Worm Algorithm for Continuous-Space Path Integral Monte Carlo Simulations

We present a new approach to path integral Monte Carlo (PIMC) simulations based on the worm algorithm, originally developed for lattice models and extended here to continuous-space many-body systems. The scheme allows for efficient computation of thermodynamic properties, including winding numbers and off-diagonal correlations, for systems of much greater size than that accessible to conventional PIMC simulations. As an illustrative application of the method, we simulate the superfluid transition of ${}^4\mathrm{He}$ in two dimensions.

Figure 1

Schematic illustration of swap move described in the text. (a) before the move. (b) after the move.

Figure 2

Superfluid fraction ${\rho }_S(T)$ computed for 2D ${}^4\mathrm{He}$ on systems with different numbers $N$ of ${}^4\mathrm{He}$ atoms. The system density is $\rho =0.0432{\AA }^{-2}$. Dotted lines represent fits to the numerical data (in the critical region) obtained using the procedure illustrated in Ref. 14. The leftmost dotted line is the extrapolation to the infinite system. Open squares show results obtained in Ref. 14 for the same system, with $N=25$.

Figure 3

One-particle density matrix computed for 2D ${}^4\mathrm{He}$ at a density $\rho =0.0432{\AA }^{-2}$ for a system of 200 atoms, at $T=0.675\mathrm{K}$ (upper curve) and $T=1.0\mathrm{K}$ (lower curve). Statistical errors on the curves are very small, and not shown for clarity. In the inset we present data (on a log-log scale) for the $N=2500$ system at $T=0.675\mathrm{K}$, with clear signatures of the Kosterlitz-Thouless behavior in the vicinity of the critical point.