line, $\alpha =0.22\pi$); $N=4$ bosons (dashed line, $\alpha =0.2\pi$, $0.22\pi$, and $\pi /2$ from top to bottom).

Figure 6

Pair distribution function ${\rho }_2({\theta }_1,{\theta }_2)$ for $N=4$ bosons at $\alpha =0.20\pi$ and $n{r}_d=0.026$ (left), $n{r}_d=0.803$ (center) and $n{r}_d=0.808$ (right). Black corresponds to ${\rho }_2=0$ and white to the highest values.