Exact ground state Monte Carlo method for Bosons without importance sampling

TL;DR

This paper demonstrates that Path Integral Ground State (PIGS) Monte Carlo simulations for Bosons can produce unbiased ground state results without importance sampling, simplifying computations and confirming their accuracy for both liquid and solid helium.

Contribution

The study shows that importance sampling is not necessary in PIGS for Bosons, as finite imaginary time evolution alone yields accurate ground state properties.

Findings

PIGS results converge to those with optimized variational wave functions.

Unbiased results obtained even with incorrect initial wave functions.

Convergence applies to both diagonal and off-diagonal properties.

Abstract

Generally ``exact'' Quantum Monte Carlo computations for the ground state of many Bosons make use of importance sampling. The importance sampling is based, either on a guiding function or on an initial variational wave function. Here we investigate the need of importance sampling in the case of Path Integral Ground State (PIGS) Monte Carlo. PIGS is based on a discrete imaginary time evolution of an initial wave function with a non zero overlap with the ground state, that gives rise to a discrete path which is sampled via a Metropolis like algorithm. In principle the exact ground state is reached in the limit of an infinite imaginary time evolution, but actual computations are based on finite time evolutions and the question is whether such computations give unbiased exact results. We have studied bulk liquid and solid 4He with PIGS by considering as initial wave function a constant, i.e. the ground state of an ideal Bose gas. This implies that the evolution toward the ground state is driven only by the imaginary time propagator, i.e. there is no importance sampling. For both the phases we obtain results converging to those obtained by considering the best available variational wave function (the Shadow wave function) as initial wave function. Moreover we obtain the same results even by considering wave functions with the wrong correlations, for instance a wave function of a strongly localized Einstein crystal for the liquid phase. This convergence is true not only for diagonal properties such as the energy, the radial distribution function and the static structure factor, but also for off-diagonal ones, such as the one--body density matrix. From this analysis we conclude that zero temperature PIGS calculations can be as unbiased as those of finite temperature Path Integral Monte Carlo.