The Fermion Monte Carlo revisited

TL;DR

This paper revisits the Fermion Monte Carlo algorithm, proving its exactness, analyzing its stability, and demonstrating its advantages over traditional methods, while also discussing its limitations through a simple model.

Contribution

The paper provides a detailed proof of the FMC method's exactness and analyzes its stability, showing it reduces the Bose-Fermi gap compared to standard DMC schemes.

Findings

FMC is an exact method with proven correctness.

FMC has improved stability over traditional DMC methods.

FMC calculations exhibit inherent uncontrolled aspects in simple models.

Abstract

In this work we present a detailed study of the Fermion Monte Carlo algorithm (FMC), a recently proposed stochastic method for calculating fermionic ground-state energies [M.H. Kalos and F. Pederiva, Phys. Rev. Lett. vol. 85, 3547 (2000)]. A proof that the FMC method is an exact method is given. In this work the stability of the method is related to the difference between the lowest (bosonic-type) eigenvalue of the FMC diffusion operator and the exact fermi energy. It is shown that within a FMC framework the lowest eigenvalue of the new diffusion operator is no longer the bosonic ground-state eigenvalue as in standard exact Diffusion Monte Carlo (DMC) schemes but a modified value which is strictly greater. Accordingly, FMC can be viewed as an exact DMC method built from a correlated diffusion process having a reduced Bose-Fermi gap. As a consequence, the FMC method is more stable than any transient method (or nodal release-type approaches). We illustrate the various ideas presented in this work with calculations performed on a very simple model having only nine states but a full sign problem. Already for this toy model it is clearly seen that FMC calculations are inherently uncontrolled.