Monte Carlo methods for the many-fermion problem

Stefan Rombouts

Siu A. Chin

Roland Assaraf, Michel Caffarel, Anatole Khélif

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. 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). It is shown that the most recent ingredient of the FMC approach (Kalos M H and Pederiva F 2000 Phys. Rev. Lett. 85 3547), namely the introduction of non-symmetric guiding functions, does not necessarily improve the stability of the algorithm. We argue that the stability observed with such guiding functions is in general a finite-size population effect disappearing for a very large population of walkers. The counterpart of this stability is a control population error which is different in nature from the standard diffusion Monte Carlo algorithm and which is at the origin of an uncontrolled approximation in FMC. 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.

B. L. G. Bakker, M. I. Polikarpov, Alexander I. Veselov

Anonymous, Martina Gisti, David J. Luitz +2 more

Strongly correlated fermionic systems are of great interest in condensed matter physics and numerical methods are indispensable tools for their study. However, existing approaches such as exact diagonalization (ED) and stochastic quantum Monte Carlo methods each suffer from fundamental limitations: ED is hindered by exponential scaling in system size, while Monte Carlo methods are plagued by sign problems and long autocorrelation times. These limitations restrict the accessible parameter space and developing algorithms that efficiently alleviate them remains a central challenge in computational physics. In this work, we propose a algorithm that combines ED and Hamiltonian Monte Carlo (HMC) to simulate two-dimensional (2D) arrays of coupled quantum wires, modeled as interacting fermionic Hubbard chains. We demonstrate how our hybrid implementation of HMC, which we dub H 2 MC , outperforms either method alone across several key simulation facets. When compared to pure ED, H 2 MC has a much more favorable computational scaling, which allows us to push simulations to much larger 2D arrays. H 2 MC also greatly alleviates the sign problem and reduces autocorrelation times when compared to pure HMC formulations utilizing either real or imaginary auxiliary fields. Our formalism demonstrates how complementary strengths of seemingly disparate methods can be leveraged to enable feasible simulations in an extended parameter space.

Andrei Alexandru, Paulo F. Bedaque, Andrea Carosso +1 more

Monte Carlo calculations of fermionic systems with continuous auxiliary fields frequently suffer from a diverging variance for fermionic observables. If the simulations have an infinite variance problem, one cannot estimate observables reliably even with an arbitrarily large number of samples. In this paper, we explore a method to solve this problem using sampling based on the distribution of a system with an extra time slice. The necessary reweighting factor is computed both perturbatively and through a secondary Monte Carlo. We show that the Monte Carlo reweighting coupled to the use of an unbiased estimator of the reweighting factor leads to a method that eliminates the infinite variance problem at a very small extra cost. We compute the double occupancy in the Hubbard model at half-filling to demonstrate the method and compare the results to well-established results obtained by other methods.