First principles Quantum Fields Theory for cold and dense matter

冷致密物质的量子场理论是20多年来的重大挑战之一。原因是计算机模拟受到臭名昭著的“符号问题”的阻碍。近年来，在解决这一问题方面取得了一些显著的成功。这些方法包括复杂朗格万模拟、泰勒展开法、二元化技术和项目主管开发的一种方法——状态密度法（Phys.Rev.Lett）。109(2012) 111601)。本文将批判性地回顾当前的事态，然后将重点放在状态密度方法的进展。这些理论的基础已经被彻底研究过了（参见eur . physj）。C76（2016）号6, 306)。博士项目将从使用SU(2) Yang-Mills理论中的Wilson环路期望值的方法性能的补充研究开始，该理论以非常差的信噪比而闻名。此外，还计算了波利科夫环表电位的电位。后一项结果将用于与海德堡大学科学家的持续合作。然后，主要的重点将是继续研究QCD中的冷和致密物质，强相互作用理论。该项目还将解决石墨烯的模拟问题。这里的研究将有助于与德国吉森大学的同事一起成功运行项目。马尔可夫链蒙特卡罗（MCMC）模拟面临两个限制：由于遍历性问题（确保分布收敛于不变分布，无论我们选择的起点是什么），以及在非正吉布斯因子的情况下，即逆温度。代替MCMC模拟，我们可以考虑一类新的非马尔可夫随机漫步模拟，它不依赖于根据重要抽样对正吉布斯因子的配置更新。一个特殊的设置与我们的局限性的qft特别相关，它使用逆态密度作为更新配置的度量。该测度在定义上是半正定的，其目的是即使在低概率测度的“被剥夺”区域也能产生随机行走的构形空间。这种方法应该减少“符号问题”的负面影响（数值方法由于积分的正负贡献被抵消而失败）。态密度的关键是估计任意给定磁化值下密度对数的导数。多亏了大学的秃鹰系统，我可以在多个磁化值下获得这个导数，然后我可以对它进行数值积分。这已经完成了一个100x100的晶格，我正在努力改变代码与不同的晶格大小的工作，与个人的愿望，调查这将如何与一个三维晶格（例如100*100*100）工作。

First principles Quantum Fields Theory for cold and dense matter is one of the grand challenges since more then 20 years. The reason is that the computer simulations are hampered by the notorious "sign problem". The recent years have seen some remarkable successes to tackle this issue. These include the Complex Langevin Simulations, the Taylor expansion method, dualisation techniques and one approach developed by the supervisor of the project - the density-of-states method (Phys.Rev.Lett. 109 (2012) 111601). The thesis will critical review the current state of affairs and will then focus on the advance of the density-of-states method. The foundations have been thoroughly investigated (see Eur.Phys.J. C76 (2016) no.6, 306). The PhD project will start with a complementing study of the performance of the method using Wilson loop expectation values in SU(2) Yang-Mills theory, which are known for a very poor signal-to-noise ratio. Also, the potential of the Polykov loop gauge potential will be calculated. The latter result will feed into an ongoing collaboration with scientists from the University of Heidelberg. Main focus will then be a continuation of the study of the cold and dense matter in QCD, the theory of strong interactions. The project will also address the simulation of Graphene. This research here will contribute to a successfully running project with colleagues at the University of Giessen, GermanyMarkov chain Monte-Carlo (MCMC) simulations face two limitations: due to ergodicity problems (making sure the distribution converges to the invariant distribution regardless of our choice of starting point) and in cases of a non-positive Gibbs factor, that is the inverse temperature. Instead of MCMC simulations, we can consider a new class of non-Markovian Random Walk simulation, which do not rely on an update of the configurations according to Importance Sampling with respect to a positive Gibbs factor. One particular set-up that is especially relevant for QFTs with our limitations uses the inverse density-of-states as a measure for updating configuration. This measure is semi-positive definite by definition and aims to generate a random walk-in configuration space even in 'deprived' regions with low probabilistic measure. This approach should reduce the negative affect of the 'sign problem' (numerical methods failing due to the cancellation of the positive and negative contributions to the integral).The key to density of states is to estimate the derivative of the logarithm of the density at any given magnetisation value. Thanks to the universities condor system, I can acquire this derivative at multiple magnetisation values, which I can then numerically integrate. This has been done for a 100x100 lattice, and I am working on altering the code to work with different lattice sizes, with a personal aspiration to investigate how this would work with a 3-dimensional lattice (e.g. 100*100*100)