Path integral molecular dynamics for bosons and fermions: From ultracold atoms to quantum dots

Whether particles are bosons or fermions is a most fundamental property of quantum-mechanical systems. It is also particularly important for accurately describing systems of ultracold trapped atoms, electrons in quantum dots, ortho- and para-hydrogen and many more. Path integral molecular dynamics (PIMD) simulations are widely used to study quantum effects in chemistry and physics. However, they completely neglect this property assuming the particles are distinguishable.

We present a new method for simulating indistinguishable particles using PIMD. For bosons, the main difficulty is enumerating all particle permutations, which scale exponentially with system size. We show that the potential and forces can be evaluated using a recurrence relation that avoids enumerating all permutations while providing the correct thermal expectation values. The resulting algorithm scales cubically with system size allowing the first application of PIMD to large bosonic systems [1].

For fermions, the infamous sign problem presents an additional formidable challenge. By adding an auxiliary, short-range repulsive potential and correcting the results using a variational principle for the free energy we are able to alleviate the sign problem for small systems.

The method is tested and applied to ultracold atoms and agrees with analytical and numerically-exact results. Preliminary results for few-electron quantum dots will also be discussed as well as an analysis of the role of exchange effects at different temperatures, through the relative probability of different ring-polymer configurations.

[1] B. Hirshberg, V. Rizzi and M. Parrinello, Proc. Natl. Acad. Sci. USA (2019) 116, 21445-21449.