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Fermion Sign Problem: Mechanism, Guarantee, and Solution

卓冰 蒋

Published Jun 7, 2026• 0 citations
Publisher

Abstract

We prove that when two identical localized sources attempt to occupy the same normal mode of a nonlinear elastic medium, the doubling of the displacement field couples with the density-dependent stiffness to produce an excess elastic energy. This excess energy is derived analytically as a function of the separation distance, yielding an effective finite-range repulsive potential that decays exponentially in the far field and is regularized by nonlinear hardening in the near field. The result provides a mechanical formulation of Pauli repulsion without invoking algebraic antisymmetrization, offering an alternative mathematical pathway for many-body ground-state calculations that avoids the fermion sign problem at the foundational level.

DOI
10.5281/zenodo.20580501

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Nowadays the term 'sign problem' is used to identify two different problems. The ideas to overcome the first type of the 'sign problem' of strongly oscillating complex valued imtegrand in the Feynman path integrals comes from Picard-Lefschetz theory and a complex version of Morse theory. The main idea is to select Lefschetz thimbles as the cycle approaching the critical point at the path-integration, where the imaginary part of the complex action stays constant. Since the imaginary part of the action is constant on each thimble, the sign problem disappears and the integral can be calculated much more effectively. Here based on the Metropolis -- Hastings algorithm a new method of calculations of the integral of the strongly oscillating integrands has been prosed. Some simple test calculation and comparison with available analytical results have been carried out. The second type of the 'sign problem' arises at studies Fermi systems by path integral approach and is caused by the requirement of antisymmetrization of the real valued matrix elements of the density matrix. An explicit analytical expression for effective pair pseudopotential in phase space has been discussed in Wigner formulation of quantum mechanics. Obtained pseudopotential allow to account for Fermi statistical effects as realizes the Pauli blocking of fermions due to the repulsion between identical fermions, which prevents their occupation of same phase space cell. To test this approach, calculations of the momentum distribution function of the ideal system of Fermi particles have been presented over a wide range of momentum and degeneracy parameter.

arXiv:2003.01549•Mar 2, 2020•0 citations
We prove that when two identical localized sources attempt to occupy the same normal mode of a nonlinear elastic medium, the doubling of the displacement field couples with the density-dependent stiffness to produce an excess elastic energy. This excess energy is derived analytically as a function of the separation distance, yielding an effective finite-range repulsive potential that decays exponentially in the far field and is regularized by nonlinear hardening in the near field. The result provides a mechanical formulation of Pauli repulsion without invoking algebraic antisymmetrization, offering an alternative mathematical pathway for many-body ground-state calculations that avoids the fermion sign problem at the foundational level.
The calculation of the ground state and thermodynamics of mass-imbalanced Fermi systems is a challenging many-body problem. Even in one spatial dimension, analytic solutions are limited to special configurations and numerical progress with standard Monte Carlo approaches is hindered by the sign problem. The focus of the present work is on the further development of methods to study imbalanced systems in a fully nonperturbative fashion. We report our calculations of the ground-state energy of mass-imbalanced fermions using two different approaches which are also very popular in the context of the theory of the strong interaction (quantum chromodynamics, QCD): (a) the hybrid Monte Carlo algorithm with imaginary mass imbalance, followed by an analytic continuation to the real axis; and (b) the complex Langevin algorithm. We cover a range of on-site interaction strengths that includes strongly attractive as well as strongly repulsive cases which we verify with nonperturbative renormalization group methods and perturbation theory. Our findings indicate that, for strong repulsive couplings, the energy starts to flatten out, implying interesting consequences for short-range and high-frequency correlation functions. Overall, our results clearly indicate that the complex Langevin approach is very versatile and works very well for imbalanced Fermi gases with both attractive and repulsive interactions.
The completely antisymmetric solution ${\ensuremath{\Psi}}^{A}$ to the problem of the scattering of a fermion by a finite system of identical fermions is studied by means of the expansion ${\ensuremath{\Psi}}^{A}=\ensuremath{\Sigma}{\ensuremath{\phi}}_{\ensuremath{\alpha}}{\ensuremath{\psi}}_{\ensuremath{\alpha}}$, where ${{\ensuremath{\phi}}_{\ensuremath{\alpha}}}$ is a complete set of antisymmetric states for the target and ${\ensuremath{\psi}}_{\ensuremath{\alpha}}$ are one-particle functions. Coupled equations for the ${\ensuremath{\psi}}_{\ensuremath{\alpha}}$ are found that obey the proper boundary conditions. This is done by means of the integral equation for ${\ensuremath{\Psi}}^{A}$ and the use of projection operators. The elastic-scattering (optical-model) wave function ${\ensuremath{\psi}}_{0}$ is shown to obey an inhomogeneous differential equation, rather than a Schr\"odinger equation. The homogeneous solution is identical to the elastic wave function obtained when the projectile is distinguishable, while the inhomogeneous solution is due entirely to exchange effects. The function ${\ensuremath{\psi}}_{0}$ is identical to the optical-model wave function found by Bell and Squires, who showed that ${\ensuremath{\psi}}_{0}$ obeys a Schr\"odinger equation with an optical potential containing direct and exchange contributions. It is shown that ${\ensuremath{\psi}}_{0}$ yields the exact elastic amplitude including direct and exchange contributions, and a phase-shift analysis of the exchange term is given. The more standard form of solution ${\ensuremath{\Psi}}^{A}=\ensuremath{\Sigma}\mathcal{a}{{\ensuremath{\phi}}_{\ensuremath{\alpha}}{f}_{\ensuremath{\alpha}}}$, where a\mathcal{a}a is an antisymmetrizer, is briefly discussed. The extension to the cases of inelastic scattering and deuteron elastic scattering is made.
Fermionic Molecular Dynamics (FMD) models a system of fermions by means of many-body states which are composed of antisymmetrized products of single-particle states. These consist of one or several Gaussians localized in coordinate and momentum space. The parameters specifying them are the dynamical variables of the model. As the repulsive core of the nucleon-nucleon interaction induces short range correlations which cannot be accommodated by a Slater determinant, a novel approach, the unitary correlation operator method (UCOM), is applied. The unitary correlator moves two particles away from each other whenever their relative distance is within the repulsive core. The time-dependent variational principle yields the equations of motion for the variables. Energies of the stationary ground states are calculated and compared to exact many-body results for nuclei up to Ca 48. Time-dependent solutions are shown for collisions between nuclei.
Calculational procedures are developed for the ground-state energy of a many-fermion system. The entire theory is developed from the many-body Schr\"odinger equation via perturbation theory. The linked cluster expansion is derived by elementary methods. The analytic singularities of the energy function are analyzed, and procedures designed to avoid or overcome them. Results for the calculation of the energy in the presence of a purely repulsive two-body force are reviewed, and the situation is found to be reasonably satisfactory, though good results are restricted to weak potentials when the system is not dilute. Results for an attractive potential surrounding a strongly repulsive core are reviewed, and procedures discribed for calculations in this case. A sample calculation with an assessment of the probable errors is reviewed.
Enforcing the Pauli principle in many-body systems of fermions to ensure an antisymmetric wave function is typically a numerically expensive task. Numerical methods, such as coupled-cluster, correlated basis function, Monte Carlo, and traditional configuration interaction, scale with powers of NNN that make the determination of the energy spectrum of these systems a challenge. In a recent paper, we successfully obtained energies for large systems of fermions in the unitary regime in which we applied the Pauli principle ``on paper'' without obtaining an explicit wave function. This approach, an extension of the symmetry invariant perturbation method to fermions, applies the Pauli principle by imposing restrictions on the quantum numbers used to obtain the energy. As a followup to this study, we perform an explicit test of the validity of this application of the Pauli principle by comparing our energy results against the energies corresponding to explicitly antisymmetrized wave functions for an exactly solvable system of harmonically confined, harmonically interacting fermions. Our results show that our simple method of applying the Pauli principle selects from a spectrum of possible many-body energies only those energies that correspond to explicitly antisymmetrized wave functions. Our results were tested for values of NNN through $\ensuremath{\approx}10 000$ and for weak and strong interactions both repulsive and attractive.
Nowadays the term 'sign problem' is used to identify two different problems. The ideas to overcome the first type of the 'sign problem' of strongly oscillating complex valued imtegrand in the Feynman path integrals comes from Picard-Lefschetz theory and a complex version of Morse theory. The main idea is to select Lefschetz thimbles as the cycle approaching the critical point at the path-integration, where the imaginary part of the complex action stays constant. Since the imaginary part of the action is constant on each thimble, the sign problem disappears and the integral can be calculated much more effectively. Here based on the Metropolis -- Hastings algorithm a new method of calculations of the integral of the strongly oscillating integrands has been prosed. Some simple test calculation and comparison with available analytical results have been carried out. The second type of the 'sign problem' arises at studies Fermi systems by path integral approach and is caused by the requirement of antisymmetrization of the real valued matrix elements of the density matrix. An explicit analytical expression for effective pair pseudopotential in phase space has been discussed in Wigner formulation of quantum mechanics. Obtained pseudopotential allow to account for Fermi statistical effects as realizes the Pauli blocking of fermions due to the repulsion between identical fermions, which prevents their occupation of same phase space cell. To test this approach, calculations of the momentum distribution function of the ideal system of Fermi particles have been presented over a wide range of momentum and degeneracy parameter.