Erhard Seiler
We discuss the problem of possible boundary terms at poles of the drift in the complex Langevin method, which spoil correctness of the method. For the simplest, however paradigmatic cases we can find complete answers. Lessons for more generic cases as well as open mathematical problems are discussed.
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The complex Langevin method is a general method to treat systems with complex action, such as QCD at nonzero density. The formal justification relies on the absence of certain boundary terms, both at infinity and at the unavoidable poles of the drift force. Here I focus on the boundary terms at these poles for simple models, which so far have not been discussed in detail. The main result is that those boundary terms (for the "un-evolved" observables) arise after running the Langevin process for a finite time and vanish again as the Langevin time goes to infinity. This is in contrast to the bou…
Lattice simulations of non-zero density QCD introduce the so-called sign problem (complex or negative probabilities), which invalidates importance sampling methods. To circumvent this, we use the Complex Langevin Equation (CLE), to measure the boundary terms and then compare these results with the ones gotten from reweighting, confirming the expectations from previous studies. We also investigate boundary terms in simulations using CLE with dynamic stabilization and compare this, to results calculated with reweighting.
On complex Langevin dynamics and zeroes of the measure I: Formal proof and simple models
在复杂的朗之万方法在非零密度的晶格模拟，零的费米子行列式导致亚纯漂移，因此需要重新审视的理论推导。我们讨论如何在漂移极点影响的正式理由的方法，然后探讨各种潜在的问题，在简单的模型，在某种程度上，是适用于重密集和完整的QCD。
In complex Langevin simulations, the insufficient decay of the probability density near infinity leads to boundary terms that spoil the formal argument for correctness. We present a formulation of this term that is cheaply measurable in lattice models, and in principle allows also the direct estimation of the systematic error of the CL method. Results for a toy model, 3d XY model and HDQCD are presented.
Selected inversion as key to a stable Langevin evolution across the QCD phase boundary
本文给出了在非零化学势下全量子色动力学的新结果。在PRD 92，094516（2015）中，当逆耦合降低并进入从非受限相到受限相的过渡区域时，复数Langevin方法被证明失效。我们发现，用于估计漂移项的随机技术可以是非常不稳定的不定矩阵。这可以通过使用狄拉克算子的完全逆来避免，然而，这对于四维晶格来说太昂贵了。在这项工作中的主要突破是通过实现所需的漂移项的逆元素可以有效地使用所选择的反演技术提供的并行稀疏直接求解器包PARDISO计算。在我们的新研究中，我们表明，没有遇到故障的复杂的朗之万方法，模拟可以跨越相边界。
We employ the Complex Langevin method for simulation of complex-valued actions. First, we show how to test for convergence of the method by explicitely computing boundary terms and demonstrate this in a model. Then we investigate the deconfinement phase transition of QCD with $N_f=2$ Wilson-fermions using the Complex Langevin Method and. We give preliminary results for the transition temperatures up to $μ/T_c(μ=0)\approx 5$ and compute the curvature coefficient $κ_2$.
On complex Langevin dynamics and zeroes of the measure II: Fermionic determinant
利用复朗之万方程（CLE）研究了非零化学势下的格点量子色动力学。CLE结果正确的条件之一是来自费米子行列式的测度的零点在组态分布之外，或者至少在一个对分布的支持受到很大抑制的区域内。我们在高温下研究了重密QCD（HDQCD）和全QCD的这个问题。在HDQCD中，我们发现组态在起始跃迁的临界化学势附近移动到最接近测量零点的位置，在那里符号问题减少了，但结果基本上不受影响。在高温下的全QCD中，对狄拉克算符谱的研究也得到了类似的观察：结果不受极点问题的影响。
In this review we present the current state-of-the-art on complex Langevin simulations and their implications for the QCD phase diagram. After a short summary of the complex Langevin method, we present and discuss recent developments. Here we focus on the explicit computation of boundary terms, which provide an observable that can be used to check one of the criteria of correctness explicitly. We also present the method of Dynamic Stabilization and elaborate on recent results for fully dynamical QCD.
One reason for the well known fact that the Complex Langevin (CL) method sometimes fails to converge or converges to the wrong limit has been identified long ago: it is insufficient decay of the probability density either near infinity or near poles of the drift, leading to boundary terms that spoil the formal argument for correctness. To gain a deeper understanding of this phenomenon, in a previous paper we have studied the emergence of such boundary terms thoroughly in a simple model, where analytic results can be compared with numerics. Here we continue this type of analysis for more physic…
As is well known the Complex Langevin (CL) method sometimes fails to converge or converges to the wrong limit. We identified one reason for this long ago: insufficient decay of the probability density either near infinity or near poles of the drift, leading to boundary terms that spoil the formal argument for correctness. To gain a deeper understanding of this phenomenon, we analyze the emergence of such boundary terms thoroughly in a simple model, where analytic results can be compared with numerics. We also show how some simple modification stabilizes the CL process in such a way that it can…
Real time simulations of scalar fields with kernelled complex Langevin equation
本研究利用带核的复Langevin方程规避符号问题，并通过机器学习优化核以延长实时模拟范围，超越现有最先进水平。
Results on the heavy-dense QCD phase diagram using complex Langevin
复杂的朗之万模拟已经能够成功地再现结果从蒙特卡洛方法在该地区的符号问题是温和的，并作出预测时，它是指数困难。本文研究了三种不同格点体积下的QCD相图以及重密夸克极限（HDQCD）中的禁闭相和退禁闭相的边界。我们还简要讨论了在我们的模拟中遇到的不稳定性。
Complex Langevin dynamics and zeroes of the fermion determinant
重子化学势不为零时的QCD由于夸克行列式的复杂性而存在符号问题。复杂的朗之万动力学可以提供一个解决方案，只要满足某些条件。这些条件之一，朗之万漂移的全纯性，在QCD中是不存在的，因为零点的行列式导致亚纯漂移。我们首先推导出漂移中的极点如何影响该方法的正式理由，然后探索简单模型中的各种可能性。从这些教训随后应用到重稠密QCD和全QCD，我们发现，所获得的结果显示出一致的图片。我们的结论是，仔细监测，该方法可以证明后验，即使在亚纯性的存在。
Complex Langevin: Correctness criteria, boundary terms and spectrum
本文针对复朗之万方法中不收敛和错误收敛两类问题，提出一类能同时排除边界项和不良谱性质的正确性判据，并分析了谱病态时无法重现目标实周线积分的原因。
The QCD phase diagram in the limit of heavy quarks using complex Langevin dynamics
复杂的朗之万模拟允许数值研究的理论，表现出一个符号的问题，如QCD，从而可能适合于确定QCD相图的第一性原理。在这里，我们研究了在较宽的温度和化学势范围内的重夸克极限下的QCD。我们的研究结果包括分析的自适应仪表冷却技术，它可以防止大的偏移到非紧凑的方向的SL（$3, \mathbb{C}$）歧管。我们发现，这种漂移可能会自发出现，并改变物理观测量的统计分布，这导致与已知结果的分歧。其偏移足够小的结果被用来映射之间的边界线限制和解除限制夸克相位。
Towards learning optimized kernels for complex Langevin
本文提出利用系统先验知识和自动微分学习复朗之万模拟的最优核，以规避符号问题，并在实时格点场论中显著扩展了正确收敛的范围。
Kernels and integration cycles in complex Langevin simulations
复数Langevin模拟有时会收敛到错误分布，本文证明仅靠边界项不足以判定正确性，并提出在复数Langevin方程中引入核以控制积分环路从而恢复正确收敛。
We investigate the possibility of spontaneous supersymmetry breaking in a class of zero-dimensional ${\cal N} = 2$ supersymmetric quantum field theories, with complex actions, using complex Langevin dynamics and stochastic quantization. Our simulations successfully capture the presence or absence of supersymmetry breaking in these models. The expectation value of the auxiliary field under twisted boundary conditions was used as an order parameter to capture spontaneous supersymmetry breaking in these models.
We apply the complex Langevin method (CLM) to overcome the sign problem in 4D SU(2) gauge theory with a theta term extending our previous work on the 2D U(1) case. The topology freezing problem can be solved by using open boundary conditions in all spatial directions, and the criterion for justifying the CLM is satisfied even for large $θ$ as far as the lattice spacing is sufficiently small. However, we find that the CP symmetry at $θ=π$ remains to be broken explicitly even in the continuum and infinite-volume limits due to the chosen boundary conditions. In particular, this prevents us from i…
Real time evolution of scalar fields with kernelled Complex Langevin equation
针对0+1维标量场实时演化的符号问题，使用复朗之万方程并引入常数矩阵核与神经网络表示的场依赖核进行优化，使可达到的实时间跨度增加约一倍。
The Role of Integration Cycles in Complex Langevin Simulations
本文研究复朗之万模拟中的“错误收敛”问题，数值证实结果由积分回路线性组合构成，并阐明核的选择影响积分回路采样，从而在玩具模型中确保正确收敛。
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