标题： DSSYK中的算子K复杂度：Krylov复杂度等于体长度 显示英文标题

标题： Operator K-complexity in DSSYK: Krylov complexity equals bulk length

摘要： 在本文中，我们研究了具有全息对偶的混沌量子系统在时间演化下的复杂性概念。 继我们之前的工作之后，我们将注意力转向在双缩放SYK模型中插入一类单粒子算符时的Krylov复杂性问题。 这样的算符由物质弦插入描述，这将理论分为左右部分，使我们能够通过弦图技术计算与算符插入相关的两种不同的复杂性概念：第一种是Krylov算符复杂性，第二种是通过算符作用于热场双态得到的状态的Krylov复杂性。 我们将提供一种解析证明和详细的数值证据，表明这两种Krylov复杂性都来自于由常数总弦数表征的递归定义的状态基。 作为结果，在所有情况下我们都能确定Krylov复杂性是由作用于理论希尔伯特空间上的长度算符的期望值给出的，该期望值用按左右弦数组织的基态表示。 我们找到了K复杂性的半经典极限的解析表达式，并研究了算符大小如何在Krylov语言中编码物质插入后的杂乱动力学。 此外，我们确定了支配K复杂性演化的有效哈密顿量，表明Krylov链上的演化可以等价地理解为一个粒子在Morse势中的运动。 一种特定的三重缩放极限允许进入理论的引力部分，在该部分中，K复杂性的几何性质由其作为总弦长的性质保证，类似于在[1]中对热场双态的K复杂性所发现的情况。 显示英文摘要

摘要： In this paper we study the notion of complexity under time evolution in chaotic quantum systems with holographic duals. Continuing on from our previous work, we turn our attention to the issue of Krylov complexity upon the insertion of a class of single-particle operators in the double-scaled SYK model. Such an operator is described by a matter-chord insertion, which splits the theory into left/right sectors, allowing us, via chord-diagram technology, to compute two different notions of complexity associated to the operator insertion: first a Krylov operator complexity, and second the Krylov complexity of a state obtained by an operator acting on the thermofield double state. We will provide both an analytic proof and detailed numerical evidence, that both Krylov complexities arise from a recursively defined basis of states characterized by a constant total chord number. As a consequence, in all cases we are able to establish that Krylov complexity is given by the expectation value of a length operator acting on the Hilbert space of the theory, expressed in terms of basis states, organized by left and right chord number. We find analytic expressions for the semiclassical limit of K-complexity, and study how the size of the operator encodes the scrambling dynamics upon the matter insertion in Krylov language. We furthermore determine the effective Hamiltonian governing the evolution of K-complexity, showing that evolution on the Krylov chain can equivalently be understood as a particle moving in a Morse potential. A particular type of triple scaling limit allows to access the gravitational sector of the theory, in which the geometrical nature of K-complexity is assured by virtue of being a total chord length, in an analogous fashion to what was found in [1] for the K-complexity of the thermofield double state.