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Recently, fictitious identical particles have provided a promising way toovercome the fermion sign problem and have been used in path integral MonteCarlo (PIMC) to accurately simulate warm dense matter with up to 1000 electrons(T. Dornheim et al., arXiv:2311.08098 (2023)). The inclusion of fictitiousidentical particles in path integral molecular dynamics (PIMD) can provideanother way to simulate fermion systems. In a recent paper (J. Chem. Phys. 159,154107 (2023)), Feldman and Hirshberg improved the recursive formula for PIMDof N identical bosons, significantly reducing the computational complexity from$O(PN^3)$ to $O(N^2+PN)$. In this paper, we extend this latest recursiveformula for bosons to PIMD of fictitious identical particles to improve theefficiency of simulating fermion systems. We also provide the virial estimatorfor calculating energy by using the recursive technique. As an example, we usethe quadratic scaling PIMD for fictitious identical particles to study thesimulation of hundreds of fermions in a two-dimensional periodic potential, inthe hope of providing a simulation tool for two-dimensional Fermi-Hubbard modeland other strongly correlated fermion systems, such as the simulation ofultracold fermionic gases in optical lattices.

Quadratic scaling path integral molecular dynamics for fictitious identical particles and its application to fermion systems