η-ensemble path integral Monte Carlo approach to the free energy of the warm dense electron gas and the uniform electron liquid

We explore the recently introduced $\eta$-ensemble approach to compute the free energy directly from \emph{ab initio} path integral Monte Carlo (PIMC) simulations [T.~Dornheim \emph{et al.}, arXiv:2407.01044] and apply it to the archetypal uniform electron gas model both in the warm dense matter and strongly coupled regimes. Specifically, we present an in-depth study of the relevant algorithmic details such as the choice of the free weighting parameter and the choice of the optimum number of intermediate $\eta$-steps to connect the real, non-ideal system ($\eta =1$) with the ideal limit ($\eta =0$). Moreover, we explore the inherent decomposition of the full free energy into its ideal bosonic, ideal-to-interacting, and bosonic-to-fermionic contributions for different parameter regimes. Finally, we compare our new free energy data with an existing free energy parametrization [Groth \emph{et al.}, Phys.~Rev.~Lett.~$\mathbf{\text{119}}$, 135001 (2017)] obtained via adiabatic connection formula evaluations, and we find very good agreement in its range of applicability, i.e., for density parameters $r_s\le 20$; in addition, we present the first PIMC results for the free energy in the low density regime of $20<r_s\le 100$. We expect our results to be of interest both for the study of matter under extreme conditions, as well as to the more general field of PIMC simulations of interacting quantum many-body systems.