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).
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).
The fermion sign problem constitutes one of the most fundamental obstacles in quantum many-body theory. Recently, it has been suggested to circumvent the sign problem by carrying out path integral simulations with a fictitious quantum statistics variable ξ, which allows for a smooth interpolation between the bosonic and fermionic limits [Xiong and Xiong, J. Chem. Phys.157, 094112 (2022)]. This ξ-extrapolation method has subsequently been applied to a variety of systems and has facilitated the analysis of an x-ray scattering measurement taken at the National Ignition Facility with unprecedented accuracy [Dornheim et al., Nat. Commun. 16, 5103 (2025)]. Yet, it comes at the cost of performing an additional 10-20 simulations, which, in combination with the required small error bars, can pose a serious practical limitation. Here, we remove this bottleneck by presenting a new reweighting estimator, which allows the study of the full ξ-dependence from a single path integral Monte Carlo (PIMC) simulation. This is demonstrated for various observables of the uniform electron gas and also warm dense beryllium. We expect our study to be useful for future PIMC simulations of Fermi systems, including ultracold atoms, electrons in quantum dots, and warm dense quantum plasmas.
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.
Being motivated by the surge of fermionic quantum Monte Carlo simulations at finite temperature, we present a detailed analysis of the permutation-cycle properties of path integral Monte Carlo (PIMC) simulations of degenerate electrons. Particular emphasis is put onto the uniform electron gas in the warm dense matter regime. We carry out PIMC simulations of up to N = 100 electrons and investigate exchange-cycle frequencies, which are found not to follow any simple exponential law even in the case of ideal fermions due to the finite size of the simulation box. Moreover, we introduce a permutation-cycle correlation function, which allows us to analyze the joint probability to simultaneously find cycles of different lengths within a single configuration. Again, we find that finite-size effects predominate the observed behavior. Finally, we briefly consider an inhomogeneous system, namely, electrons in a 2D harmonic trap. We expect our results to be of interest for the further development of fermionic PIMC methods, in particular, to alleviate the notorious fermion sign problem.
The exchange antisymmetry of identical fermions leads to an exponential computational bottleneck in ab initio simulations, known as the fermion sign problem. The thermodynamic approach of fictitious identical particles (Y. Xiong and H. Xiong, J. Chem. Phys. 157, 094112 (2022)) provides an efficient and accurate means to simulate some fermionic systems by overcoming the fermion sign problem. This method has been significantly promoted and used by National Ignition Facilities for the ab initio simulations and is believed to have wide application prospects in warm dense quantum matter (T. Dornheim et al., arXiv: 2402.19113 (2023)). By utilizing the fictitious identical particles in the bosonic regime and constant energy extrapolation method (Y. Xiong and H. Xiong, Phys. Rev. E 107, 055308 (2023); T. Morresi and G. Garberoglio, Phys. Rev. B 111, 014521 (2025)), there are promising results in simulating the energy of strongly quantum degenerate fermionic systems. The previous works mainly concern the energy of Fermi systems or only consider situations of weak quantum degeneracy. In this study, we extend the concept of the constant energy extrapolation method and demonstrate the potential of the constant density extrapolation method to accurately simulate the density distribution of fermionic systems in strongly quantum degenerate conditions. Furthermore, based on the energy derived from the constant energy extrapolation method, we present simulation results for the entropy of fermions.
Accurate thermodynamic simulations of correlated fermions using path integral Monte Carlo (PIMC) methods are of paramount importance for many applications such as the description of ultracold atoms, electrons in quantum dots, and warm-dense matter. The main obstacle is the fermion sign problem (FSP), which leads to an exponential increase in computation time both with an increase in the system size and with a decrease in the temperature. Very recently, Hirshberg et al. [J. Chem. Phys. 152, 171102 (2020)] have proposed to alleviate the FSP based on the Bogoliubov inequality. In the present work, we extend this approach by adding a parameter that controls the perturbation, allowing for an extrapolation to the exact result. In this way, we can also use thermodynamic integration to obtain an improved estimate of the fermionic energy. As a test system, we choose electrons in 2D and 3D quantum dots and find in some cases a speed-up exceeding 106, as compared to standard PIMC, while retaining a relative accuracy of ∼0.1%. Our approach is quite general and can readily be adapted to other simulation methods.
Recent advances in warm dense matter physics, e.g. laser compressed matter, lead to an increasing interest in the description of correlated, degenerate electrons at finite temperatures. Path integral Monte Carlo (PIMC) methods cannot correctly describe weakly to moderately coupled and strongly degenerate Fermi systems due to the so-called fermion sign problem. The Configuration Path Integral Monte Carlo (CPIMC) approach greatly reduces the sign problem and allows for the exact computation of thermodynamic properties in this regime. In addition, the first successful implementation of the diagrammatic worm algorithm for a general Hamiltonian in Fock space with arbitrary pair interactions gives direct access to the Matsubara Green function.
This thesis demonstrates the capabilities of the CPIMC approach for a model system of Coulomb interacting fermions in a two-dimensional harmonic trap. The correctness of the CPIMC implementation is verified by rigorous comparisons with an exact diagonalization method. Benchmark results are presented, which reveal large errors of the Hartree-Fock approximation in open shell configurations even for weak coupling and a significant deviation of multi-level blocking PIMC data in the complete basis set limit.
The application of the CPIMC method to the warm dense homogeneous electron gas (HEG) quantifies the accuracy of recently published restricted PIMC (RPIMC) results, which have been the basis for the construction of exchange-correlation free energy functionals to be used in finite-temperature density functional theory calculations of warm dense matter. It is shown that the errors of the RPIMC data exceed 10 % at intermediate densities. Additionally, highly accurate data for the exchange-correlation energy at high densities, which are inaccessible by the RPIMC method, are provided. These results are needed to significantly increase the quality of future exchange-correlation functionals to be used in finite-temperature applications.