Some remarks on Lefschetz thimbles and complex Langevin dynamics
Gert Aarts, Lorenzo Bongiovanni, Erhard Seiler, Denes Sexty
TL;DR
The work tackles the numerical sign problem in theories with a complex weight $\exp(-S(z))$ where $S(z)$ is holomorphic, exploring two complementary strategies in the complexified field space: Lefschetz thimbles and complex Langevin dynamics. It shows how thimbles deform the integration contour through fixed points $z_k$ with $\partial_z S(z_k)=0$ so that $\mathrm{Im}\,S(z)$ is constant along the thimble, while Langevin dynamics samples with drift $-\partial_z S(z)$ in the complex plane. The paper extends previous work by applying both methods to quartic, $U(1)$ and $SU(2)$ systems with a determinant, uncovering features such as thimbles ending at determinant zeros and the emergence of singular drift or degenerate fixed points. It provides evidence linking classical runaways in Langevin flow to stable thimbles and discusses caveats due to unstable fixed points, highlighting how these insights guide the use of these methods in gauge theories with determinants.
Abstract
Lefschetz thimbles and complex Langevin dynamics both provide a means to tackle the numerical sign problem prevalent in theories with a complex weight in the partition function, e.g. due to nonzero chemical potential. Here we collect some findings for the quartic model, and for U(1) and SU(2) models in the presence of a determinant, which have some features not discussed before, due to a singular drift. We find evidence for a relation between classical runaways and stable thimbles, and give an example of a degenerate fixed point. We typically find that the distributions sampled in complex Langevin dynamics are related to the thimble(s), but with some important caveats, for instance due to the presence of unstable fixed points in the Langevin dynamics.