Taylor expansions on Lefschetz thimbles (and not only that)

TL;DR

This paper proposes a method to perform Taylor expansions on Lefschetz thimbles to address the sign problem in quantum field theories, enabling the bridging of different thimble regions and potentially simplifying multi-thimble simulations.

Contribution

It introduces a novel approach using Taylor expansions and Padé approximants on Lefschetz thimbles to circumvent multi-thimble complexities in sign problem solutions.

Findings

Successful control of analytical structure in prototype computations

Effective bridging of disjoint thimble regions via Padé approximants

Potential to simplify multi-thimble simulations in quantum field theories

Abstract

Thimble regularisation is a possible solution to the sign problem, which is evaded by formulating quantum field theories on manifolds where the imaginary part of the action stays constant (Lefschetz thimbles). A major obstacle is due to the fact that one in general needs to collect contributions coming from more than one thimble. Here we explore the idea of performing Taylor expansions on Lefschetz thimbles. We show that in some cases we can compute expansions in regions where only the dominant thimble contributes to the result in such a way that these (different, disjoint) regions can be bridged. This can most effectively be done via Pad\'e approximants. In this way multi-thimble simulations can be circumvented. The approach can be trusted provided we can show that the analytic continuation we are performing is a legitimate one, which thing we can indeed show. We briefly discuss two prototypal computations, for which we obtained a very good control on the analytical structure (and singularities) of the results. All in all, the main strategy that we adopt is supposed to be valuable not only in the thimble approach, which thing we finally discuss.