First principles Quantum Fields Theory for cold and dense matter

第一性原理量子场论的冷和密集的物质是一个巨大的挑战，因为超过20年。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。这些方法包括复朗之万模拟、泰勒展开法、二元论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）。本文将对目前的研究现状进行评述，并重点介绍态密度方法的进展。对地基进行了彻底调查（见Eur. Phys. J. C76（2016）no.6，306）。该博士项目将从使用SU（2）Yang-Mills理论中的Wilson回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。同时，还计算了波里科夫环规势的势.后一项成果将用于与海德堡大学科学家正在进行的合作。然后，主要重点将是继续研究QCD中的冷和致密物质，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（确保分布收敛到不变分布，无论我们选择的起始点如何），以及在非正吉布斯因子的情况下，即逆温度。而不是MCMC模拟，我们可以考虑一类新的非马尔可夫随机游走模拟，它不依赖于更新的配置，根据重要性采样相对于一个积极的吉布斯因子。一个特定的设置，特别是相关的QFT与我们的限制使用的状态密度的倒数作为更新配置的措施。这个措施是半正定的定义，目的是产生一个随机的步行在配置空间，即使在“剥夺”的地区与低概率的措施。这种方法应该减少“符号问题”的负面影响（数值方法由于取消了对积分的正负贡献而失败）。态密度的关键是估计在任何给定磁化强度值下密度对数的导数。多亏了大学的condor系统，我可以在多个磁化强度值下获得这个导数，然后我可以数值积分。这已经完成了100 x100晶格，我正在修改代码，以适应不同的晶格大小，个人愿望是研究这将如何与三维晶格（例如100*100*100）一起工作。

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)