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

Abstract

We investigate the sign problem in field theories by using the path optimization method with use of the neural network. For theories with the sign problem, integral in the complexified variable space is a promising approach to obtain a finite (non-zero) average phase factor. In the path optimization method, the imaginary part of variables are given as functions of the real part, y i = y i ({x}), and are optimized to enhance the average phase factor. The feedforward neural network can be used to give and to optimize functions with many variables. The combined framework, the path optimization with use of the neural network, is applied to the complex φ 4 theory at finite density, the 0+1 dimensional QCD at finite density, and the Polyakov loop extended Nambu-Jona-Lasinio (PNJL) model, all of which have the sign problem. In these cases, the average phase factor is found to be enhanced significantly. In the complex φ 4 theory, it is demonstrated that the number density is calculated at a high precision. On the optimized path, the imaginary part is found to have strong correlation with the real part on the temporal nearest neighbor site. In the 0+1 dimensional QCD, we compare the results in two different treatments of the link variable: optimization after the diagonal gauge fixing and optimization without the diagonal gauge fixing. These two methods show consistent eigenvalue distribution of the link variables. In the PNJL model with homogeneous field ansatz, finite volume results approach the mean field results as expected, and the phase transition behavior can be described.

Real-time lattice gauge theory actions: unitarity, convergence, and path integral contour deformations

“…The foundation of path integral contour deformations is the complex analysis result that for holomorphic integrands Oe iS M , the integration contour of the path integral can be deformed in order to affect the StN properties of the integral without modifying the total integral value. Previously, path integral contour deformations have been used to improve the sign and associated StN problems affecting real-time (0 + 1)D quantum mechanics models [3][4][5][6][7], as well as imaginary-time theories of scalars and fermions , U (1) gauge theory [109][110][111][112][113][114], dimensionally reduced (single-or few-variable) non-Abelian gauge theory [104,[115][116][117][118][119][120], and recently large Wilson loops in (1 + 1)D SU (N ) gauge theory [9]. By modifying the integrand magnitude and phase, contour deformations also have the potential to improve the convergence problems highlighted above.…”

Review on novel methods for lattice gauge theories

“…The authors of [199] demonstrated this technique in a toy 1D integral with a severe sign problem and showed that the latter is solved, while the presence of singular points can make the Lefschetz thimbles method fail. In subsequent publications, neural networks were employed as a tool to solve the optimization problem of minimizing the cost function in a finite-density scalar field theory [200], the Polyakov loop extended Nambu-Jona-Lasinio model [201,202] and (0 + 1)-dimensional QCD [203,204]. [203] provides also a review of the path optimization technique.…”

“…In subsequent publications, neural networks were employed as a tool to solve the optimization problem of minimizing the cost function in a finite-density scalar field theory [200], the Polyakov loop extended Nambu-Jona-Lasinio model [201,202] and (0 + 1)-dimensional QCD [203,204]. [203] provides also a review of the path optimization technique. Another proposal to optimize the integration path was introduced in [205].…”