Sign problem and Monte Carlo calculations beyond Lefschetz thimbles

Abstract

We point out that Monte Carlo simulations of theories with severe sign problems can be profitably performed over manifolds in complex space different from the one with fixed imaginary part of the action ("Lefschetz thimble"). We describe a family of such manifolds that interpolate between the tangent space at one critical point (where the sign problem is milder compared to the real plane but in some cases still severe) and the union of relevant thimbles (where the sign problem is mild but a multimodal distribution function complicates the Monte Carlo sampling). We exemplify this approach using a simple 0+1 dimensional fermion model previously used on sign problem studies and show that it can solve the model for some parameter values where a solution using Lefschetz thimbles was elusive.

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

“…Thus, in this case our combined formulation interpolates the CLM and the original Lefschetz-thimble method via the flow time. In the case (iii) shown in figure 3, our formulation reduces to the GLTM [4]. At τ = 0, the calculation simply amounts to applying the reweighting method to the original model (4.1).…”

“…In the GLTM [4], we deform the integration contour from R n to an n-dimensional real manifold in C n by using the so-called holomorphic gradient flow, which makes the sign problem milder.…”

“…On the other hand, the GLTM [4] is based on a continuous deformation of the integration contour using the so-called holomorphic gradient flow. Along the deformed contour, the phase of the complex weight fluctuates only mildly for a sufficiently long flow time, and one can apply ordinary Monte Carlo methods using the absolute value of the weight in generating configurations and reweighting the phase factor when one calculates the expectation values.…”

“…However, when there are more than one thimbles, the transition between thimbles does not occur during the Monte Carlo simulation, which leads to the violation of ergodicity. The GLTM [4] avoids this problem of the original method by choosing a large but finite flow time. Recently, it has been pointed out [21,22] that the flow time can be made as large as one wishes without the ergodicity problem if one uses the parallel tempering algorithm.…”

“…One is the complex Langevin method (CLM) [1,2], and the other is the Lefschetz-thimble method [3]. An important generalization of the second approach was proposed [4] to overcome some problems in the original proposal, and we will therefore refer to this approach as the generalized Lefschetz-thimble method (GLTM) in this paper.…”