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

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.