Path integral Monte Carlo simulation of global and local superfluidity in liquid ${}^4\mathrm{He}$ reservoirs separated by nanoscale apertures

We present a path integral Monte Carlo study of the global superfluid fraction and local superfluid density in cylindrically symmetric reservoirs of liquid ${}^4\mathrm{He}$ separated by nanoaperture arrays. The superfluid response to both translations along the axis of symmetry (longitudinal response) and rotations about the cylinder axis (transverse response) are computed, together with radial and axial density distributions that reveal the microscopic inhomogeneity arising from the combined effects of the confining external potential and the ${}^4\mathrm{He}{-}^4\text{He}$ interatomic potentials. We make a microscopic determination of the length scale of decay of superfluidity at the radial boundaries of the system by analyzing the local superfluid density distribution to extract a displacement length that quantifies the superfluid mass displacement away from the boundary. We find that the longitudinal superfluid response is reduced in reservoirs separated by a septum containing sufficiently small apertures compared to a cylinder with no intervening aperture array, for all temperatures below $T_{\lambda }$. For a single aperture in the septum, a significant drop in the longitudinal superfluid response is seen when the aperture diameter is made smaller than twice the empirical temperature-dependent ${}^4\mathrm{He}$ healing length, consistent with the formation of a weak link between the reservoirs. Increasing the diameter of a single aperture or the number of apertures in the array results in an increase of the superfluid density toward the expected bulk value.

Figure 1

(a) 2D density distribution of ${}^4\mathrm{He}$ atoms at $T=1.25\text{K}$ contained inside a tube of radius $R_t=10$ Å and length $L=18$ Å, with periodic boundary conditions along $z$. The atomic density distribution $\rho (R,\varphi ,z)$ is averaged over the azimuthal angle $\varphi$ to give $\rho (R,z)$ in units of ${\AA }^{-3}$ (red: high density, blue: low density). (b) One-dimensional density distributions in (units of ${\AA }^{-3}$) computed as a function of $R$ for a range of temperatures below the bulk, $T_{\lambda }\approx 2.17$ K.

Figure 2

Global superfluid fraction of liquid ${}^4\mathrm{He}$ contained inside tubes of diameter $D_t=16$ Å (red circles) and $D_t=20$ Å (black circles) as a function of temperature. The calculations were made for $N=80$ atoms (red circles) or $N=123$ atoms (black circles) in a cylinder of length $L=18$ Å, with periodic boundary conditions in $z$. Blue dots show recommended values for the global superfluid fraction of bulk liquid ${}^4\mathrm{He}$ below $T_{\lambda }$ at saturated vapor pressure [21].

Figure 3

(a) One-dimensional radial superfluid density distributions in a cylindrically symmetric potential for four temperatures below bulk $T_{\lambda }$. Calculations were made for $N=123$ atoms in a tube of length $L=18$ Å and diameter $D_t=20$ Å with periodic boundary conditions imposed in the $z$ direction. The error bar at a given radial coordinate represents the sample standard deviation from the average superfluid density at that coordinate computed from statistically independent data blocks. (b) Healing surfaces for $T=0.625$ K (purple) and $T=2.00\text{K}$ (orange) computed using a local version of the superfluid mass displacement length (see text). The outer cylinder with radius 10 Å is a guide to the eye.

Figure 4

(a) Cross-sectional view of a single aperture in a septum separating cylindrical reservoirs of liquid ${}^4\mathrm{He}$. $2\delta$ is the thickness of the septum, $R_t$ is the radius of the container, and $R_a=D_a/2$ is the radius of the aperture allowing for transport of fluid between the reservoirs. (b) Superfluid fraction of reservoirs ($N=35,D_t=13$ Å, and $L=24$ Å) of liquid ${}^4\mathrm{He}$ connected by a single atomic-scale to nanoscale aperture as a function of aperture diameter at temperatures $T=0.625$ K (black squares) and 1.25 K (red squares). In each case, the aperture has a thickness $2\delta =3$ Å and all calculations use periodic boundary conditions in $z$.

Figure 5

(a) Comparison of global transverse superfluid fraction, computed by the projected path area estimator in Eq. (8) (squares), with the longitudinal superfluid fraction, computed by the winding number estimator (inverted triangles), for reservoirs of ${}^4\mathrm{He}$ ($N=100,D_t=20$ Å, and $L=18$ Å) separated by a septum containing a single aperture with $D_a=6$ Å. Red symbols denote results with the aperture center on the axis of symmetry; black symbols denote results with the aperture located off the axis of symmetry by 4 Å. (b) Local longitudinal superfluid densities ${\rho }_s(z,R)$ [obtained by averaging Eq. (4) over the angular coordinate $\varphi$], shown as functions of $(z,R)$ in cylinders ($L=18$ Å, $D_t=20$ Å) with a single off-center aperture having $\delta =1.5$ and $D_a=6$ Å at $T=0.25\mathrm{K}$ (left) and $T=1.00\mathrm{K}$ (right).

Figure 6

Global superfluid fractions computed by the winding number estimator, Eq. (3), of a reservoir of $N=100{}^4\mathrm{He}$ atoms bipartitioned by an aperture array defined in Eq. (7) for $N_a=2$ (green), $N_a=3$ (red), $N_a=4$ (yellow), and $N_a=5$ (blue) apertures having $D_a=5$ Å for three temperatures below $T_{\lambda }$. Configurations of apertures in the septum are shown above the plot. The $\left[x\right(\AA ),y(\AA \left)\right]$ coordinates of the aperture centers in the septum plane are as follows: $(0,4),(0,-4)$ for $N_a=2$; $(0,5),(4.33,-2.5),(-4.33,-2.5)$ for $N_a=3$; $(-5,0),(5,0),(0,5),(0,-5)$ for $N_a=4$; $(5.71,1.86),(0,6),(-5.71,1.86),(-3.53,-4.86),(3.53,-4.86)$ for $N_a=5$. For all simulations, the tube radius and tube length are given by $D_t=20$ Å and $L=18$ Å, respectively.