Crossover from one to two dimensions in liquid He4 in a nanopore

Using diffusion and path-integral Monte Carlo methods, we show that liquid $⁴\mathrm{He}$ confined in a narrow nanopore of liquid radius $R=4\phantom{\rule{0.28em}{0ex}}\text{\AA{}}$ undergoes a crossover from a one-dimensional (1D) to a two-dimensional (2D) fluid as a function of liquid density. At low liquid density, e.g., a linear density ${\ensuremath{\rho}}₀=0.15 {\text{\AA{}}}^{\ensuremath{-}1}$, the liquid energy is at a minimum when the liquid lies in a line at the center of the pore. The pair distribution function $g(x)$, the one-body density matrix $n(x)$, and the superfluid fraction ${\ensuremath{\rho}}_S/{\ensuremath{\rho}}₀$ all show 1D character that is well described by Luttinger liquid (LL) predictions. As density is increased, there is a crossover to 2D with the minimum energy configuration moving from a line at the center of the pore to a film near the pore walls. At linear density ${\ensuremath{\rho}}₀>0.40 {\text{\AA{}}}^{\ensuremath{-}1}$, the $⁴\mathrm{He}$ lies predominantly in a 2D cylindrical film midway between the center and the nanopore walls. The $g(x), n(x)$, and ${\ensuremath{\rho}}_S/{\ensuremath{\rho}}₀$ all show 2D character and the film has a low but finite transition temperature. $⁴\mathrm{He}$ at a bulk liquid density corresponds to ${\ensuremath{\rho}}₀=0.6 {\text{\AA{}}}^{\ensuremath{-}1}$ in the pore which lies in the 2D regime.