Superfluidity of strongly correlated bosons in two- and three-dimensional traps

We analyze the superfluid crossover of harmonically confined bosons with long-range interaction in both two and three dimensions in a broad parameter range from weak to strong coupling. We observe that the onset of superfluidity occurs in three dimensions at significantly lower temperatures compared to that in two dimensions. This is demonstrated to be a quantum degeneracy effect. In addition, the spatial distribution of superfluidity across the shells of the clusters is investigated. It is found that superfluidity is substantially reduced in the outer layers due to increased correlation effects.

Figure 1

Superfluid crossover in 2D and 3D: The superfluid fraction ${\gamma }_{s}f$ is plotted versus the inverse temperature $\beta$ for $N=150$ particles in 2D (red triangles) and 3D (blue squares) for the coupling constants (a) $\lambda =10$, (b) $\lambda =3$, and (c) $\lambda =1$.

Figure 2

Probability distribution of exchange cycles in 2D and 3D. The probability for a single bead to be involved in an exchange cycle of length $L$, $P\left(L\right)L$, is plotted versus the former for $N=150$ particles and $\lambda =3$. The top image corresponds to $\beta =2$ with ${\gamma }_{s}f\left(2\mathrm{D}\right)\approx 0.92$ and ${\gamma }_{s}f\left(3\mathrm{D}\right)\approx 0.34$, and for the bottom image the inverse temperature has been adjusted such that ${\gamma }_{s}f\approx 0.6$ in both dimensionalities, i.e., $\beta \left(2\mathrm{D}\right)=1.35$ and $\beta \left(3\mathrm{D}\right)=2.27$.

Figure 3

Degeneracy parameter. The degeneracy parameter $\chi$ is plotted versus the inverse temperature $\beta$ for the systems from Fig. 1. The critical value ${\chi }_{c}rit$ (orange triangles) marks the superfluid crossover, i.e., ${\gamma }_{s}f=0.5$. Image (a) covers the entire inverse temperature range of our simulations, and (b) shows a magnified segment around the superfluid crossover.

Figure 4

Local superfluid fraction in 2D and 3D. The spatially resolved ratio of the superfluid and total density ${\gamma }_{s}f\left(r\right)=n_{s}f\left(r\right)/n\left(r\right)$ is plotted versus the distance to the center of the trap $r$ for $N=150$ particles in 2D and 3D for a fixed superfluid fraction ${\gamma }_{s}f\approx 0.6$. The larger errors around the center of the trap are due to the worse statistics in small angular and spherical segments in 2D and 3D, respectively.

Figure 5

Spatially resolved superfluidity. The radial density $n$ (red triangles) and superfluid density $n_{s}f$ (blue circles) are plotted with respect to the distance to the center of the trap $r$ for a fixed total superfluid fraction ${\gamma }_{s}f\approx 0.6$ for the systems from Figs. 1 and 3. The black curves correspond to a constant local SF ratio ${\gamma }_{s}f\left(r\right)=0.6$.