Optimisation of complex integration contours at higher order

TL;DR

This paper develops and tests advanced contour deformation techniques for complex integrals in quantum field theory, significantly reducing the sign problem and improving computational efficiency in higher dimensions.

Contribution

It derives explicit second-order contour expressions, generalizes them to fast-evaluating ansatzes, and analyzes their effectiveness across various parameters, advancing the practical handling of the sign problem.

Findings

Contours reduce the sign problem substantially.

Efficiency improves with higher space-time dimensions.

Correlations among contributions influence phase factor behavior.

Abstract

We continue our study of contour deformation as a practical tool for dealing with the sign problem using the $d$-dimensional Bose gas with non-zero chemical potential as a toy model. We derive explicit expressions for contours up to the second order with respect to a natural small parameter and generalise these contours to an ansatz for which the evaluation of the Jacobian is fast ($O(1)$). We examine the behaviour of the various proposed contours as a function of space-time dimensionality, the chemical potential, and lattice size and geometry and use the mean phase factor as a measure of the severity of the sign problem. In turns out that this method leads to a substantial reduction of the sign problem and that it becomes more efficient as space-time dimensionality is increased. Correlations among contributions to $\operatorname{Im}\langle S \rangle$ play a key role in determining the mean phase factor and we examine these correlations in detail.