Superfluid Filaments of Dipolar Bosons in Free Space

We systematically investigate the zero temperature phase diagram of bosons interacting via dipolar interactions in three dimensions in free space via path integral Monte Carlo simulations with a few hundreds of particles and periodic boundary conditions based on the worm algorithm. Upon increasing the strength of the dipolar interaction and at sufficiently high densities we find a wide region where filaments are stabilized along the direction of the external field. Most interestingly by computing the superfluid fraction we conclude that the superfluidity is anisotropic and is greatly suppressed along the orthogonal plane. Finally, we perform simulations at finite temperature confirming the stability of the filaments against thermal fluctuations and provide an estimate of the superfluid fraction in the weak coupling limit in the framework of the Landau two-fluid model.

Figure 1

Zero-temperature phases of dipolar bosons in three dimensions in free space: superfluid (SF), superfluid filaments (FP), and cluster phases (CP) for varying dimensionless interaction strength $U$ (or equivalently dipolar length $a_d/r_0$) and scaled density $n{r}_0^3$. For $a_d/r_0\lesssim 0.9$ the blue SF region corresponds to a uniform superfluid, which agrees with the mean-field prediction that predicts stability for $a_d/a_s<1$. FPs are encountered for high densities $n{r}_0^3\gtrsim 5\times {10}^{-4}$ and for $a_d/r_0\gtrsim 1$. For small densities $n{r}_0^3\lesssim 5\times {10}^{-4}$ the system breaks up into tiny clusters. Phase boundaries in the gray region are not resolved. The blue line corresponds to $n=5\times {10}^{20}{\mathrm{m}}^{-3}$ and $a_d=130a_0$. The black [red] dots are the experimentally measured background scattering lengths $a_s=122a_0$ (${}^{162}\mathrm{Dy}$) [$a_s=92a_0$ (${}^{164}\mathrm{Dy}$)]. The green point at $a_s=29a_0$ [36].

Figure 2

Characterization of the phases at $n{r}_0^3={10}^{-2}$. Upper panels. PIMC density distribution at different strengths of the dipolar interaction: (a) the SF at $a_d=0.6r_0$, (b) the superfluid FP at $a_d=2.6r_0$ and (c) the CP at $a_d=6.0r_0$. Lower panels. (a) radial correlation functions $g(r)$ in the SF. (b),(c) Radial correlation functions $g_{\parallel }(r)$ along the vertical direction (red) and $g_{\perp }(r)$ along the orthogonal plane (blue). $g_{\parallel }(r)$ has a fluidlike profile in the FP. In the CP $g_{\parallel }(r)$ shows a high peak due to strong attractive interactions appears at distances $r\gtrsim r_0$. $g_{\perp }(r)$ in the FP shows a strong suppression in between two filaments whereas in the CP it flattens at intermediate distances. Simulations are done with 100 particles and 500 slices. Correlation functions are averaged over 100 configurations.

Figure 3

Superfluid fraction $f_s$ across the transition from superfluid to filament at high densities. Upper panel. $f_s$ as a function of the dipolar length along the vertical direction and the orthogonal direction for $n{r}_0^3={10}^{-2}$. Dipoles and filaments are aligned along the $z$ axis. In the SF the superfluid fraction converges to $f_s=1$ isotropically. In the filaments we observe $f_s=1$ along the vertical axis and vanishing on the orthogonal plane. Error bars are statistical uncertainties. Lower panel. Relative frequency of permutation cycles of length $L$ again for $n{r}_0^3={10}^{-2}$ and $a_d=0$, 0.6, 2.8, $6.0r_0$ in the SF (blue and yellow), the FP (gray), and the CP (purple).

Figure 4

Superfluid fraction $f_s^{(z)}$ as a function of the scaled temperature $T/T_0$ at $n{r}_0^3={10}^{-2}$ for three values of the dipolar lengths: (a) the SF with $a_d=0$ (hard-core bosons), (b) the SF with $a_d=0.6r_0$, (c) the FP with $a_d=2.6r_0$. The lines refer to the analytical prediction of the temperature dependence of the superfluid fraction at $n{r}_0^3={10}^{-2}$ from Eq. (3) at $a_d=0$ (black continuous line) and $a_d=0.6r_0$ (red dashed line). Inset: low temperature limit of $f_s^{(z)}$ from Eq. (3) and Eq. (5) in the SF at $a_d=0$ and $a_d=0.6r_0$.