Periodic boundary conditions for bosonic path integral molecular dynamics

We develop an algorithm for bosonic path integral molecular dynamics (PIMD) simulations with periodic boundary conditions (PBC) that scales quadratically with the number of particles. Path integral methods are powerful tools for simulating bosonic condensed phases, which exhibit fundamental physical phenomena such as Bose-Einstein condensation and superfluidity. Recently, we developed a quadratic-scaling algorithm for bosonic PIMD but employed an ad hoc treatment of PBC. Here, we rigorously enforce PBC in bosonic PIMD. This requires summing over the spring energies of all periodic images in the partition function, and a naïve implementation scales exponentially with system size. We present an algorithm for bosonic PIMD simulations of periodic systems that scales only quadratically. We benchmark our implementation on the free Bose gas and a model system of cold atoms in optical lattices. We also study an approximate treatment of PBC based on the minimum-image convention and derive a numerical criterion to determine when it is valid.