Combining Harmonic Sampling with the Worm Algorithm to Improve the Efficiency of Path Integral Monte Carlo

Sourav Karmakar, Sutirtha Paul, Adrian Del Maestro, Barak Hirshberg

arXiv:2511.04597

Abstract

We propose an improved Path Integral Monte Carlo (PIMC) algorithm called Harmonic PIMC (H-PIMC) and its generalization, Mixed PIMC (M-PIMC). PIMC is a powerful tool for studying quantum condensed phases. However, it often suffers from a low acceptance ratio for solids and dense confined liquids. We develop two sampling schemes especially suited for such problems by dividing the potential into its harmonic and anharmonic contributions. In H-PIMC, we generate the imaginary time paths for the harmonic part of the potential exactly and accept or reject it based on the anharmonic part. In M-PIMC, we restrict the harmonic sampling to the vicinity of local minimum and use standard PIMC otherwise, to optimize efficiency. We benchmark H-PIMC on systems with increasing anharmonicity, improving the acceptance ratio and lowering the auto-correlation time. For weakly to moderately anharmonic systems, at βℏω=16, H-PIMC improves the acceptance ratio by a factor of 6-16 and reduces the autocorrelation time by a factor of 7-30. We also find that the method requires a smaller number of imaginary time slices for convergence, which leads to another two- to four-fold acceleration. For strongly anharmonic systems, M-PIMC converges with a similar number of imaginary time slices as standard PIMC, but allows the optimization of the auto-correlation time. We extend M-PIMC to periodic systems and apply it to a sinusoidal potential. Finally, we combine H- and M-PIMC with the worm algorithm, allowing us to obtain similar efficiency gains for systems of indistinguishable particles.

Description

This repository includes links, code, scripts, and data to generate the figures in a paper.

Requirements

The data in this project was generated via the H-PIMC publicly available here. Processed data is included in the data directory

Support

This work was performed with support from the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award Number DE-SC0024333, the Israel Science Foundation (grants No. 1037/22 and 1312/22) and the Center for Computational Molecular and Materials Science at Tel Aviv University.

Figures

Figure 02: PIMC vs H-PIMC

Figure 03: PIMC vs H-PIMC for different anharmonicities

Figure 04: Many body PIMC vs H-PIMC for different anharmonicities

Figure 05: Speedup vs anharmonicity

Figure 06: WallClocktime

Figure 07: M-PIMC with periodic boundaries

Figure 08: 2-body Speedup vs anharmonicity

These figures are released under CC BY-SA 4.0 and can be freely copied, redistributed and remixed.