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arXiv:2603.28000

Simulation of strongly quantum-degenerate uniform electron gas using the pseudo-fermion method

Yunuo Xiong, Tommaso Morresi, Hongwei Xiong

Published Mar 30, 2026• 0 citations
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Abstract

For strongly quantum-degenerate systems at finite temperatures, the fermion sign problem remains the major obstacle to first-principles simulations. In this work, we apply the recently proposed pseudo-fermion method - designed to overcome the sign problem - to strongly quantum-degenerate uniform electron gases. We find that the pseudo-fermion method can efficiently and highly accurately infer the energy of the uniform electron gas while being free from the fermion sign problem. For example, in the strongly quantum-degenerate regime where RPIMC fails (33 spin-polarized electrons at the density parameter $r_s = 0.5$), the relative deviation between the pseudo-fermion method and the exact CPIMC result is only 0.6%. In particular, the pseudo-fermion method bridges the gap where neither CPIMC nor RPIMC can accurately simulate the regime $1 \le r_s \le 2$ at the reduced temperature $θ= 0.0625$. This work demonstrates that the pseudo-fermion method opens a new pathway for studying strongly quantum-degenerate systems in a sign-problem-free manner.

DOI
10.48550/arxiv.2603.28000

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For strongly quantum-degenerate systems at finite temperatures, the fermion sign problem remains the major obstacle to first-principles simulations. In this work, we apply the recently proposed pseudo-fermion method - designed to overcome the sign problem - to strongly quantum-degenerate uniform electron gases. We find that the pseudo-fermion method can efficiently and highly accurately infer the energy of the uniform electron gas while being free from the fermion sign problem. For example, in the strongly quantum-degenerate regime where RPIMC fails (33 spin-polarized electrons at the density parameter rs=0.5r_s = 0.5rs​=0.5), the relative deviation between the pseudo-fermion method and the exact CPIMC result is only 0.6%. In particular, the pseudo-fermion method bridges the gap where neither CPIMC nor RPIMC can accurately simulate the regime 1≤rs≤21 \le r_s \le 21≤rs​≤2 at the reduced temperature θ=0.0625θ= 0.0625θ=0.0625. This work demonstrates that the pseudo-fermion method opens a new pathway for studying strongly quantum-degenerate systems in a sign-problem-free manner.
In this work, within the framework of path integral Monte Carlo, we construct a pseudo-fermion propagator by replacing the original fermionic determinant with its absolute value. This modified propagator defines an auxiliary system free from the fermion sign problem, enabling efficient simulations of fermionic systems. We found that by shifting the pseudo-fermion energy based on the energy of a non-interacting fermion system, we can efficiently and reliably infer the energy of fermionic systems in various situations, from strong quantum degeneracy to weak quantum degeneracy. We have performed numerical simulations of quantum dots confined in a two-dimensional harmonic potential and found excellent agreement with benchmark results provided by other established methods. We believe that this pseudo-fermion propagator framework opens up new possibilities for numerical simulations of fermionic systems.
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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).
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.
In general, Quantum Monte Carlo methods suffer from a sign problem when simulating fermionic systems. This causes the efficiency of a simulation to decrease exponentially with the number of particles and inverse temperature. To circumvent this issue, a nodal constraint is often implemented, restricting the Monte Carlo procedure from sampling paths that cause the many-body density matrix to change sign. Unfortunately, this high-dimensional nodal surface is not a priori known unless the system is exactly solvable, resulting in uncontrolled errors. We will discuss two possible routes to extend the applicability of finite-temperatue path integral Monte Carlo. First we extend the regime where signful simulations are possible through a novel permutation sampling scheme. Afterwards, we discuss a method to variationally improve the nodal surface by minimizing a free energy during simulation. Applications of these methods will include both free and interacting electron gases, concluding with discussion concerning extension to inhomogeneous systems.
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.