Loop algorithms for quantum simulations of fermion models on lattices

TL;DR

This paper introduces two novel loop-based cluster algorithms for quantum Monte Carlo simulations of fermion models, significantly reducing autocorrelation times compared to traditional local move algorithms, especially in challenging low-temperature regimes.

Contribution

The paper presents the loop-flip and loop-exchange algorithms, new non-local cluster algorithms that improve simulation efficiency for fermion models on lattices.

Findings

Significantly shorter autocorrelation times with the new algorithms.

Effective in low-temperature, large-U regimes.

Algorithms are ergodic and applicable to grand canonical ensemble.

Abstract

Two cluster algorithms, based on constructing and flipping loops, are presented for worldline quantum Monte Carlo simulations of fermions and are tested on the one-dimensional repulsive Hubbard model. We call these algorithms the loop-flip and loop-exchange algorithms. For these two algorithms and the standard worldline algorithm, we calculated the autocorrelation times for various physical quantities and found that the ordinary worldline algorithm, which uses only local moves, suffers from very long correlation times that makes not only the estimate of the error difficult but also the estimate of the average values themselves difficult. These difficulties are especially severe in the low-temperature, large-$U$ regime. In contrast, we find that new algorithms, when used alone or in combinations with themselves and the standard algorithm, can have significantly smaller autocorrelation times, in some cases being smaller by three orders of magnitude. The new algorithms, which use non-local moves, are discussed from the point of view of a general prescription for developing cluster algorithms. The loop-flip algorithm is also shown to be ergodic and to belong to the grand canonical ensemble. Extensions to other models and higher dimensions is briefly discussed.