Ab initio simulations of the thermodynamic properties and phase transition of Fermi systems based on fictitious identical particles and physics-informed neural networks

Fictitious identical particle thermodynamics has emerged as a powerful tool to overcome the fermion sign problem, enabling highly accurate simulations of one thousand fermions in warm dense matter (T. Dornheim et al., J. Phys. Chem. Lett. 15, 1305 (2024)). However, inferring the thermodynamic properties of Fermi systems from a large number of exact numerical simulations of the bosonic sector still poses subtle challenges, especially in the regime of high quantum degeneracy and in the presence of phase transitions. In this work, we demonstrate that physics-informed neural networks (PINNs), trained on data from extensive and sign-problem-free numerical simulations of the bosonic sector, offer a valuable means to infer the thermodynamic properties of Fermi systems. PINNs can play a particularly crucial role in capturing phase transitions. To illustrate the methodology of fictitious identical particles combined with PINNs for simulating the thermodynamics of Fermi systems, we explore its application in realistic scenarios, including ultracold Fermi gases in periodic potentials, and phase transitions of pair condensation formed in the unitary limit in a three-dimensional harmonic trap. For the spatially continuous Fermi-Hubbard model, we efficiently and reliably simulated hundreds of fermions here. For the Fermi gas in the unitary limit, based on the fictitious identical particle combined with PINNs, our approach confirms the universal result of the critical temperature with the increasing of the number of fermions, and is consistent with the experimental observations.