Re-weighting estimator for ab initio path integral Monte Carlo simulations of fictitious identical particles

The fermion sign problem constitutes one of the most fundamental obstacles in quantum many-body theory. Recently, it has been suggested to circumvent the sign problem by carrying out path integral simulations with a fictitious quantum statistics variable $\xi$, which allows for a smooth interpolation between the bosonic and fermionic limits [\textit{J.~Chem.~Phys.}~\textbf{157}, 094112 (2022)]. This $\xi$-extrapolation method has subsequently been applied to a variety of systems and has facilitated the analysis of an x-ray scattering measurement taken at the National Ignition Facility with unprecedented accuracy [\textit{Nature Commun.}~\textbf{16}, 5103 (2025)]. Yet, it comes at the cost of performing an additional $10-20$ simulations, which, in combination with the required small error bars, can pose a serious practical limitation. Here, we remove this bottleneck by presenting a new re-weighting estimator, which allows the study of the full $\xi$-dependence from a single path integral Monte Carlo (PIMC) simulation. This is demonstrated for various observables of the uniform electron gas and also warm dense beryllium. We expect our work to be useful for future PIMC simulations of Fermi systems, including ultracold atoms, electrons in quantum dots, and warm dense quantum plasmas.