Integrability of Fermionic Dynamics

Barak Hirshberg, Michele Invernizzi, Michele Parrinello

We present a method for performing path integral molecular dynamics (PIMD) simulations for fermions and address its sign problem. PIMD simulations are widely used for studying many-body quantum systems at thermal equilibrium. However, they assume that the particles are distinguishable and neglect bosonic and fermionic exchange effects. Interacting fermions play a key role in many chemical and physical systems, such as electrons in quantum dots and ultracold trapped atoms. A direct sampling of the fermionic partition function is impossible using PIMD since its integrand is not positive definite. We show that PIMD simulations for fermions are feasible by employing our recently developed method for bosonic PIMD and reweighting the results to obtain fermionic expectation values. The approach is tested against path integral Monte Carlo (PIMC) simulations for up to seven electrons in a two-dimensional quantum dot for a range of interaction strengths. However, like PIMC, the method suffers from the sign problem at low temperatures. We propose a simple approach for alleviating it by simulating an auxiliary system with a larger average sign and obtaining an upper bound to the energy of the original system using the Bogoliubov inequality. This allows fermions to be studied at temperatures lower than would otherwise have been feasible using PIMD, as demonstrated in the case of a three-electron quantum dot. Our results extend the boundaries of PIMD simulations of fermions and will hopefully stimulate the development of new approaches for tackling the sign problem.

E. B. Davies

Yunuo Xiong, Hongwei Xiong

Most recently, path integral molecular dynamics (PIMD) has been successfully applied to perform simulations of identical bosons and fermions by Hirshberg et al. In this work, we demonstrate that PIMD can be developed to calculate Green's function and extract momentum distributions for spin-polarized fermions. In particular, we show that the momentum distribution calculated by PIMD has potential applications to numerous quantum systems, e.g., ultracold fermionic atoms in optical lattices.

Artur Sowa

Michael A. Clark F G, Anthony D. Kennedy

We consider how to accelerate fermionic molecular dynamics algorithms by introducing n pseudofermion fields coupled with the nth root of the fermionic kernel. This reduces the maximum pseudofermionic force, and thus allows a larger molecular dynamics integration step size without hitting an instability in the integrator. 1.