Physics and Mathematics of Quantum Many-Body Systems
Physics and Mathematics of Quantum Many-Body Systems
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DOI：
10.1007/978-3-030-41265-4
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发表时间：
2020-05
期刊：
影响因子：
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通讯作者：
H. Tasaki
中科院分区：
文献类型：
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作者：
H. Tasaki
这是一本关于量子多体系统的独立高级教科书，旨在为物理，数学，量子信息科学及相关领域的学生和研究人员提供。先决条件是量子力学，微积分和线性代数的本科水平的基础知识。我们详细地讨论在量子自旋系统和晶格电子系统中的选定的题目，并且也描述基本概念和重要的基本结果必要了解书的先进题目（当然，还有文献中的其他相关结果）。更具体地说，我们关注的是二维或更高维反铁磁海森堡模型中的长程有序和自发对称破缺（第一部分），在反铁磁量子自旋链中的Haldock现象和对称性保护拓扑相的相关概念（第二部分），以及在强相互作用晶格电子系统中磁性的起源，即各种版本的Hubbard模型（第三部分）。虽然题目的选择肯定会因我们的研究兴趣而有所偏颇，但我们相信每个题目本身都很有趣，值得研究。更重要的是，每个主题都代表了我们在各种量子多体系统中普遍遇到的某些非平凡现象或特征，包括量子场论，凝聚态系统，冷原子和为未来量子计算机设计的人工量子系统。换句话说，虽然我们在书中讨论的大多数系统都是广义上的磁性模型，但我们的兴趣并不局限于磁性。我们更感兴趣的是量子多体系统的普遍行为。正如标题所示，我们在这里采取的观点，数学物理。我们的主要目标是讨论数学上严格的结果，这些结果从物理学家的角度来看是非常重要和有趣的。我们还将详细讨论数学结果背后的物理直观和图像。我们相信，坚持数学上的严格证明（如果有的话）是至关重要的，因为多体系统中的一些现象是如此复杂和微妙，以至于我们不容易仅仅基于天真的物理直觉得出正确的结论。同样值得强调的是，在某些情况下（但不是所有情况下），人们通过欣赏一个数学证明来获得对“物理学”更深刻和更清晰的理解
This is a self-contained advanced textbook on quantum many-body systems, which is intended to be accessible to students and researchers in physics, mathematics, quantum information science, and related fields. The prerequisite is undergraduatelevel basic knowledge of quantum mechanics, calculus, and linear algebra. We discuss in detail selected topics in quantum spin systems and lattice electron systems, and also describe fundamental concepts and important basic results necessary to understand the advanced topics of the book (and, of course, other related results in the literature).More specifically, we focus on long-range order and spontaneous symmetry breaking in the antiferromagnetic Heisenberg model in two or higher dimensions (Part I), the Haldane phenomena in antiferromagnetic quantum spin chains and the related notion of symmetry protected topological phase (Part II), and the origin of magnetism in strongly interacting lattice electron systems, namely various versions of the Hubbard model (Part III). Although the selection of the topics is certainly biased by our research interests, we believe that each topic is, by itself, interesting and worth studying. More importantly, each of the topics represents certain nontrivial phenomena or features that we universally encounter in a variety of quantum many-body systems, including quantum field theory, condensed matter systems, cold atoms, and artificial quantum systems designed for future quantum computers. In other words, although most of the systems that we treat in the book are models of magnetism in a broad sense, our interest is not limited to magnetism. We are more interested in universal behaviors of quantum many-body systems. As the title suggests, we here take the point of view of mathematical physics. Our major goal is to discuss mathematically rigorous results which are of essential importance and interest from the physicists’ point of view. We shall also discuss in detal physical intuitions and pictures behind the mathematical results. We believe it is crucial to insist on mathematically rigorous proofs (when available) since some phenomena in many-body systems are so intricate and subtle that it is not easy for us to reach the right conclusions based only on naive physical intuitions. It is also worth stressing that, in some (but not all) cases, one gets a deeper and clearer understanding of “physics” by appreciating a mathematical proof