标题： 关于Krylov复杂度 显示英文标题

标题： On Krylov Complexity

摘要： 本论文探讨了Krylov复杂性作为量子混沌的探测器以及全息复杂性的候选者这一概念。 第一部分致力于介绍在此领域进行研究所需的基本概念。 具体而言，提供了对Lanczos算法的广泛介绍，包括其性质和相关代数结构，以及与其实际实现相关的技术细节。 随后，提供了关于Krylov复杂性及其与混沌和全息理论关系的开创性参考文献和主要争论的概述。 本文第一部分的内容结合了综述材料与原创分析，这些分析旨在对文献中的结果进行背景说明、比较和批评，或是基于导致本论文所依据的出版物的研究成果。 这些研究项目是手稿第二部分的主题。 在这些研究中，开发了在有限多体系统中高效实现Lanczos算法的方法，使得能够数值计算如SYK或XXZ自旋链模型的Krylov复杂性，达到系统尺寸指数级大的时间尺度。 观察到，在SYK中，算子Krylov复杂性轮廓与全息期望一致，而XXZ等可积模型的复杂性则受到所谓的Kry洛空间中一种新的局域化效应的影响，从而阻碍了其增长。 最后，建立了双缩放SYK模型低能区无限温度热场双态的Krylov复杂性与JT引力理论中体长度之间的精确解析对应关系。 显示英文摘要

摘要： This Thesis explores the notion of Krylov complexity as a probe of quantum chaos and as a candidate for holographic complexity. The first Part is devoted to presenting the fundamental notions required to conduct research in this area. Namely, an extensive introduction to the Lanczos algorithm, its properties and associated algebraic structures, as well as technical details related to its practical implementation, is given. Subsequently, an overview of the seminal references and the main debates regarding Krylov complexity and its relation to chaos and holography is provided. The text throughout this first Part combines review material with original analyses which either intend to contextualize, compare and criticize results in the literature, or are the fruit of the investigations leading to the publications on which this Thesis is based. These research projects are the subject of the second Part of the manuscript. In them, methods for the efficient implementation of the Lanczos algorithm in finite many-body systems were developed, allowing to compute numerically the Krylov complexity of models like SYK or the XXZ spin chain up to time scales exponentially large in system size. It was observed that the operator Krylov complexity profile in SYK, a paradigmatic low-dimensional chaotic system with a holographic dual, agrees with holographic expectations, while in the case of integrable models like XXZ complexity is affected by a novel localization effect in the so-called Krylov space which hinders its growth. Finally, an exact, analytical, correspondence between the Krylov complexity of the infinite-temperature thermofield double state in the low-energy regime of the double-scaled SYK model and bulk length in the theory of JT gravity is established.