Path optimization for U(1) gauge theory with complexified parameters

Abstract

In this article, we apply the path optimization method to handle the complexified parameters in the 1 þ 1 dimensional pure Uð1Þ gauge theory on the lattice. Complexified parameters make it possible to explore the Lee-Yang zeros which helps us to understand the phase structure and thus we consider the complex coupling constant with the path optimization method in the theory. We clarify the gauge fixing issue in the path optimization method; the gauge fixing helps to optimize the integration path effectively. With the gauge fixing, the path optimization method can treat the complex parameter and control the sign problem. It is the first step to directly tackle the Lee-Yang zero analysis of the gauge theory by using the path optimization method.

Exploration of Efficient Neural Network for Path Optimization Method

“…The path optimization method (POM) [2,3], also called the sign-optimized manifold [4,5], utilizes machine learning to find the optimal path which maximally weakens the sign problem. The POM has been applied to several models [2,4,[6][7][8][9][10][11][12][13][14][15]. The POM works well in a small system, such as finite-density 0 + 1-dimensional QCD [10].…”

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.…”

Path integral contour deformations for observables in $SU(N)$ gauge theory

“…This can be seen by noting that the end-points of the shifted contour can be connected to the original endpoints using a pair of segments parallel to the imaginary axis; these segments differ by a real 2π shift and a change of orientation so that integrals of periodic functions along these segments exactly cancel. This approach to deforming periodic variables with identified endpoints has previously been applied to path integrals involving U (1) variables [48,[56][57][58]. If the integrand h(Ω)f (U (Ω)) is a holomorphic function of all components of Ω, the value of the integral is unchanged by the deformations described above.…”