SIAM Journal on Scientific Computing, Volume 43, Issue 1, Page A685-A719, January 2021.
The complex Langevin (CL) method is a classical numerical strategy to alleviate the numerical sign problem in the computation of lattice field theories. Mathematically, it is a simple numerical tool to compute a wide class of high-dimensional and oscillatory integrals. However, it is often observed that the CL method converges but the limiting result is incorrect. The literature has several unclear or even conflicting statements, making the method look mysterious. By an in-depth analysis of a model problem, we reveal the mechanism of how the CL result turns biased as the parameter changes, and it is demonstrated that such a transition is difficult to capture. Our analysis also shows that the method works for any observables only if the probability density function generated by the CL process is localized. To generalize such observations to lattice field theories, we formulate the CL method on general groups using rigorous mathematical languages for the first time, and we demonstrate that such localized probability density function does not exist in the simulation of lattice field theories for general compact groups, which explains the unstable behavior of the CL method. Fortunately, we also find that the gauge cooling technique creates additional velocity that helps confine the samples, so that we can still see localized probability density functions in certain cases. Thereby, the gauge cooling method significantly broadens the application of the CL method. The limitations of gauge cooling are also discussed. In particular, we prove that gauge cooling has no effect for Abelian groups, and we provide an example showing that biased results still exist when gauge cooling is insufficient to confine the probability density function.

中文翻译：

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复杂兰格文方法在路径积分计算中的有效性

SIAM科学计算杂志，第43卷，第1期，第A685-A719页，2021年1月。
复杂的Langevin（CL）方法是一种经典的数值策略，可缓解晶格场理论计算中的数字符号问题。从数学上讲，它是一个简单的数值工具，可以计算各种高维和振荡积分。但是，经常观察到CL方法收敛，但限制结果不正确。文献中有一些不清楚甚至冲突的陈述，使该方法看起来很神秘。通过对模型问题的深入分析，我们揭示了CL结果如何随参数的变化而产生偏向的机制，并证明了这种转变很难捕获。我们的分析还表明，仅当CL过程生成的概率密度函数是局部的时，该方法才适用于任何可观测的对象。为了将这些观察结果推广到晶格场理论，我们首次使用严格的数学语言在通用群上建立了CL方法，并证明了这种局部概率密度函数在一般紧致群的晶格场理论的模拟中不存在，这解释了CL方法的不稳定行为。幸运的是，我们还发现量规冷却技术产生了有助于限制样本的额外速度，因此在某些情况下我们仍然可以看到局部概率密度函数。因此，规范冷却方法大大拓宽了CL方法的应用范围。还讨论了量规冷却的局限性。特别是，我们证明了轨距冷却对亚伯利亚族群没有影响，