The exchange antisymmetry between identical fermions gives rise to the well known fermion sign problem, in the form of large cancellation between positive and negative contribution to the partition function, making any simulation methods which directly sample this partition function exponentially difficult to converge. In this work, we employ path integral molecular dynamics (PIMD) and build upon the recently discovered fictitious particle model to investigate the fermion sign problem further. We consider the validity and invalidity condition for the method of parametrized path integral formulation of the partition function and extrapolation to circumvent the fermion sign problem. For the valid region of our method, our simulation shows that we may give accurate prediction of the energy for large fermion systems, which is much beyond the capability of the direct sampling in the traditional method. In particular, we find and verify a simple universal relation for high temperature noninteracting particles or strongly repulsive interacting particles at low temperatures.
The exchange antisymmetry between identical fermions gives rise to the well known fermion sign problem, in the form of large cancellation between positive and negative contribution to the partition function, making any simulation methods which directly sample this partition function exponentially difficult to converge. In this work, we employ path integral molecular dynamics (PIMD) and build upon the recently discovered fictitious particle model to investigate the fermion sign problem further. We consider the validity and invalidity condition for the method of parametrized path integral formulation of the partition function and extrapolation to circumvent the fermion sign problem. For the valid region of our method, our simulation shows that we may give accurate prediction of the energy for large fermion systems, which is much beyond the capability of the direct sampling in the traditional method. In particular, we find and verify a simple universal relation for high temperature noninteracting particles or strongly repulsive interacting particles at low temperatures.
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.
By generalizing the recently developed path integral molecular dynamics for identical bosons and fermions, we consider the finite-temperature thermodynamic properties of fictitious identical particles with a real parameter ξ interpolating continuously between bosons (ξ = 1) and fermions (ξ = -1). Through general analysis and numerical experiments, we find that the average energy may have good analytical properties as a function of this real parameter ξ, which provides the chance to calculate the thermodynamical properties of identical fermions by extrapolation with a simple polynomial function after accurately calculating the thermodynamic properties of the fictitious particles for ξ ≥ 0. Using several examples, it is shown that our method can efficiently give accurate energy values for finite-temperature fermionic systems. Our work provides a chance to circumvent the fermion sign problem for some quantum systems.
High Resolution Image Download MS PowerPoint Slide We combine the recent η-ensemble path integral Monte Carlo approach to the free energy [Dornheim et al. Phys. Rev. B 2025 111, L041114] with a recent fictitious partition function technique based on inserting a continuous variable that interpolates between the bosonic and Fermionic limits [Xiong and Xiong J. Chem. Phys. 2022 157, 094112] to deal with the Fermion sign problem. As a practical example, we apply our setup to the warm, dense, uniform electron gas over a broad range of densities and temperatures. We obtain accurate results for the exchange–correlation free energy down to half the Fermi temperature and find excellent agreement with the state-of-the-art parametrization by Groth et al. [ Phys. Rev. Lett. 2017 119, 135001]. Our work opens up new avenues for the future study of a host of interacting Fermi systems, including warm dense matter, ultracold atoms, and electrons in quantum dots, and for Fermionic free energy calculations with unprecedented system size.
We formulate a path-integral Monte Carlo algorithm for simulating lattice systems consisting of fictitious particles governed by a generalized exchange statistics. This method, initially proposed for continuum systems, introduces a continuous parameter ξ in the partition function that interpolates between bosonic (ξ=1) and fermionic (ξ=−1) statistics. We generalize this approach to discrete lattice models and apply it to the two-dimensional Hubbard model of fictitious particles, including the Bose- and Fermi-Hubbard models as special cases. By combining reweighting and ξ-extrapolation techniques, we access both half-filled and doped regimes. In particular, we demonstrate that the method remains effective even in strongly correlated, doped systems where the fermion sign problem hinders conventional quantum Monte Carlo approaches. Our results validate the applicability of the fictitious particle framework on lattice models and establish it as a promising tool for sign-problem mitigation in strongly interacting fermionic systems.
Recently, fictitious identical particles have provided a promising way to overcome the fermion sign problem and have been used in path integral Monte Carlo to accurately simulate warm dense matter with up to 1000 electrons [T. Dornheim et al., J. Phys. Chem. Lett. 15, 1305 (2024)1948-718510.1021/acs.jpclett.3c03193]. The inclusion of fictitious identical particles in path integral molecular dynamics (PIMD) can provide another way to simulate fermion systems. In a recent paper [Y. M. Y. Feldman et al., J. Chem. Phys. 159, 154107 (2023)0021-960610.1063/5.0173749], Feldman and Hirshberg improved the recursive formula for PIMD of identical bosons, significantly reducing the computational complexity. In this paper, we extend this latest recursive formula for bosons to PIMD of fictitious identical particles to improve the efficiency of simulating fermion systems. We also provide the virial estimator for calculating energy of fictitious identical particles by using the new recursive technique, which can help suppress fluctuations in energy calculation. As an example, we use the quadratic scaling PIMD for fictitious identical particles to study the simulation of hundreds and even thousands of fermions in a two-dimensional periodic potential, in the hope of providing a simulation tool for the large-scale Fermi-Hubbard model and other strongly correlated fermion systems, such as the simulation of ultracold fermionic gases in optical lattices and the extended Fermi-Hubbard model using quantum dots.
The ab initio path integral Monte Carlo (PIMC) method is one of the most successful methods in statistical physics, quantum chemistry and related fields, but its application to quantum degenerate Fermi systems is severely hampered by an exponential computational bottleneck: the notorious fermion sign problem. Very recently, Xiong and Xiong [J. Chem. Phys. 157, 094112 (2022)] have suggested to partially circumvent the sign problem by carrying out simulations of fictitious systems guided by an interpolating continuous variable ξ ∈ [-1, 1], with the physical Fermi- and Bose-statistics corresponding to ξ = -1 and ξ = 1. It has been proposed that information about the fermionic limit might be obtained by calculations within the bosonic sector ξ > 0 combined with an extrapolation throughout the fermionic sector ξ < 0, essentially bypassing the sign problem. Here, we show how the inclusion of the artificial parameter ξ can be interpreted as an effective penalty on the formation of permutation cycles in the PIMC simulation. We demonstrate that the proposed extrapolation method breaks down for moderate to high quantum degeneracy. Instead, the method constitutes a valuable tool for the description of large Fermi-systems of weak quantum degeneracy. This is demonstrated for electrons in a 2D harmonic trap and for the uniform electron gas (UEG), where we find excellent agreement (∼0.5%) with exact configuration PIMC results in the high-density regime while attaining a speed-up exceeding 11 orders of magnitude. Finally, we extend the idea beyond the energy and analyze the radial density distribution (2D trap), as well as the static structure factor and imaginary-time density-density correlation function (UEG).