arXiv 1603.064582016-03-23hep-lat

格点场论中符号问题的数值方法

Numerical methods for the sign problem in Lattice Field Theory

Lorenzo Bongiovanni

备注：: 131 pages,34 figures

摘要：

在格点场论研究中使用的绝大多数算法都是基于蒙特卡罗重要抽样方法，即玻尔兹曼权的概率解释。不幸的是，在许多令人感兴趣的理论中，人们不能把一个实的和正的权重与每一个构形联系起来，这是因为它们的作用量是显式复的，或者因为权重乘以了某个非正项。在这种情况下，人们说，理论上的格是影响的迹象问题。有限化学势下的量子色动力学是符号问题阻碍量子场论研究的一个突出例子。当符号问题出现时，标准蒙特卡罗方法的应用就有问题，一般来说，需要新的方法来探索复理论的相图。在这里，我们将回顾三个主要的候选方法来处理的符号问题，即复杂的朗之万动力学，莱夫谢茨顶针和状态密度方法。我们将首先在一维Polyakov线模型上结合规范冷却方法研究复Langevin动力学，然后将其应用于带有拓扑θ项的纯规范Yang-Mills理论。在四次模型、u依赖行列式的U（1）单环节模型和复beta参数的SU（2）非交换单环节模型上，比较了复Langevin动力学和Lefschetz顶针方法的异同。最后，我们介绍了基于LLR算法的态密度方法，并将其应用于有限化学势下相对论玻色气体的研究。

英文摘要：

The great majority of algorithms employed in the study of lattice field theory are based on Monte Carlo's importance sampling method, i.e. on probability interpretation of the Boltzmann weight. Unfortunately in many theories of interest one cannot associated a real and positive weight to every configuration, that is because their action is explicitly complex or because the weight is multiplied by some non positive term. In this cases one says that the theory on the lattice is affected by the sign problem. An outstanding example of sign problem preventing a quantum field theory to be studied, is QCD at finite chemical potential. Whenever the sign problem is present, standard Monte Carlo methods are problematic to apply and, in general, new approaches are needed to explore the phase diagram of the complex theory. Here we will review three of the main candidate methods to deal with the sign problem, namely complex Langevin dynamics, Lefschetz thimbles and density of states method. We will first study complex Langevin dynamics, combined with the gauge cooling method, on the one-dimensional Polyakov line model, and then we will apply it to pure gauge Yang-Mills theory with a topological theta-term. It follows a comparison between complex Langevin dynamics and the Lefschetz thimbles method on three toy models, which are the quartic model, the U(1) one-link model with a mu dependent determinant, and the SU(2) non abelian one-link model with complex beta parameter. Lastly, we introduce the density of state method, based on the LLR algorithm, and we will employ it in the study of the relativistic Bose gas at finite chemical potential.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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