Normal liquid $^3$He studied by Path Integral Monte Carlo with a parametrized partition function

We compute the energy per particle of normal liquid ${}^3$He in the temperature range $0.15-2$ K using Path Integral Monte Carlo simulations, leveraging a recently proposed method to overcome the sign problem -- a long-standing challenge in many-body fermionic simulations. This approach is based on introducing a parameter $\xi$ into the partition function, which allows a generalization from bosons ($\xi=1$) to fermions ($\xi=-1$). By simulating systems with $\xi \geq 0$, where the sign problem is absent, one can then extrapolate to the fermionic case at $\xi = -1$. Guided by an independent particle model that uncovers non-analytic behavior due to the superfluid transition, which is moderated by finite-size effects, we develop a tailored extrapolation strategy for liquid ${}^3$He that departs from the extrapolation schemes shown to be accurate in those cases were quantum degeneracy effects are weak, and enables accurate results in the presence of Bose-Einstein Condensation and superfluidity for $\xi > 0$. Our approach extends the previously proposed framework and yields energy per particle values in good agreement with experimental data.