Normalizing Flows and the Real-Time Sign Problem

TL;DR

This paper extends normalizing flows to tackle sign problems in lattice field theory, relating them to contour deformations and demonstrating their potential through numerical results on a scalar field model.

Contribution

It introduces a generalized class of normalizing flows for theories with sign problems, connecting them to contour deformations and showing their existence and effectiveness in specific models.

Findings

Normalizing flows can be generalized to handle sign problems.

Exact normalizing flows are likely to exist for many physical theories.

Numerical results show effectiveness in a scalar field model.

Abstract

Normalizing flows have recently been applied to the problem of accelerating Markov chains in lattice field theory. We propose a generalization of normalizing flows that allows them to applied to theories with a sign problem. These complex normalizing flows are closely related to contour deformations (i.e. the generalized Lefschetz thimble method), which been applied to sign problems in the past. We discuss the question of the existence of normalizing flows: they do not exist in the most general case, but we argue that exact normalizing flows are likely to exist for many physically interesting problems, including cases where the Lefschetz thimble decomposition has an intractable sign problem. Finally, normalizing flows can be constructed in perturbation theory. We give numerical results on their effectiveness across a range of couplings for the Schwinger-Keldysh sign problem associated to a real scalar field in $0+1$ dimensions.