Superfluidity, Bose-Einstein condensation and dimensions of liquid $^4$He in nanopores

Path integral Monte Carlo (PIMC) calculations of the superfluid fraction, $\rho_S/\rho$, and the one-body density matrix (OBDM) (Bose-Einstein condensation (BEC)) of liquid $^4$He confined in nanopores are presented. The goal is to determine the effective dimensions of the liquid in the nanopore. We simulate a cylinder of liquid of diameter $d_L$~ surrounded by 5 \AA~ of inert solid $^4$He in a nanopore of diameter $d$; $d$ = $d_L$ + 10 \AA~ [1]. The PIMC $\rho_S(T)/\rho$ and OBDM scales as a 1D Luttinger Liquid at extremely small liquid pore diameters only, $d_L$~ = 6 \AA~ where the liquid atoms form a 1D line at the center of the pore. In the range 8 $\leq d_L \leq$ 22 \AA~ the PIMC $\rho_S(T)/\rho$ scales as a 2D liquid. In this $d_L$ range the liquid fills the pores in cylindrical layers. There is a cross over from 2D to 3D scaling at larger $d_L$ $\simeq$ 22 \AA. In the range 8 $\leq d_L \leq$ 22 \AA~, the $T_C$ predicted using the Kosterlitz-Thouless 2D scaling criterion of the OBDM agrees well with the $T_C$ obtained from $\rho_S(T)/\rho$. Superflow observed in pores of diameter (18 $< d <$ 32 \AA) is apparently standard static superflow with the low $T_C$ arising from its 2D character.\\ 1. L. Vranje\v{s} Marki\'c and H. R. Glyde, Phys. Rev. B92, 064510 (2015)