arXiv 2004.139822020-04hep-lathep-th

环面上含$θ$项的二维U（1）规范理论的复朗之万分析

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

Mitsuaki Hirasawa, Akira Matsumoto, Jun Nishimura, Atis Yosprakob

备注：: 46 pages, 18 figures; (v2) reference added

摘要：

由于符号问题，具有$θ$项的规范理论的蒙特卡罗模拟是已知的极其困难的。近年来，基于动力学变量复化的思想，在解决这一问题上取得了重大进展。在这里，我们考虑复朗之万方法（CLM），这是一个很有前途的方法，其计算成本低。然而，这种方法的缺点是存在一个必须满足的条件，以便结果是正确的。作为第一步，我们将该方法应用于具有$θ$项的环面上的二维U（1）规范理论，它可以解析地求解。我们发现，一个天真的实现方法失败，因为拓扑性质的$θ$长期。为了避免这个问题，我们模拟了相同的理论上的穿孔环面，这是等价于原来的模型中的无限体积限制$ |θ| < π$。而令人惊讶的是，我们发现，CLM的作品，并再现了确切的结果，穿孔环面，即使在大$θ$，其中附近的链接变量穿刺变得非常远离是单一的。

英文摘要：

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.

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英文摘要：

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