Taylor expansions on Lefschetz thimbles (and not only that)

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 Padé 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.

arXiv comment15 pages, 3 figures (out of 6 figure files)

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• Contribution to understanding the phase structure of strong interaction matter: Lee-Yang edge singularities from lattice QCD
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• Settling an old story: solution of the Thirring model in thimble regularization
• Is it worth the effort to find Lefschetz thimbles? Integration contours with numerically optimal signal-to-noise ratios in simple fermionic toy models
• Lee-Yang edge singularities in lattice QCD : A systematic study of singularities in the complex ${\mu }_B$ plane using rational approximations