A simple approach towards the sign problem using path optimisation

TL;DR

This paper introduces a simple, optimized contour deformation method to mitigate the sign problem in lattice simulations, significantly extending the feasible lattice size for simulations.

Contribution

It proposes a novel contour optimization approach that reduces the sign problem, with practical algorithms and validation on a Bose gas toy model.

Findings

Increases the lattice size limit for sign problem from ~32 to ~700.

Develops a fast, stable Jacobian evaluation method for contour deformation.

Demonstrates the approach's effectiveness with minimal computational resources.

Abstract

We suggest an approach for simulating theories with a sign problem that relies on optimisation of complex integration contours that are not restricted to lie along Lefschetz thimbles. To that end we consider the toy model of a one-dimensional Bose gas with chemical potential. We identify the main contribution to the sign problem in this case as coming from a nearest neighbour interaction and approximately cancel it by an explicit deformation of the integration contour. We extend the obtained expressions to more general ones, depending on a small set of parameters. We find the optimal values of these parameters on a small lattice and study their range of validity. We also identify precursors for the onset of the sign problem. A fast method of evaluating the Jacobian related to the contour deformation is proposed and its numerical stability is examined. For a particular choice of lattice parameters, we find that our approach increases the lattice size at which the sign problem becomes serious from $L \approx 32$ to $L \approx 700$. The efficient evaluation of the Jacobian ($O(L)$ for a sweep) results in running times that are of the order of a few minutes on a standard laptop.