Path optimization method for the sign problem

Abstract

We propose a path optimization method (POM) to evade the sign problem in the Monte-Carlo calculations for complex actions. Among many approaches to the sign problem, the Lefschetz-thimble path-integral method and the complex Langevin method are promising and extensively discussed. In these methods, real field variables are complexified and the integration manifold is determined by the flow equations or stochastically sampled. When we have singular points of the action or multiple critical points near the original integral surface, however, we have a risk to encounter the residual and global sign problems or the singular drift term problem. One of the ways to avoid the singular points is to optimize the integration path which is designed not to hit the singular points of the Boltzmann weight. By specifying the one-dimensional integration-path as z = t + i f (t)( f ∈ R) and by optimizing f (t) to enhance the average phase factor, we demonstrate that we can avoid the sign problem in a one-variable toy model for which the complex Langevin method is found to fail. In this proceedings, we propose POM and discuss how we can avoid the sign problem in a toy model. We also discuss the possibility to utilize the neural network to optimize the path.

Application of a neural network to the sign problem via the path optimization method

“…In Ref. [16], the mono-layer neural network is adopted, and the optimized path is found to agree with that in Ref. [15].…”

Path optimization method with use of neural network for the sign problem in field theories

“…In the last lattice meeting [33] and in Ref. [34], we have proposed another method, the path optimization method (POM), in which the integration path is optimized to evade the sign problem, i.e.…”

“…The path optimization with use of the neural network has been applied to the one-dimensional integral [33]. The obtained optimized path is close to the thimble and the path optimized by using the standard gradient descent method around the fixed points.…”

Path Optimization for the Sign Problem in Field Theories Using Neural Network

“…This sign problem is one of the serious problems in theoretical physics. A promising idea to attack the sign problem is to complexify the integration variables, and several complexified variable methods have been proposed, such as the Lefschetz thimble method [1,2], the complex Langevin method [3], and the path optimization method [4][5][6][7][8][9][10][11].…”

“…The path optimization method with the neural network has been applied to the one-dimensional integral [5], 1+1 dimensional complex ϕ 4 theory [6], the Polyakov loop extended Nambu-Jona-Lasinio (PNJL) model [7], and the 0+1 dimensional QCD [8,9], as discussed in the presentation at QNP 2018. In these cases, we find that the sign problem is weakened and observables are obtained correctly and efficiently.…”