Infinite Variance in Fermion Quantum Monte Carlo Calculations

TL;DR

This paper identifies an infinite variance problem in fermion quantum Monte Carlo methods that can undermine the reliability of results, and proposes a practical solution involving a 'bridge link' to fix it.

Contribution

It reveals the infinite variance issue in standard QMC algorithms for fermion problems and introduces a simple, broadly applicable fix without major algorithm changes.

Findings

Infinite variance causes unreliable error estimates in QMC.

The proposed 'bridge link' method effectively resolves the variance issue.

Results on the Hubbard model demonstrate the solution's effectiveness.

Abstract

For important classes of many-fermion problems, quantum Monte Carlo (QMC) methods allow exact calculations of ground-state and finite-temperature properties, without the sign problem. The list spans condensed matter, nuclear physics, and high-energy physics, including the half-filled repulsive Hubbard model, the spin-balanced atomic Fermi gas, lattice QCD calculations at zero density with Wilson Fermions, and is growing rapidly as a number of problems have been discovered recently to be free of the sign problem. In these situations, QMC calculations are relied upon to provide definitive answers. Their results are instrumental to our ability to understand and compute properties in fundamental models important to multiple sub-areas in quantum physics. It is shown, however, that the most commonly employed algorithms in such situations turn out to have an infinite variance problem. A diverging variance causes the estimated Monte Carlo statistical error bar to be incorrect, which can render the results of the calculation unreliable or meaningless. We discuss how to identify the infinite variance problem. An approach is then proposed to solve the problem. The solution does not require major modifications to standard algorithms, adding a ``bridge link" to the imaginary-time path-integral. The general idea is applicable to a variety of situations where the infinite variance problem may be present. Illustrative results are presented for the ground state of the Hubbard model at half-filling.