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Path integral molecular dynamics for indistinguishable particles

Whether particles are bosons or fermions is a most fundamental property of quantum-mechanical systems. It is particularly important for accurately describing, for example, ultracold trapped atoms, electrons in quantum dots and nuclear spin isomers of hydrogen. 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 scales 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 reduces the scaling from exponential to cubic, allowing the first applications of PIMD to large bosonic systems ^[1].

For fermions, the infamous sign problem presents an additional formidable challenge, limiting applications to moderate temperatures and strongly-repelling systems. By harnessing the power of free-energy methods, we are able to alleviate the sign problem and study small systems at temperatures three times lower than using standard PIMD ^[2-3].

Applications ranging from models of ultracold trapped atoms ^[1] and electrons in two-dimensional quantum dots ^[2-3] to a supersolid phase of deuterium under high-pressure and low temperature ^[4] will be discussed. I will also present an analysis of the condensate fraction and the role of exchange effects at different temperatures, through the relative probability of different ring-polymer configurations.

References

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

2. B. Hirshberg, M. Invernizzi and M. Parrinello, J. Chem. Phys. (2020) 152, 171102. 3. T. Dornheim, M. Invernizzi, J. Vorberger and B. Hirshberg, J. Chem. Phys. (2020) 153, 234104.

4. C.W. Myung, B. Hirshberg, and M. Parrinello Phys. Rev. Lett. (2022) 128, 045301.

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