Worm algorithm and diagrammatic Monte Carlo: A new approach to continuous-space path integral Monte Carlo simulations

A detailed description is provided of a new worm algorithm, enabling the accurate computation of thermodynamic properties of quantum many-body systems in continuous space, at finite temperature. The algorithm is formulated within the general path integral Monte Carlo (PIMC) scheme, but also allows one to perform quantum simulations in the grand canonical ensemble, as well as to compute off-diagonal imaginary-time correlation functions, such as the Matsubara Green function, simultaneously with diagonal observables. Another important innovation consists of the expansion of the attractive part of the pairwise potential energy into elementary (diagrammatic) contributions, which are then statistically sampled. This affords a complete microscopic account of the long-range part of the potential energy, while keeping the computational complexity of all updates independent of the size of the simulated system. The computational scheme allows for efficient calculations of the superfluid fraction and off-diagonal correlations in space-time, for system sizes which are orders of magnitude larger than those accessible to conventional PIMC. We present illustrative results for the superfluid transition in bulk liquid ${}^4\mathrm{He}$ in two and three dimensions, as well as the calculation of the chemical potential of hcp ${}^4\mathrm{He}$.

Figure 1

World line beads on the $\beta$-cylinder and nearest-neighbor associations between them.

Figure 2

An illustration of the Swap update and the notation introduced in the main text.

Figure 3

Construction of the estimator for the Green function $G({\mathbf{r}}_p,\epsilon j)$, where without loss of generality we assume that ${\mathbf{r}}_{\mathcal{M}}=0$, $j_{\mathcal{M}}=0$.

Figure 4

Configuration space with diagrammatic bonds between the equal-time beads.

Figure 5

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

Figure 6

(Color online) One-particle density matrix computed for 2D ${}^4\mathrm{He}$ at a density $n=0.0432{\mathrm{\AA }}^{-2}$ 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. (Reproduced from Ref. 11.)

Figure 7

(Color online). One-particle density matrix $n\left(r\right)$ close to the SVP critical point at $T=2.14\mathrm{K}$ for two system sizes $N=64$ (filled squares) and $N=2048$ (open squares). The solid line is the theoretical prediction based on long-wavelength phase fluctuations, Eq. (6.1).

Figure 8

One-particle density matrix $n\left(r\right)$ for $N=1024$ particles at $T=1\mathrm{K}$ and SVP pressure. The solid line is the theoretical prediction based on the long-wave phase fluctuations, Eq. (6.1).

Figure 9

(Color online). Superfluid fraction ${\rho }_s\left(T\right)$ as a function of temperature at SVP, computed for different system sizes, namely $N=64$ (filled circles), $N=128$ (open circles), $N=256$ (filled squares), $N=512$ (diamonds), $N=1024$ (triangles down), and $N=2048$ (triangles left). The solid line is the experimental curve.

Figure 10

(Color online) Finite-size scaling plot for $2\lambda n{\rho }_{s}L∕T=⟨W^2⟩∕3$ at SVP. The solid line is the U(1) universality class value of $0.516\left(1\right)$.

Figure 11

(Color online) Condensate fraction at the saturated vapor pressure. The dashed line is used to guide the eye.

Figure 12

Liquid density as a function of chemical potential at low temperature $T=0.25\mathrm{K}$. The slope of the solid line is deduced from the simulated values of compressibility averaged over four points shown in the plot. The critical value of the chemical potential deduced from this plot is estimated as ${\mu }_0=0.06\pm 0.04$.

Figure 13

Zero-momentum Green function for the superfluid state of $N\approx 1000$ atoms at $T=1\mathrm{K}$, $\mu =-7.35\mathrm{K}$, and density $n=0.02184{\mathrm{\AA }}^{-3}$. It is normalized to unity at the origin. Note the small vertical scale: The $\tau$ dependence is a finite-size effect.