Tensor networks for quantum matter, quantum fields, and holography

量子力学为我们提供了微观物体在原子和亚原子尺度上行为的基本规则。然而，当我们把许多这样的量子物体放在一起时，我们无法从理论上预测它们的共同行为。例如，尽管我们知道一类高温超导材料(称为铜酸盐)中的电子遵守量子力学定律，并相信我们有一个关于它们如何相互作用的准确微观模型，但我们还无法从这个微观模型推导出超导的出现。张量网络是一种新的描述量子多体系统集体态的方法，它建立在我们对量子纠缠理解的最新进展的基础上。*这项发现拨款将在三个重要方向上推进和进一步发展张量网络形式主义。首先，对于晶格上的量子物质(例如，反铁磁材料的量子自旋的晶格模型)。张量网络最初是为量子物质的晶格模型开发的，这种模型被认为是捕捉多体系统本质特征的简化模型。其次，对于直接在连续统中的量子场。用于连续系统的张量网络目前正在建设中，它可能会彻底改变我们对强相互作用量子场的思考和计算方式。第三，张量网络将作为全息原理的实现进行研究，全息原理将量子引力理论与量子场论联系起来。*因此，通过一个大胆的、多学科的理论物理研究计划，探索基金旨在进一步加强加拿大在凝聚态、量子信息、计算物理和高能物理领域的领先地位和国际知名度。这笔补助金还将对高素质人才的培训做出重大贡献，这些人才将接触到世界上最负盛名的理论物理研究所之一的领先专家的前沿研究。***********

Quantum Mechanics provides us with the basic rules of how microscopic objects behave at the atomic and subatomic scales. However, when we place many such quantum objects together, we are not capable of theoretically predicting how they will behave collectively. For instance, even though we know that the electrons in a class of high-temperature superconducting materials (called cuprates) obey the laws of Quantum Mechanics, and believe that we have an accurate microscopic model for how they interact with each other, we have not yet been able to derive the emergence of superconductivity from that microscopic model. Tensor networks are a new way of representing the collective state of quantum many-body systems, based on recent improvements in our understanding of quantum entanglement. ***This Discovery Grant will advance and further develop the tensor network formalism in three important directions. Firstly, for quantum matter on the lattice (e.g. a lattice model of quantum spins for antiferromagnetic materials). Tensor networks were originally developed for lattice models of quantum matter, which are simplified models believed to capture the essential features of a many-body system. Secondly, for quantum fields directly in the continuum. Tensor networks for continuous systems are currently under construction, and could revolutionize how we think about and compute with strongly interacting quantum fields. Thirdly, tensor networks will be investigated as a realization of the holographic principle, which relates a theory of quantum gravity with a quantum field theory.***Thus, through a bold, multi-disciplinary research program in theoretical physics, this Discovery Grant aims to further strengthening Canada's leading role and international visibility in the areas of condensed matter, quantum information, computational physics, and high energy physics. The Grant will also contribute significantly to the training of highly qualified personnel, which will be exposed to fore-front research by leading experts in one of the world's most prestigious institutes for theoretical physics. ***********