Systematic errors in Gaussian Quantum Monte Carlo and a systematic study of the symmetry projection method

TL;DR

This paper investigates the systematic errors in Gaussian Quantum Monte Carlo methods applied to the Hubbard model, analyzing their origins, limitations, and potential solutions through symmetry projection and different sampling techniques.

Contribution

The study systematically examines the sources of errors in GQMC for Hubbard ladders and explores methods to mitigate these errors, including symmetry projection and alternative sampling algorithms.

Findings

Systematic errors are linked to boundary terms in the Fokker-Planck derivation.

Symmetry projection improves ground state property calculations but depends on large overlap.

Different sampling methods affect the reliability and accuracy of results.

Abstract

Gaussian Quantum Monte Carlo (GQMC) is a stochastic phase space method for fermions with positive weights. In the example of the Hubbard model close to half filling it fails to reproduce all the symmetries of the ground state leading to systematic errors at low temperatures. In a previous work [Phys. Rev. B {\bf 72}, 224518 (2005)] we proposed to restore the broken symmetries by projecting the density matrix obtained from the simulation onto the ground state symmetry sector. For ground state properties, the accuracy of this method depends on a {\it large overlap} between the GQMC and exact density matrices. Thus, the method is not rigorously exact. We present the limits of the approach by a systematic study of the method for 2 and 3 leg Hubbard ladders for different fillings and on-site repulsion strengths. We show several indications that the systematic errors stem from non-vanishing boundary terms in the partial integration step in the derivation of the Fokker-Planck equation. Checking for spiking trajectories and slow decaying probability distributions provides an important test of the reliability of the results. Possible solutions to avoid boundary terms are discussed. Furthermore we compare results obtained from two different sampling methods: Reconfiguration of walkers and the Metropolis algorithm.