Periodic Boundary Conditions for Bosonic Path Integral Molecular Dynamics

TL;DR

This paper introduces a quadratic-scaling algorithm for bosonic path integral molecular dynamics with periodic boundary conditions, enabling efficient simulations of condensed bosonic phases like superfluidity.

Contribution

The paper rigorously enforces periodic boundary conditions in bosonic PIMD, improving upon previous ad hoc methods and ensuring quadratic computational scaling.

Findings

Algorithm scales quadratically with system size

Validated on free Bose gas and cold atom systems

Derived criteria for minimum-image approximation validity

Abstract

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 a powerful tool to simulate 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. It requires summing over the spring energies of all periodic images in the partition function, and a naive implementation scales exponentially with the 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.