Tensor Network Theory for strongly correlated quantum systems

由于其组成部分之间的相互作用而显示出强烈相关性的物理系统在我们的日常生活中无处不在。例如，由于汽车之间的强烈相互作用，不允许其中两辆车占据同一条道路，每天早上我们的道路上都会出现交通拥堵。然而，排队行进的蚂蚁永远不会形成这样的交通拥堵，尽管面临着非常相似的限制，即不允许坐在彼此的顶部。这两个例子表明，相互作用的精确微观性质的细微差异如何导致不同的宏观观察性质，这给他们的理论研究带来了重大挑战。在量子情况下，强相互作用导致一些最不为人所知的凝聚态现象，如高温超导、受挫和拓扑相，如分数量子霍尔物理，它们只出现在具有主导二维特征的材料中。这些系统的一个惊人特征是，人们可以很容易地写下看起来简单的模型，这些模型被认为在宏观层面上捕捉到了主要的物理学。然而，由于强相互作用，即使是这些简单的模型也很难求解。尽管在这一领域进行了近40年的研究，但对于许多看似简单的模型，对强相互作用产生的热力学平衡中的宏观性质仍然缺乏详细的理论理解。此外，最近的实验进展使得对远离平衡的驱动强关联量子系统的动力学研究成为可能，这为量子增强器件的应用提供了新的机遇，也给理论物理研究带来了新的挑战。在这个项目中，我们将开发高性能的软件，这将使我们能够解决有关量子和经典情况下强关联系统模型的基本问题。所谓的张量网络算法是在过去20年里发展起来的，但直到现在才知道这些算法的统一框架。这证明有理由开发高性能的计算机软件，该软件将包括现有的和经过充分测试的算法，但也具有足够的通用性，以形成这一研究领域未来发展的基础。事实上，在这个项目中开发的软件将可供英国各地的研究人员使用，并构成未来十年甚至更长时间内基于张量网络算法的数值研究的支柱。开发这一用于增强模拟方法的强大新工具，将使诸如优化量子增强效果等事情在前景看好的新一代技术中得以实现。改进的数值算法将使传感器以及能量转移和存储设备能够在动态量子效应至关重要的范围内利用物理增强的过程。特别是，该软件将被要求研究高度相关的模型，而不会受到一些其他方法固有的边界效应或减号问题的阻碍。这一研究成果可能会导致新型超导材料的出现，或在纳米材料的应用中开发动态非平衡性质。此外，这些也可能适用于日常经典的强相互作用系统，例如交通拥堵的形成或在票房或证券交易所的订单簿上形成的排队动力学。

Physical systems that display strong correlations as a result of interactions between their constituents are present everywhere around us in our daily lives. For example traffic jams form on our roads every day in the morning due to strong interactions between cars that do not allow two of them to occupy the same piece of road. However, ants marching in a line never form such traffic jams despite facing very similar restrictions of not being allowed to sit on top of each other. These two examples demonstrate how subtle differences in the precise microscopic nature of interactions may lead to qualitatively different macroscopically observed properties and this poses major challenges for their theoretical study. In the quantum case strong interactions lead to some of the least well understood phenomena of condensed matter like high-Tc superconductivity, frustration, and topological phases such fractional quantum Hall physics which only appear in materials with a dominant two dimensional character. An amazing feature of these systems is that one can readily write down simple looking models which are believed to capture the main physics on a macroscopic level. However, because of the strong interactions even these simple models turn out to be very hard to solve. Despite almost four decades of research in this area a detailed theoretical understanding of macroscopic properties in thermodynamic equilibrium emerging from strong interactions is still lacking for many of these seemingly simple models. In addition recent experimental progress now allows for the dynamical study of driven strongly correlated quantum systems far away from equilibrium and this poses new opportunities for applications in quantum enhanced devices as well as new challenges for theoretical physics research. In this project we will develop high performance software which will enable tackling basic questions about models for strongly correlated systems in the quantum and also in the classical case. The underlying so-called tensor network algorithms have been developed over the past two decades but it is only now that a unified framework for these algorithms is known. This justifies the development of high performance computer software which will encompass existing and well-tested algorithms but is also sufficiently versatile to form the basis for future developments in this field of research. Indeed, the software developed in this project will be available to researchers throughout the UK and form the backbone of numerical studies based on tensor network algorithms for the next decade and possibly beyond. Developing this powerful new tool for enhancing simulation methods will enable such things as the optimisation of quantum enhanced effects in promising new generations of technology. Improved numerical algorithms will enable sensors as well as energy transfer and storage devices to utilise physically enhanced processes at the scale where dynamical quantum effects are crucial. In particular the software will be required to study strongly correlated models without being hindered by boundary effects or minus-sign problems inherent in some other methods. The insights gained from this research could lead to novel superconducting materials or the exploitation of dynamical non-equilibrium properties in applications of nano-materials. Furthermore these may also be applicable to everyday classical strongly interacting systems like e.g. the formation of traffic jams or the dynamics of queues forming at box offices or in order books at the stock exchange.