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相关性的物理系统在我们的日常生活中无处不在。例如，每天早上在我们的道路上形成交通堵塞，这是由于汽车之间的强烈相互作用，不允许两辆汽车占据同一条道路。然而，排成一行行进的蚂蚁从来不会形成这样的交通堵塞，尽管它们面临着非常相似的限制，即不允许坐在彼此的上面。这两个例子表明，相互作用的精确微观性质的细微差异可能导致宏观观察性质的质的不同，这对他们的理论研究提出了重大挑战。在量子情况下，强相互作用导致一些最不容易理解的凝聚态现象，如高tc超导，挫折和拓扑相，如分数量子霍尔物理，这些现象只出现在具有主要二维特征的材料中。这些系统的一个惊人的特点是，人们可以很容易地写下看起来简单的模型，这些模型被认为可以在宏观层面上捕捉到主要的物理现象。然而，由于强烈的相互作用，即使是这些简单的模型也很难解决。尽管在这一领域进行了近四十年的研究，但对于许多这些看似简单的模型，仍然缺乏对强相互作用产生的热力学平衡宏观性质的详细理论理解。此外，最近的实验进展现在允许对远离平衡态的驱动强相关量子系统进行动力学研究，这为量子增强器件的应用提供了新的机会，也为理论物理研究提出了新的挑战。在这个项目中，我们将开发高性能软件，这将使解决量子和经典情况下强相关系统模型的基本问题成为可能。在过去的二十年里，所谓的张量网络算法已经被开发出来，但直到现在，这些算法的统一框架才为人所知。这证明了高性能计算机软件的发展是合理的，这些软件将包括现有的和经过良好测试的算法，但也足够通用，为这一研究领域的未来发展奠定基础。事实上，在这个项目中开发的软件将提供给整个英国的研究人员，并在未来十年甚至更长时间内形成基于张量网络算法的数值研究的支柱。开发这种强大的新工具来增强模拟方法，将使诸如优化量子增强效应等有前途的新一代技术成为可能。改进的数值算法将使传感器以及能量传输和存储设备能够在动态量子效应至关重要的尺度上利用物理增强的过程。特别是，该软件将需要研究强相关模型，而不受边界效应或某些其他方法固有的负号问题的阻碍。从这项研究中获得的见解可能导致新的超导材料或在纳米材料应用中开发动态非平衡特性。此外，这些也可能适用于日常经典的强相互作用系统，例如交通堵塞的形成或售票处或证券交易所订单簿中排队的动态。

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.