Complex Langevin analysis of 2D U(1) gauge theory on a torus with a θ term

Abstract

Monte Carlo simulation of gauge theories with a θ term is known to be extremely difficult due to the sign problem. Recently there has been major progress in solving this problem based on the idea of complexifying dynamical variables. Here we consider the complex Langevin method (CLM), which is a promising approach for its low computational cost. The drawback of this method, however, is the existence of a condition that has to be met in order for the results to be correct. As a first step, we apply the method to 2D U(1) gauge theory on a torus with a θ term, which can be solved analytically. We find that a naive implementation of the method fails because of the topological nature of the θ term. In order to circumvent this problem, we simulate the same theory on a punctured torus, which is equivalent to the original model in the infinite volume limit for |θ| < π. Rather surprisingly, we find that the CLM works and reproduces the exact results for a punctured torus even at large θ, where the link variables near the puncture become very far from being unitary.

Complex Langevin calculations in QCD at finite density

“…Rapid growth of the unitarity norm typically signals the occurrence of the excursion problem. We note, however, that recent work [49] on 2D U(1) theory with a θ term suggests that the CLM works even if the unitarity norm becomes large as far as the drift histogram falls off fast. 1 Therefore, one cannot tell the validity of the CLM by looking at the unitarity norm alone.…”

Is N = 2 Large?

“…This program has to overcome notoriously difficult problem, the sign problem. Since recent development in methodology is remarkable [11][12][13], such direct studies appear to be within reach in the near future.…”

“…Recent and related developments towards direct simulations are found, for example, in refs [11][12][13]…”

Tensor renormalization group and the volume independence in 2D U(N) and SU(N) gauge theories

“…This model is of particular interest since ordinary Monte Carlo simulation becomes extremely difficult due to the sign problem. Recently the complex Langevin method [43] and the density-of-state method [44] were applied to this model successfully. (See also ref.…”