ON THE VALIDITY OF COMPLEX LANGEVIN METHOD FOR PATH INTEGRAL COMPUTATIONS

doi:10.1137/20M1363224

CSpace

	ON THE VALIDITY OF COMPLEX LANGEVIN METHOD FOR PATH INTEGRAL COMPUTATIONS
	Cai, Zhenning 1; Dong, Xiaoyu 2; Kuang, Yang 1
	2021
发表期刊	SIAM JOURNAL ON SCIENTIFIC COMPUTING
ISSN	1064-8275
卷号	43 期号:1 页码:A685-A719
摘要	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 indepth 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.
关键词	complex Langevin method gauge cooling lattice field theory
DOI	10.1137/20M1363224
收录类别	SCI
语种	英语
资助项目	Academic Research Fund of the Ministry of Education of Singapore[R-146-000-291-114]
WOS研究方向	Mathematics
WOS类目	Mathematics, Applied
WOS记录号	WOS:000623833100040
出版者	SIAM PUBLICATIONS
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/58279
专题	中国科学院数学与系统科学研究院
通讯作者	Cai, Zhenning
作者单位	1.Natl Univ Singapore, Dept Math, Singapore 119076, Singapore 2.Chinese Acad Sci, Acad Math & Syst Sci, State Key Lab Sci & Engn Comp, Beijing 100190, Peoples R China
推荐引用方式 GB/T 7714	Cai, Zhenning,Dong, Xiaoyu,Kuang, Yang. ON THE VALIDITY OF COMPLEX LANGEVIN METHOD FOR PATH INTEGRAL COMPUTATIONS[J]. SIAM JOURNAL ON SCIENTIFIC COMPUTING,2021,43(1):A685-A719.
APA	Cai, Zhenning,Dong, Xiaoyu,&Kuang, Yang.(2021).ON THE VALIDITY OF COMPLEX LANGEVIN METHOD FOR PATH INTEGRAL COMPUTATIONS.SIAM JOURNAL ON SCIENTIFIC COMPUTING,43(1),A685-A719.
MLA	Cai, Zhenning,et al."ON THE VALIDITY OF COMPLEX LANGEVIN METHOD FOR PATH INTEGRAL COMPUTATIONS".SIAM JOURNAL ON SCIENTIFIC COMPUTING 43.1(2021):A685-A719.