Lefschetz thimbles and stochastic quantisation: Complex actions in the complex plane
Gert Aarts
TL;DR
This paper tackles the numerical sign problem in theories with complex actions by comparing two approaches: Lefschetz thimbles and complex Langevin dynamics. It analytically identifies the single Lefschetz thimble ${\cal J}_0$, showing that $\operatorname{Im} S(z)$ is constant along the thimble and that observables are computed with a real weight $e^{-\operatorname{Re} S(z)}$ up to a residual phase from the Jacobian, while complex Langevin dynamics explores the full complex plane via $\dot z = -\partial_z S(z) + \eta$ and is tested on a solvable toy model with a known $Z$. The study contrasts the sampling distributions produced by the two methods and discusses the role of the residual phase and multi-thimble contributions, offering insights into mitigating the sign problem and guiding practical implementations of these approaches. Overall, the work clarifies how thimble-based contour deformations and stochastic quantisation complement each other in handling complex actions and provides analytic benchmarks for cross-validation.
Abstract
Lattice field theories with a complex action can be studied numerically by allowing a complexified configuration space to be explored. Here we compare the recently introduced formulation on a Lefschetz thimble with the result from stochastic quantisation (or complex Langevin dynamics) in the case of a simple model and contrast the distributions being sampled. We also study the role of the residual phase on the Lefschetz thimble.