Crossover from Two- to Three-Dimensional Behavior in Superfluids

TL;DR

This study investigates the dimensional crossover in superfluids from two to three dimensions by analyzing superfluid density in different lattice geometries using Monte Carlo simulations and scaling laws.

Contribution

It introduces a numerical approach to analyze superfluid density scaling and critical behavior across different film thicknesses, confirming theoretical predictions.

Findings

Superfluid density obeys expected scaling with film thickness H.

Critical temperature T_c scales with H as predicted by theory.

Coefficient b in the scaling law matches theoretical H-dependence.

Abstract

We have studied the superfluid density $\rho_{s}$ on various size-lattices in the geometry $L \times L \times H$ by numerical simulation of the $x-y$ model using the Cluster Monte Carlo method. Applying the Kosterlitz-Thouless-Nelson renormalization group equations for the superfluid density we have been able to extrapolate to the $L \to \infty$ limit for a given value of $H$. In the superfluid phase we find that the superfluid density faithfully obeys the expected scaling law with $H$, using the experimental value for the critical exponent $\nu=0.6705$. For the sizes of film thickness studied here the critical temperature $T_{c}$ and the coefficient $b$ entering the equation $T/(\rho_{s} H) \propto 1-b(1-T/T_{c})^{1/2}$ are in agreement with the expected $H$-dependence deduced from general scaling ideas.