# Combining the complex Langevin method and the generalized Lefschetz-thimble method

## TL;DR

This paper explores a combined approach to the sign problem by integrating the complex Langevin method with the generalized Lefschetz-thimble method, revealing their relationship and potential interpolation between them.

## Contribution

It proposes a novel formulation that merges the two methods, applying Langevin dynamics to the parameters of the Lefschetz-thimble deformation, and analyzes different treatments of the residual sign problem.

## Key findings

- One version interpolates between the complex Langevin and Lefschetz-thimble methods.
- The combined approach clarifies the relationship between the two methods.
- Application to a single-variable model demonstrates the method's potential.

## Abstract

The complex Langevin method and the generalized Lefschetz-thimble method are two closely related approaches to the sign problem, which are both based on complexification of the original dynamical variables. The former can be viewed as a generalization of the stochastic quantization using the Langevin equation, whereas the latter is a deformation of the integration contour using the so-called holomorphic gradient flow. In order to clarify their relationship, we propose a formulation which combines the two methods by applying the former method to the real variables that parametrize the deformed integration contour in the latter method. Three versions, which differ in the treatment of the residual sign problem in the latter method, are considered. By applying them to a single-variable model, we find, in particular, that one of the versions interpolates the complex Langevin method and the original Lefschetz-thimble method.