First principles Quantum Fields Theory for cold and dense matter

冷致密物质的第一原理量子场理论是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着的成功。其中包括复杂的朗之万法、泰勒展开法、对偶化技术和由项目主管开发的一种方法--状态密度法(Phys.Rev.Lett)。109(2012年)111601)。本文将批判性地回顾事件的现状，然后将重点放在状态密度方法的进展上。已对基础进行了彻底调查(见Eur.Phys.J.C76(2016)第6,306号)。博士项目将首先使用SU(2)杨-米尔斯理论中的威尔逊循环期望值对该方法的性能进行补充研究，这些值以非常低的信噪比而闻名。此外，还将计算Polykov回路规范势的势。后一项研究结果将与海德堡大学的科学家进行持续的合作。主要焦点将是继续研究QCD中的冷密物质，即强相互作用理论。该项目还将解决石墨烯的模拟问题。这项研究将有助于与德国吉森大学的同事一起成功运行的项目，因为马尔科夫链蒙特卡罗(MCMC)模拟面临两个限制：由于遍历性问题(无论我们选择什么起点，确保分布收敛到不变分布)和在非正Gibbs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游走模拟，其不依赖于关于正Gibbs因子的根据重要性抽样的配置更新。与我们的局限性特别相关的一种特殊设置是使用逆态密度作为更新配置的度量。根据定义，该度量是半正定的，旨在生成一个随机走入配置空间，即使在具有低概率度量的被剥夺区域也是如此。这种方法应该可以减少“符号问题”(由于对积分的正、负贡献的抵消而导致数值方法失败)的负面影响。态密度的关键是估计在任何给定磁化值时密度的对数的导数。多亏了大学秃鹰系统，我可以获得多个磁化值的导数，然后我可以对其进行数值积分。这已经在100x100的晶格上完成了，我正在修改代码以使用不同的晶格大小，个人渴望研究这将如何与3维晶格一起工作(例如，100*100*100)

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)