Metropolis Monte Carlo on the Lefschetz thimble: application to a one-plaquette model

TL;DR

This paper introduces a Metropolis Monte Carlo algorithm tailored for path integrals on Lefschetz thimbles, demonstrating its effectiveness on a simple model with no sign problem, aligning with analytical results.

Contribution

It presents a novel Monte Carlo method on Lefschetz thimbles with an explicit residual phase calculation, applied successfully to a one-plaquette model.

Findings

Algorithm accurately reproduces analytical results.

Residual phase does not cause a sign problem in the tested model.

Method offers a new approach for quantum field theories on Lefschetz thimbles.

Abstract

We propose a new algorithm based on the Metropolis sampling method to perform Monte Carlo integration for path integrals in the recently proposed formulation of quantum field theories on the Lefschetz thimble. The algorithm is based on a mapping between the curved manifold defined by the Lefschetz thimble of the full action and the flat manifold associated with the corresponding quadratic action. We discuss an explicit method to calculate the residual phase due to the curvature of the Lefschetz thimble. Finally, we apply this new algorithm to a simple one-plaquette model where our results are in perfect agreement with the analytic integration. We also show that for this system the residual phase does not represent a sign problem.